Coefficients and integral mean estimates for harmonic mappings via quasi-subordination

Jasbir Parashar; Sairam Kaliraj Anbareeswaran

doi:10.4153/S0008439526102239

Coefficients and integral mean estimates for harmonic mappings via quasi-subordination

Part of: Two-dimensional theory Manifolds Geometric function theory

Published online by Cambridge University Press: 21 July 2026

Jasbir Parashar and

Sairam Kaliraj Anbareeswaran

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Jasbir Parashar: Affiliation:
Indian Institute of Technology Ropar , India e-mail: jasbir.20maz0013@iitrpr.ac.in
Sairam Kaliraj Anbareeswaran*: Affiliation:
Indian Institute of Technology Ropar , India e-mail: jasbir.20maz0013@iitrpr.ac.in
*: e-mail: sairamkaliraj@gmail.com

Article contents

Abstract
Introduction and main results
Preliminaries and auxiliary results
Proofs of main results
Footnotes
References

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Abstract

In this article, we study normalized univalent harmonic mappings associated with quasi-subordination. For the class $\mathcal {S}_{H_q}^0(\mathcal {S})$, we establish the harmonic analog of the Bieberbach conjecture together with the corresponding sharp growth estimate. Under suitable additional assumptions on the quasi-subordination data, we obtain integral mean estimates for this class. We also derive analogous results for several significant subclasses of $\mathcal {S}^{0}_{H}$, including the class of mappings with m-fold symmetric analytic and co-analytic parts, as well as the class of mappings admitting minimal surface lifts. By leveraging the flexibility of quasi-subordination, these findings extend and unify several earlier results in the study of univalent harmonic mappings.

Keywords

Univalent harmonic mappings quasi-subordination coefficient bounds integral means Hardy spaces

MSC classification

Primary: 30C45: Special classes of univalent and multivalent functions (starlike, convex, bounded rotation, etc.) 31A05: Harmonic, subharmonic, superharmonic functions

Secondary: 30C50: Coefficient problems for univalent and multivalent functions 49Q05: Minimal surfaces

Information

Type: Article
Information: Canadian Mathematical Bulletin , First View , pp. 1 - 22

DOI: https://doi.org/10.4153/S0008439526102239 [Opens in a new window]
Copyright: © The Author(s), 2026. Published by Cambridge University Press on behalf of Canadian Mathematical Society

1 Introduction and main results

Let $\mathbb {D} := \{ z \in \mathbb {C} : \vert z \vert < 1 \}$ . A complex-valued function f on $\mathbb {D}$ is harmonic if $\Delta f = 0$ , where $\Delta = 4 \partial _z \partial _{\bar {z}} = \partial _{xx} + \partial _{yy}$ . The class of all such functions is denoted by $\mathcal {H}$ . Every $f \in \mathcal {H}$ admits a decomposition $f = h + \overline {g}$ , where h and g are analytic in $\mathbb {D}$ , normalized by $h(0) = g(0) = 0$ and $h'(0) = 1$ , with power series expansions:

(1.1) $$ \begin{align} h(z) = z + \sum_{n=2}^{\infty} a_n z^n, \quad g(z) = \sum_{n=1}^{\infty} b_n z^n, \quad z \in \mathbb{D}. \end{align} $$ h left parenthesis z right parenthesis equals z plus sigma summation Underscript n equals 2 Overscript infinity Endscripts a Subscript n Baseline z Superscript n Baseline comma g left parenthesis z right parenthesis equals sigma summation Underscript n equals 1 Overscript infinity Endscripts b Subscript n Baseline z Superscript n Baseline comma z element of double struck upper D period

The function f is sense-preserving in $\mathbb {D}$ if its Jacobian $J_f(z) = \vert h'(z)\vert ^2 - \vert g'(z)\vert ^2 = |f_z(z)|^2 - |f_{\bar z}(z)|^2> 0$ for all $z \in \mathbb {D}$ . This is equivalent to the existence of an analytic function w, known as the analytic dilatation of f, such that $g'(z) = w(z) h'(z)$ and $\vert w(z) \vert < 1$ for all $z \in \mathbb {D}$ . The subclass of $\mathcal {H}$ consisting of sense-preserving univalent harmonic functions is denoted by $\mathcal {S}_H$ . Although $\mathcal {S}_H$ is not compact, its subclass ${\mathcal {S}_H^0 := \{ f \in \mathcal {S}_H : g'(0) = 0 \}}$ is compact. A function $ f \in \mathcal {S}_H $ is called starlike (resp. convex and close-to-convex) in $\mathbb {D}$ if $f(\mathbb {D})$ is a starlike (resp. convex and close-to-convex) domain (see [Reference Clunie and Sheil-Small4, Reference Duren10, Reference Wang, Liang and Zhang27]). In 1984, Clunie and Sheil-Small [Reference Clunie and Sheil-Small4] proposed the following conjecture concerning the coefficient bounds for normalized univalent harmonic functions.

Conjecture A (Clunie and Sheil-Small)

Let $ f(z) = h(z) + \overline {g(z)} \in \mathcal {S}_{H}^{0} $ have the series representation as in (1.1). Then, for all integers $n \geq 2$ , the following sharp estimates hold:

(1.2) $$ \begin{align} \big \vert \vert a_n \vert - \vert b_n\vert \big \vert \leq n, \quad \vert a_n \vert\leqslant \frac{(n+1)(2n+1)}{6}, \quad \vert b_n \vert \leqslant \frac{(n-1)(2n-1)}{6}. \end{align} $$

These bounds are attained by the harmonic Koebe function $K(z)$ , defined by

(1.3) $$ \begin{align} K(z) =H(z)+\overline{G(z)} = \frac{z - \frac{1}{2}z^2 + \frac{1}{6}z^3}{(1-z)^3} + \overline{\left( \frac{\frac{1}{2}z^2 + \frac{1}{6}z^3}{(1-z)^3} \right)}. \end{align} $$

This coefficient conjecture proposed by Clunie and Sheil-Small [Reference Clunie and Sheil-Small4] serves as the harmonic analog of the classical Bieberbach conjecture. For the analytic class

$$\begin{align*}\mathcal{S} = \{ f = h+\overline{g} \in \mathcal{S}_H : g \equiv 0 \text{ in } \mathbb{D} \}, \end{align*}$$

the conjecture reduces to the celebrated result $|a_n| \leq n$ , settled by de Branges [Reference de Branges6]. While the harmonic analog of the Bieberbach conjecture remains a formidable open problem for the full class $\mathcal {S}_H^0$ , it has been validated for several geometric subclasses, including starlike, convex, close-to-convex, and typically real mappings [Reference Clunie and Sheil-Small4, Reference Duren10, Reference Wang, Liang and Zhang27].

Much like its analytic predecessor, this conjecture has become a primary catalyst for advancements in planar univalent harmonic mappings. A significant step was taken by Ponnusamy and Sairam Kaliraj [Reference Ponnusamy and Sairam Kaliraj25], who confirmed the coefficient estimates for the subclass:

$$\begin{align*}\mathcal{S}^0_H(\mathcal{S}) = \left\{ h + \overline{g} \in \mathcal{S}^0_H : h + e^{i\theta}g \in \mathcal{S} \text{ for some } \theta \in \mathbb{R} \right\}. \end{align*}$$

Although it was initially conjectured that $\mathcal {S}^0_H(\mathcal {S}) = \mathcal {S}^0_H$ , this was later disproved by Aydogan et al. [Reference Aydogan, Bshouty, Lyzzaik and Sakar1], revealing that the relationship between these classes is more complex than previously thought.

1.1 Quasi-subordination

To bridge this gap, we employ the framework of quasi-subordination. Recall that for analytic functions $\phi $ and $\psi $ , we say that $\phi $ is quasi-subordinate to $\psi $ (denoted by $\phi \prec _q \psi $ ) if there exist analytic functions $\xi $ and $\eta $ defined on $\mathbb {D}$ such that

(1.4) $$ \begin{align} \begin{aligned} \phi(z) &= \xi(z)\,\psi(\eta(z)), \quad |\xi(z)| \le 1, \quad |\eta(z)| < 1, \quad \text{and} \quad \eta(0) = 0. \end{aligned} \end{align} $$

This notion is a powerful generalization: it reduces to subordination ( $\phi \prec \psi $ ) when $\xi \equiv 1$ , and to majorization when $\eta (z) = z$ . For a comprehensive treatment, see [Reference Pommerenke22, Reference Robertson26].

In this article, we introduce and investigate a significantly broader class defined via this flexible framework:

$$\begin{align*}\mathcal{S}_{H_q}^{0}(\mathcal{S}) = \left\{ f = h + \overline{g} \in \mathcal{S}_{H}^{0} : h + e^{i\theta}g \prec_q \psi \text{ for some } \theta \in \mathbb{R}, \psi \in \mathcal{S} \right\}. \end{align*}$$

By relaxing the condition $h + e^{i\theta }g \in \mathcal {S}$ to a quasi-subordination relation, $\mathcal {S}_{H_q}^{0}(\mathcal {S})$ contains $\mathcal {S}_{H}^{0}(\mathcal {S})$ . Our first main result establishes that the Clunie and Sheil-Small conjecture indeed holds for this enlarged class and derives its growth estimate.

Theorem 1 Let $f = h + \overline {g} \in \mathcal {S}_{H_q}^0(\mathcal {S})$ be a harmonic function with the series expansion given in (1.1). Then for all integers $n \geq 2$ , the coefficients satisfy inequalities in (1.2). Furthermore,

$$\begin{align*}|f(z)| \le \frac{r(r^2 + 3)}{3(1 - r)^3}, ~~~\quad \text{for}~~ |z| \leq r <1.~ \end{align*}$$ StartAbsoluteValue f left parenthesis z right parenthesis EndAbsoluteValue less than or equals StartFraction r left parenthesis r squared plus 3 right parenthesis Over 3 left parenthesis 1 minus r right parenthesis cubed EndFraction comma for StartAbsoluteValue z EndAbsoluteValue less than or equals r less than 1 period

These estimates are optimal for the class $\mathcal {S}_{H_q}^0(\mathcal {S})$ , with equality attained by the harmonic Koebe function $K(z)$ defined in (1.3).

1.2 Hardy spaces

For $0 < p \le \infty $ , let $H^p$ (resp. $h^p$ ) denote the Hardy space of complex-valued analytic (resp. harmonic) functions $\Phi $ on the unit disk $\mathbb {D}$ such that $\|\Phi \|_p < \infty $ . The norm is defined by

$$\begin{align*}\|\Phi\|_p = \begin{cases} \sup_{0<r<1} M_p(r, \Phi) & \text{for } 0 < p < \infty, \\ \sup_{z \in \mathbb{D}} |\Phi(z)| & \text{for } p = \infty, \end{cases} \end{align*}$$

where the integral means $M_p(r, \Phi )$ are given by

$$\begin{align*}M_p(r, \Phi) = \left( \frac{1}{2\pi} \int_0^{2\pi} |\Phi(re^{i\theta})|^p \, d\theta \right)^{1/p}. \end{align*}$$

The problem of determining the range of p for which univalent harmonic functions belong to $h^p$ has been a subject of significant interest. Notably:

• Nowak [Reference Nowak19] proved that convex and close-to-convex harmonic mappings belong to $h^p$ for $p < 1/2$ and $p < 1/3$ , respectively.
• Kayumov et al. [Reference Kayumov, Ponnusamy and Le Xuan16] established that if $f \in \mathcal {S}_H^0(\mathcal {S})$ , then $f \in h^p$ for all $p < 1/3$ .

For a comprehensive treatment of Hardy spaces, we refer to [Reference Duren8, Reference Pavlović21]. We now establish the integral mean estimate for the class defined via quasi-subordination.

Theorem 2 Suppose that $f=h+\overline {g}\in \mathcal {S}_{H_q}^{0}(\mathcal {S})$ . Then there exists a constant $\epsilon $ with $|\epsilon |=1$ , and a function $\psi \in \mathcal {S}$ , such that $ h+\epsilon g=\xi (z)\,\psi (\eta (z)), $ where the functions $\xi $ and $\eta $ are defined in (1.4). Assume further that $\xi $ extends continuously to $\overline {\mathbb {D}}$ , is absolutely continuous on $\mathbb {T}$ , and that $ \eta ' \in H^{\infty }(\mathbb {D}). $ Then

(i) $f \in h^p$ for all $p < 1/3$ .
(ii) $\displaystyle \iint _{\mathbb {D}} \big (|h'(z)| + |g'(z)|\big )^p \, dA < \infty $ for all $p < 1/2$ , where $dA$ denotes the area element.

1.3 Harmonic mappings associated with m-fold symmetric functions

An analytic function $\varphi $ in the unit disk $\mathbb {D}$ is m-fold symmetric ( $\!\!\!m \in \mathbb {N}, m \ge 2$ ) if it satisfies $\varphi (e^{2\pi i/m} z) = e^{2\pi i/m} \varphi (z)$ for all $z \in \mathbb {D}$ . Such functions admit the power series representation $\varphi (z) = z + \sum _{n=1}^{\infty } c_{mn+1} z^{mn+1}$ . Let $\mathcal {S}^{(m)}$ denote the class of m-fold symmetric functions in $\mathcal {S}$ , where each $\varphi \in \mathcal {S}^{(m)}$ arises as the m-th root transform $\varphi (z) = \{f(z^m)\}^{1/m}$ of some $f \in \mathcal {S}$ . Notably, $\mathcal {S}^{(2)}$ corresponds to the class of odd functions in $\mathcal {S}$ .

Recently, Jaglan and Sairam Kaliraj [Reference Jaglan and Sairam Kaliraj15] investigated several properties of odd univalent harmonic mappings in well-known geometric subclasses of $\mathcal {S}_H^0$ . Based on their results on Hardy space inclusion of odd harmonic mappings, they proposed the following conjecture.

Conjecture B If $f(z) = z + \sum _{n=1}^{\infty } a_{2n+1} z^{2n+1} + \sum _{n=1}^{\infty } \overline {b_{2n+1} z^{2n+1}} \in \mathcal {S}_H^0$ , then $f \in h^p$ for all $p < 1/2$ .

Inspired by this conjecture, we define the class $\mathcal {S}_{H_q}^{0,m}(\mathcal {S}^{(m)})$ as the collection of all functions $f = h + \overline {g} \in \mathcal {S}_{H}^{0} $ such that $h + e^{i\theta }g \prec _q \varphi \text { for some } \theta \in \mathbb {R} ~\text {and} ~ \varphi \in \mathcal {S}^{(m)},$ where $h(z)= z + \sum _{n=1}^{\infty } a_{ mn+1} z^{m n+1}$ and $g(z)= \sum _{n=1}^{\infty } b_{ mn+1} z^{m n+1}$ are m-fold symmetric analytic functions (not necessarily univalent) in the unit disk $\mathbb {D}$ . In our next result, we find integral mean estimate of such mappings.

Theorem 3 Let $f = h + \overline {g} \in \mathcal {S}_{H_q}^{0,m}(\mathcal {S}^{(m)})$ . Then there exists a constant $\epsilon $ with $|\epsilon |=1$ , and a function $\varphi \in \mathcal {S}^{(m)}$ , such that $ h+\epsilon g=\xi (z)\,\varphi (\eta (z)), $ where the functions $\xi $ and $\eta $ are defined in (1.4). Assume further that $\xi $ extends continuously to $\overline {\mathbb {D}}$ , is absolutely continuous on $\mathbb {T}$ , and that $ \eta ' \in H^{\infty }(\mathbb {D}). $ Then $f \in h^p$ for all $p < \frac {m}{m+2}$ .

Remark 1 In the special case $m=2$ , Theorem 3 confirms that Conjecture B holds for the generalized class, thereby extending the results of [Reference Jaglan and Sairam Kaliraj15].

1.4 Harmonic mappings and minimal surfaces

Harmonic mappings play a central role in the theory of minimal surfaces. The relationship between univalent harmonic functions and minimal surfaces is elegantly characterized by the Weierstrass–Enneper representation theorem, stated below.

Theorem A [Reference Duren10, p. 177]

If a minimal graph $ \{(u, v, F(u, v)) : u + iv \in \Omega \}$ is parameterized by sense-preserving isothermal parameters $ z = x + iy \in \mathbb {D} $ , the projection onto its base plane defines a univalent harmonic function $w = u + iv = f(z) $ of $\mathbb {D}$ onto $ \Omega $ whose dilatation is the square of an analytic function. Conversely, if $ f = h + \overline {g}$ is a sense-preserving univalent harmonic function of $ \mathbb {D}$ onto some domain $ \Omega $ with dilatation $ \omega = q^2 $ for some function $ q $ analytic in $\mathbb {D} $ , then the minimal surface over $ \Omega $ has the following isothermal representation:

$$ \begin{align*}u = \text{Re} \{ f(z) \}, \quad v = \text{Im} \{ f(z) \}, ~\text{ and } ~F(u, v) = 2\text{Im} \int_{0}^{z} q(\zeta)h'(\zeta) \, d\zeta .\end{align*} $$

In [Reference Parashar and Sairam Kaliraj20], the authors investigated convex and close-to-convex harmonic mappings under the constraint that their dilatations are squares of analytic functions. They derived sharp coefficient bounds associated with these classes of functions and further proposed the following conjecture.

Conjecture C Let $ f(z) = h(z) + \overline {g(z)} \in \mathcal {S}_{H}^{0} $ have the series representation as in (1.1), and suppose the dilatation $ w(z) = g'(z)/h'(z) = q^2(z) $ for some analytic function $ q $ in $ \mathbb {D} $ . Then, for all integers $n \geq 2$ , the following sharp estimates hold:

(1.5) $$ \begin{align} \big \vert \vert a_n \vert - \vert b_n\vert \big \vert \leq n, \quad \vert a_n \vert\leqslant \frac{(n+1)(n+2)}{6}, \quad \vert b_n \vert \leqslant \frac{(n-1)(n-2)}{6}. \end{align} $$

These bounds are attained by the harmonic function $F(z)$ , defined by

(1.6) $$ \begin{align} F(z) =H_1(z)+\overline{G_1(z)} = \frac{1-(1-z)^3}{3(1-z)^3} + \overline{\frac{z^3}{3(1-z)^3}}. \end{align} $$

For more information on harmonic mappings and their connection to minimal surfaces, one can refer to [Reference Dorff, Halverson and Lawlor7, Reference Brilleslyper, Dorff, McDougall, Rolf, Schaubroeck, Stankewitz and Stephenson3, Reference Duren10, Reference Hengartner and Schober13, Reference Jaglan and Sairam Kaliraj14, Reference Ponnusamy, Rasila and Sairam Kaliraj24]. Our next objective is to show that Conjecture C holds for functions in $\mathcal {S}_{H_q}^{0}(\mathcal {S})$ that admit a minimal surface lift and to derive the corresponding growth estimates for these mappings.

Theorem 4 Let $f = h + \overline {g} \in \mathcal {S}_{{H}_{q}}^{0}(\mathcal {S})$ have the series representation as in (1.1), and suppose the dilatation $ w(z) = g'(z)/h'(z) = q^2(z) $ for some analytic function q in $ \mathbb {D} $ . Then, for all integers $n \geq 2$ , the coefficients satisfy the inequalities in (1.5). Furthermore,

$$\begin{align*}|f(z)| \le \frac{1}{3} \left( \frac{2r^{3} - 3r^{2} + 3r}{(1-r)^{3}} \right) \qquad (\lvert z \rvert \le r < 1). \end{align*}$$ StartAbsoluteValue f left parenthesis z right parenthesis EndAbsoluteValue less than or equals one third left parenthesis StartFraction 2 r cubed minus 3 r squared plus 3 r Over left parenthesis 1 minus r right parenthesis cubed EndFraction right parenthesis left parenthesis StartAbsoluteValue z EndAbsoluteValue less than or equals r less than 1 right parenthesis period

These inequalities are sharp and are attained by the function F defined in (1.6).

2 Preliminaries and auxiliary results

This section summarizes several foundational results from the literature that are essential for the proofs of our main results. We begin with a generalization of de Branges theorem for the class $\mathcal {S}$ within the framework of quasi-subordination.

Theorem B [Reference Hayman12, p. 249]

Let $\xi $ and $\eta $ be analytic functions in $\mathbb {D}$ such that $|\xi (z)| \le 1$ , $|\eta (z)| < 1$ , and $\eta (0) = 0$ . If $\psi \in \mathcal {S}$ and

$$\begin{align*}\phi(z) = \xi(z)\psi(\eta(z)) = \sum_{n=1}^\infty A_n z^n, \end{align*}$$ phi left parenthesis z right parenthesis equals xi left parenthesis z right parenthesis psi left parenthesis eta left parenthesis z right parenthesis right parenthesis equals sigma summation Underscript n equals 1 Overscript infinity Endscripts upper A Subscript n Baseline z Superscript n Baseline comma

then $|A_n| \le n$ for all $n \ge 1$ . Equality holds if and only if $\phi (z)$ is a rotation of the Koebe function.

The following result, known as the shearing method due to Clunie and Sheil-Small [Reference Clunie and Sheil-Small4], provides a criterion for the univalency of harmonic mappings and is a key tool for constructing extremal functions.

Theorem C [Reference Clunie and Sheil-Small4]

A harmonic function $f = h + \overline {g}$ is univalent and maps $\mathbb {D}$ onto a domain convex in the direction $\alpha $ if and only if the analytic function $h - e^{2i\alpha } g$ is univalent and maps $\mathbb {D}$ onto a domain convex in the direction $\alpha $ .

To establish our main results on integral mean estimates, we employ the star-function technique introduced by Baernstein [Reference Baernstein2]. For a comprehensive account of star-functions and their properties, we refer to [Reference Duren9, Chapter 7] and [Reference Baernstein2, Reference Leung17]. We now recall a few fundamental properties of the star-function.

Lemma A [Reference Duren9, p. 218]

For $g, h \in L^1[-\pi , \pi ]$ , the following statements are equivalent:

(a) For every convex non-decreasing function $\phi $ on $(-\infty , \infty )$ ,
$$ \begin{align*}\int_{-\pi}^{\pi} \phi(g(x)) \, dx \leq \int_{-\pi}^{\pi} \phi(h(x)) \, dx.\end{align*} $$
(b) $ g^*(\theta ) \leq h^*(\theta ) $ , $ 0 \leq \theta \leq \pi $ .

Lemma B [Reference Duren9, p. 230]

For $ g, h \in L^1[-\pi , \pi ] $ ,

$$ \begin{align*}[g(\theta) + h(\theta)]^* \leq g^*(\theta) + h^*(\theta). \end{align*} $$

Equality holds if $ g $ and $ h $ are both symmetric in $[- \pi , \pi ]$ and non-increasing in $[0, \pi ]$ .

Lemma C [Reference Duren9, p. 230]

If $ g $ and $ h $ are subharmonic functions in $\mathbb {D}$ and $g $ is subordinate to $h $ , then for each $ r \in (0, 1) $ ,

$$ \begin{align*}g^*(re^{i\theta}) \leq h^*(re^{i\theta}), \quad 0 \leq \theta \leq \pi. \end{align*} $$

Next, we recall Littlewood’s subordination theorem and a lemma concerning the integral means of harmonic functions, which are vital for our Hardy space estimates.

Theorem D (Littlewood’s subordination theorem) [Reference Duren9]

Let f and g be analytic functions in $\mathbb {D}$ . If $g \prec f$ , then for $0 < p < \infty $ and $0 \le r < 1$ ,

$$\begin{align*}M_p(r, g) \le M_p(r, f). \end{align*}$$

Lemma D [Reference Das and Kaliraj5]

Let $0 < p \le 1$ . If $f = h + \overline {g}$ is a locally univalent, sense-preserving harmonic function in $\mathbb {D}$ with $f(0) = 0$ , then there exists a constant C such that

$$\begin{align*}M_p^p(r, f) \le C \int_0^r (r - s)^{p-1} M_p^p(s, h') \, ds. \end{align*}$$

The following theorem establishes that the Koebe function attains the maximal integral mean among all functions in the class $\mathcal {S}$ .

Theorem E [Reference Duren9, p. 215]

If $\phi (z)\in \mathcal {S}$ and $\Phi (x)$ is a convex nondecreasing function on $(-\infty ,\infty )$ , then

$$\begin{align*}\int_{-\pi}^{\pi} \Phi\!\left(\log |\phi(re^{i\theta})|\right)\, d\theta \leq \int_{-\pi}^{\pi} \Phi\!\left(\log |k(re^{i\theta})|\right)\, d\theta, \end{align*}$$

where $ k(z)=\frac {z}{(1-z)^2} $ is the Koebe function. Consequently,

$$\begin{align*}M_p(r,\phi)\leq M_p(r,k), \qquad 0<p<\infty. \end{align*}$$

Finally, we utilize the following asymptotic estimate for the growth of integral means.

Lemma E [Reference Pommerenke23, p. 176]

For $p> 1$ and $r \to 1^-$ , the following estimate holds:

$$\begin{align*}\int_{0}^{2\pi} |1 - r e^{it}|^{-p} \, dt = O\big((1-r)^{1-p}\big). \end{align*}$$

Lemma F [Reference Duren8, p. 57]

For arbitrary positive numbers $ a $ and $ b $ ,

$$ \begin{align*}(a + b)^p \leq \begin{cases} a^p + b^p, & 0 < p \leq 1, \\ 2^{p-1}(a^p + b^p), & p> 1. \end{cases} \end{align*} $$

3 Proofs of main results

3.1 Proof of Theorem 1

Since $f = h + \overline {g} \in \mathcal {S}_{H_q}^0(\mathcal {S})$ . Then there exists some $\epsilon $ with $\vert \epsilon \vert = 1$ and a function $\psi \in \mathcal {S}$ such that $ \phi _{\epsilon } = h + \epsilon g \prec _q \psi $ . This implies that there exist analytic functions $\xi (z)$ and $\eta (z)$ in the unit disk $\mathbb {D}$ satisfying $\vert \xi (z) \vert \leq 1$ , $\vert \eta (z) \vert < 1$ for all $z \in \mathbb {D}$ , with $\eta (0) = 0$ such that

$$ \begin{align*}\phi_{\epsilon}(z) = \xi(z)\psi(\eta(z)) =z+ \sum_{n=2}^\infty {\epsilon}_nz^n. \end{align*} $$ phi Subscript epsilon Baseline left parenthesis z right parenthesis equals xi left parenthesis z right parenthesis psi left parenthesis eta left parenthesis z right parenthesis right parenthesis equals z plus sigma summation Underscript n equals 2 Overscript infinity Endscripts epsilon Subscript n Baseline z Superscript n Baseline period

By Theorem B, we have $\vert \epsilon _n\vert \leq n,$ that is, $\big \vert \vert a_n \vert - \vert b_n \vert \big \vert \leq \vert a_n+\epsilon b_n \vert \leq n$ for all $n \geq 2$ . Since f is sense-preserving in $\mathbb {D}$ , there exists an analytic function $w(z)$ in $\mathbb {D}$ such that $w (0) = 0$ and $\vert w (z)\vert = \left \vert g'(z)/h'(z)\right \vert < 1$ for all $z \in \mathbb {D}$ , from which we obtain that

(3.1) $$ \begin{align} h^{\prime}(z)=\frac {\phi_{\epsilon}'(z)}{1 + \epsilon w(z)} \quad \text{and} \quad g'(z) = \dfrac{w(z)}{1 + \epsilon w(z)} \, \phi_{\epsilon}'(z). \end{align} $$

The Taylor coefficients of $w(z)/(1 + \epsilon w(z))$ and $\phi _\epsilon '(z)$ are dominated in modulus by those of $z/(1-z)$ and the derivative of the Koebe function $ k(z) = z/(1 - z)^2$ , respectively. In accordance with the terminology from [Reference Goodman11, Chapter 7, p. 82], we may express this as

$$ \begin{align*}g^{\prime}(z)=\dfrac{w(z)}{1 + \epsilon w(z)} \, \phi_{\epsilon}'(z) \ll \frac{z}{1 - z} \cdot \frac{1 + z}{(1 - z)^3} = \frac{1}{6} \sum_{n=2}^\infty (n - 1)n(2n - 1)z^{n-1}, \end{align*} $$

which implies $\vert b_n\vert \leq (n - 1)(2n - 1)/6,~ \text {for all } n \geq 2.$ Similarly, we have

$$\begin{align*}h^{\prime}(z)=\frac {\phi_{\epsilon}'(z)}{1 + \epsilon w(z)} \ll \frac{1}{1-z} \cdot \frac{1 + z}{(1 - z)^3} = 1 + \frac{1}{6}\sum_{n=2}^{\infty} (n + 1)n(2n +1)z^{n-1}.\end{align*}$$

This implies $\vert a_n \vert \leq (n + 1)(2n + 1)/6, ~\text {for all } n \geq 2.$

Since

$$ \begin{align*}h(re^{i\theta}) = \int_0^r h'(s e^{i\theta}) e^{i\theta} \, ds, \quad \quad g(re^{i\theta}) = \int_0^r g'(s e^{i\theta}) e^{i\theta} \, ds, \end{align*} $$

and

$$\begin{align*}\vert f(z)\vert = \vert h(z) + \overline{g(z)}\vert \le \vert h(z)\vert + \vert g(z)\vert, \end{align*}$$

we obtain

(3.2) $$ \begin{align} \vert f(z)\vert \le \int_0^r \vert f_z(s e^{i\theta})\vert \, ds + \int_0^r \vert f_{\bar{z}}(s e^{i\theta})\vert \, ds. \end{align} $$

Since $f = h + \overline {g} \in \mathcal {S}_{H_q}^{0}(\mathcal {S})$ , it follows from Theorem B that

$$ \begin{align*}\phi_{\epsilon}(z) = h(z) + \epsilon g(z) \ll \frac{z}{(1-z)^{2}}, \qquad \text{for some } \epsilon \text{ with } |\epsilon| = 1. \end{align*} $$

Furthermore, by [Reference Goodman11, p. 82, Theorem 5], we have

$$ \begin{align*}\phi_{\epsilon}'(z)\ll \frac{1+z}{(1-z)^{3}} \qquad \text{and} \qquad |\phi_{\epsilon}'(z)| \le \frac{1+r}{(1-r)^{3}}, \quad |z| = r < 1. \end{align*} $$

Therefore,

(3.3) $$ \begin{align} \vert f_z(z) \vert = \left\vert \frac{\phi_{\epsilon}'(z)}{1 +\epsilon w(z)} \right\vert \le \frac{\vert \phi_{\epsilon}'(z) \vert}{1 - \vert w(z) \vert} \le \frac{\vert\phi_{\epsilon}'(z) \vert}{1 - \vert z \vert}\le \frac{1+r}{(1-r)^4}, \end{align} $$

and

(3.4) $$ \begin{align} \vert f_{\bar{z}}(z) \vert = \left\vert \frac{w(z)\phi_{\epsilon}'(z)}{1 +\epsilon w(z)} \right\vert \le \frac{ \vert z \vert \vert \phi_{\epsilon}'(z) \vert}{1 - \vert w(z) \vert} \le \frac{\vert z \vert \vert\phi_{\epsilon}'(z) \vert}{1 - \vert z \vert}\le \frac{r(1+r)}{(1-r)^4}. \end{align} $$

The inequalities (3.3) and (3.4), together with (3.2), can now be used to obtain the desired inequality

$$ \begin{align*}| f(z)| \le \int_0^r \frac{1+s}{(1-s)^4} \, ds + \int_0^r \frac{s(1+s)}{(1-s)^4} \, ds = \frac{r(r^2 + 3)}{3(1-r)^3}. \end{align*} $$

To show that the bounds are optimal, it suffices to verify that $K(z) \in \mathcal {S}_{H_q}^{0}(\mathcal {S})$ . This follows directly from the construction of $K(z) $ , together with the fact that every function in the class $\mathcal {S}$ is subordinate to itself. This completes the proof.

3.2 Proof of Theorem 2

We require the following lemma to prove Theorem 2.

Lemma 1 Let $\phi $ be analytic in $\mathbb {D}$ with $\phi (0)=0$ , and suppose that $\phi \prec _q \psi $ for some $\psi \in \mathcal {S}$ , where the quasi-subordination relation is defined by (1.4). Assume further that $\xi $ extends continuously to $\overline {\mathbb {D}}$ , is absolutely continuous on $\mathbb {T}$ , and that $ \eta ' \in H^{\infty }(\mathbb {D}). $ Let $\omega $ be a Schwarz function. Then, for every $2/7 < p < 1$ and $0<r<1$ , there exists a constant $C_p>0$ , depending only on p, $\|\xi ^{\prime }\|_{H^1}$ , and $\|\eta ^{\prime }\|_{\infty }$ , such that

$$ \begin{align*}\int_0^{2\pi} \left| \frac{\phi'(re^{it})}{1 - \omega(re^{it})} \right|^p{} \, dt \leq \frac{C_p}{(1 - r)^{4p - 1}}. \end{align*} $$

Proof Since

$$ \begin{align*}\phi(z) = \xi(z)\psi(\eta(z)), \end{align*} $$

we have

$$ \begin{align*}\phi'(z) = \xi'(z)\psi(\eta(z)) + \xi(z)\psi'(\eta(z))\eta'(z). \end{align*} $$

As $2/7<p<1$ , Lemma F yields

$$ \begin{align*}|\phi'(z)|^p \le |\xi'(z)\psi(\eta(z))|^p + |\xi(z)\psi'(\eta(z))\eta'(z)|^p.\end{align*} $$

Set

$$ \begin{align*}I(r):= \int_0^{2\pi} \left| \frac{\phi'(re^{it})}{1-\omega(re^{it})} \right|^pdt. \end{align*} $$ upper I left parenthesis r right parenthesis colon equals integral Subscript 0 Superscript 2 pi Baseline StartAbsoluteValue StartFraction phi prime left parenthesis r e Superscript i t Baseline right parenthesis Over 1 minus omega left parenthesis r e Superscript i t Baseline right parenthesis EndFraction EndAbsoluteValue Superscript p Baseline d t period

Then

$$ \begin{align*}I(r) \le I_1(r)+I_2(r), \end{align*} $$

where

$$ \begin{align*}I_1(r):= \int_0^{2\pi} \left| \frac{\xi'(re^{it})\psi(\eta(re^{it}))}{1-\omega(re^{it})} \right|^pdt \end{align*} $$

and

$$ \begin{align*}I_2(r):= \int_0^{2\pi} \left| \frac{\xi(re^{it})\psi'(\eta(re^{it}))\eta'(re^{it})}{1-\omega(re^{it})} \right|^pdt. \end{align*} $$

We estimate $I_1(r)$ and $I_2(r)$ separately.

Estimate of $I_{1}(r)$ . By Hölder’s inequality with exponents $1/p$ and $1/(1-p)$ , we obtain

$$ \begin{align*}I_{1}(r)\leq \left(\int_{0}^{2\pi} |\xi'(re^{it})|\,dt\right)^{p} \left(\int_{0}^{2\pi} \left| \frac{\psi(\eta(re^{it}))}{1-\omega(re^{it})} \right|^{\frac{p}{1-p}} \,dt\right)^{1-p}. \end{align*} $$

upper I 1 left parenthesis r right parenthesis less than or equals left parenthesis integral Subscript 0 Superscript 2 pi Baseline StartAbsoluteValue xi prime left parenthesis r e Superscript i t Baseline right parenthesis EndAbsoluteValue d t right parenthesis Superscript p Baseline left parenthesis integral Subscript 0 Superscript 2 pi Baseline StartAbsoluteValue StartFraction psi left parenthesis eta left parenthesis r e Superscript i t Baseline right parenthesis right parenthesis Over 1 minus omega left parenthesis r e Superscript i t Baseline right parenthesis EndFraction EndAbsoluteValue Superscript StartFraction p Over 1 minus p EndFraction Baseline d t right parenthesis Superscript 1 minus p Baseline period

Since $\xi $ extends continuously to $\overline {\mathbb {D}}$ and is absolutely continuous on $\mathbb {T}$ , it follows from [Reference Mashreghi18, Theorem 8.3] that

$$ \begin{align*}\left(\int_{0}^{2\pi} |\xi'(re^{it})|\,dt\right)^{p} \leq C_p, \end{align*} $$ left parenthesis integral Subscript 0 Superscript 2 pi Baseline StartAbsoluteValue xi prime left parenthesis r e Superscript i t Baseline right parenthesis EndAbsoluteValue d t right parenthesis Superscript p Baseline less than or equals upper C Subscript p Baseline comma

where $C_p$ depends only on p and $\|\xi ^{\prime }\|_{H^1}$ . Hence,

$$ \begin{align*}I_1(r) \lesssim \left( \int_0^{2\pi} \left\vert \frac{\psi(\eta(re^{it}))}{1-\omega(re^{it})} \right\vert^q dt \right)^{1-p}, \qquad q:=\frac{p}{1-p}. \end{align*} $$

Using Lemmas A–C, together with Theorem E, we obtain

$$ \begin{align*} \left( \log \left| \frac{\psi(\eta(re^{it}))}{1-\omega(re^{it})} \right| \right)^* &\leq (\log |\psi(\eta(re^{it}))|)^* + \left( \log \left| \frac{1}{1-\omega(re^{it})} \right| \right)^* \\[1ex] &\leq \left( \log |\psi(re^{it})| \right)^* + \left( \log \left| \frac{1}{1-re^{it}} \right| \right)^* \\[1ex] &\leq \left( \log |k(re^{it})| \right)^* + \left( \log \left| \frac{1}{1-re^{it}} \right| \right)^* \\[1ex] &= \left( \log \frac{r}{|1-re^{it}|^2\,|1-re^{it}|} \right)^*. \end{align*} $$

StartLayout 1st Row 1st Column left parenthesis log StartAbsoluteValue StartFraction psi left parenthesis eta left parenthesis r e Superscript i t Baseline right parenthesis right parenthesis Over 1 minus omega left parenthesis r e Superscript i t Baseline right parenthesis EndFraction EndAbsoluteValue right parenthesis Superscript asterisk 2nd Column less than or equals left parenthesis log StartAbsoluteValue psi left parenthesis eta left parenthesis r e Superscript i t Baseline right parenthesis right parenthesis EndAbsoluteValue right parenthesis Superscript asterisk Baseline plus left parenthesis log StartAbsoluteValue StartFraction 1 Over 1 minus omega left parenthesis r e Superscript i t Baseline right parenthesis EndFraction EndAbsoluteValue right parenthesis Superscript asterisk Baseline 2nd Row 1st Column Blank 2nd Column less than or equals left parenthesis log StartAbsoluteValue psi left parenthesis r e Superscript i t Baseline right parenthesis EndAbsoluteValue right parenthesis Superscript asterisk Baseline plus left parenthesis log StartAbsoluteValue StartFraction 1 Over 1 minus r e Superscript i t Baseline EndFraction EndAbsoluteValue right parenthesis Superscript asterisk Baseline 3rd Row 1st Column Blank 2nd Column less than or equals left parenthesis log StartAbsoluteValue k left parenthesis r e Superscript i t Baseline right parenthesis EndAbsoluteValue right parenthesis Superscript asterisk Baseline plus left parenthesis log StartAbsoluteValue StartFraction 1 Over 1 minus r e Superscript i t Baseline EndFraction EndAbsoluteValue right parenthesis Superscript asterisk Baseline 4th Row 1st Column Blank 2nd Column equals left parenthesis log StartFraction r Over StartAbsoluteValue 1 minus r e Superscript i t Baseline EndAbsoluteValue squared StartAbsoluteValue 1 minus r e Superscript i t Baseline EndAbsoluteValue EndFraction right parenthesis Superscript asterisk Baseline period EndLayout

The last equality follows from the equality case in Lemma B, since for each fixed $r \in (0,1)$ , the functions

$$ \begin{align*}\frac{r}{\left\vert 1-r e^{it}\right\vert{}^2} ~\qquad \text{and}~~\qquad \left\vert \frac{1}{1-r e^{it}} \right\vert \end{align*} $$

are symmetric on $[-\pi ,\pi ]$ and non-increasing on $[0,\pi ]$ .

Now choosing $\Phi (x)=e^{qx}$ in Lemma A, we get

$$ \begin{align*}\int_{0}^{2\pi} \left| \frac{\psi(\eta(re^{it}))}{1-\omega(re^{it})} \right|^{q} dt \leq \int_{0}^{2\pi} \frac{r^{q}} {|1-re^{it}|^{3q}} \,dt. \end{align*} $$

Since $p>\frac 27,$ we have

$$ \begin{align*}3q=\frac{3p}{1-p}>1. \end{align*} $$ 3 q equals StartFraction 3 p Over 1 minus p EndFraction greater than 1 period

Therefore, by Lemma E,

$$ \begin{align*}\int_{0}^{2\pi} \frac{r^{q}} {|1-re^{it}|^{3q}} \,dt \lesssim \frac{1}{(1-r)^{3q-1}}= \frac{1} {(1-r)^{\frac{4p-1}{1-p}}}.\end{align*} $$

Consequently,

$$ \begin{align*}I_1(r)\lesssim \frac{1} {(1-r)^{4p-1}}. \end{align*} $$

Estimate of $I_{2}(r)$ . Since $|\xi (z)|\le 1$ for all $z\in \mathbb D$ and $\eta '\in H^\infty (\mathbb D)$ , we have

$$ \begin{align*}I_2(r) \leq \|\eta^{\prime}\|_\infty^p \int_0^{2\pi} \left| \frac{\psi'\big(\eta(re^{it})\big)}{1 - \omega(re^{it})} \right|^p \, dt. \end{align*} $$

Applying Hölder’s inequality with conjugate exponents $\alpha $ and $\beta $ , we obtain

$$ \begin{align*}I_2(r)\lesssim \left( \int_0^{2\pi} \left| \psi'\big(\eta(re^{it})\big) \right|^{p \alpha}{} \, dt \right)^{1/\alpha} \left( \int_0^{2\pi} \left| 1 - \omega(re^{it}) \right|^{-p \beta}{} \, dt \right)^{1/\beta}. \end{align*} $$

Put $a=p\alpha $ and $ b=p\beta $ . Then

$$ \begin{align*}\alpha=\frac{a}{p}>1, \quad \beta=\frac{b}{p}>1, \quad \text{and} \quad \frac{1}{\alpha}+\frac{1}{\beta}=\frac{p}{a}+\frac{p}{b}=1.\end{align*} $$ alpha equals StartFraction a Over p EndFraction greater than 1 comma beta equals StartFraction b Over p EndFraction greater than 1 comma and StartFraction 1 Over alpha EndFraction plus StartFraction 1 Over beta EndFraction equals StartFraction p Over a EndFraction plus StartFraction p Over b EndFraction equals 1 period

Thus,

$$ \begin{align*}I_2(r)\lesssim \left( \int_0^{2\pi} \left| \psi'\big(\eta(re^{it})\big) \right|^{a}{} \, dt \right)^{p/a} \left( \int_0^{2\pi} \left| 1 - \omega(re^{it}) \right|^{-b}{} \, dt \right)^{p/b}. \end{align*} $$

Since $\eta $ is a Schwarz function, we have $\psi '\circ \eta \prec \psi '$ . Therefore, by Theorem D,

$$ \begin{align*}\int_{0}^{2\pi} |\psi'(\eta(re^{it}))|^{a}\,dt \leq \int_{0}^{2\pi} |\psi'(re^{it})|^{a}\,dt. \end{align*} $$

Now Feng–MacGregor’s estimate [Reference Duren9, p. 229] implies that

$$ \begin{align*}\int_{0}^{2\pi} |\psi'(re^{it})|^{a}\,dt \lesssim \frac{1}{(1-r)^{3a-1}}, ~\quad a>\frac{2}{5}. \end{align*} $$ integral Subscript 0 Superscript 2 pi Baseline StartAbsoluteValue psi prime left parenthesis r e Superscript i t Baseline right parenthesis EndAbsoluteValue Superscript a Baseline d t less than or equivalent to StartFraction 1 Over left parenthesis 1 minus r right parenthesis Superscript 3 a minus 1 Baseline EndFraction comma a greater than two fifths period

Therefore,

$$ \begin{align*}\left( \int_{0}^{2\pi} \left|\psi'(\eta(re^{it}))\right|{}^{a}\,dt \right)^{p/a}\lesssim \frac{1}{(1-r)^{\frac{(3a-1)p}{a}}}. \end{align*} $$

For the second factor, an application of Lemma E, together with Theorem D, yields

$$ \begin{align*}\left( \int_{0}^{2\pi} |1-\omega(re^{it})|^{-b}\,dt \right)^{p/b} \lesssim \frac{1}{(1-r)^{\frac{(b-1)p}{b}}},~\quad b>1. \end{align*} $$

Moreover,

$$ \begin{align*}b>1 \iff a<\frac{p}{1-p}.\end{align*} $$

Thus, combining the above estimates, we obtain

$$ \begin{align*}I_2(r)\lesssim \frac{1}{(1-r)^{\frac{(3a-1)p}{a}}} \cdot \frac{1}{(1-r)^{\frac{(b-1)p}{b}}}, \end{align*} $$

provided that

$$ \begin{align*}\max\left\{\frac25,p\right\}<a<\frac{p}{1-p}. \end{align*} $$ max left brace two fifths comma p right brace less than a less than StartFraction p Over 1 minus p EndFraction period

Such a choice of a is possible, since

$$ \begin{align*}\frac25<\frac{p}{1-p} \iff p>\frac27, \end{align*} $$ two fifths less than StartFraction p Over 1 minus p EndFraction long left right double arrow p greater than two sevenths comma

and

$$ \begin{align*}p<\frac{p}{1-p} \iff 0<p<1. \end{align*} $$

Hence, for $2/7<p<1,$ the interval

$$ \begin{align*}\left(\max\left\{\frac25,p\right\},\,\frac{p}{1-p}\right) \end{align*} $$ Start 1 By 1 Matrix 1st Row max left brace two fifths comma p right brace comma StartFraction p Over 1 minus p EndFraction EndMatrix

is non-empty.

Finally, since

$$ \begin{align*}\frac{(3a-1)p}{a}+\frac{(b-1)p}{b} = 3p-\frac{p}{a}+p-\frac{p}{b} = 4p-1, \end{align*} $$

we get

$$ \begin{align*}I_2(r)\lesssim \frac{1}{(1-r)^{4p-1}}. \end{align*} $$

Combining the estimates for $I_1(r)$ and $I_2(r)$ , we conclude that

$$ \begin{align*}I(r)\le I_1(r)+I_2(r) \le \frac{C_p}{(1-r)^{4p-1}}. \end{align*} $$ StartLayout 1st Row upper I left parenthesis r right parenthesis less than or equals upper I 1 left parenthesis r right parenthesis plus upper I 2 left parenthesis r right parenthesis less than or equals StartFraction upper C Subscript p Baseline Over left parenthesis 1 minus r right parenthesis Superscript 4 p minus 1 Baseline EndFraction period EndLayout

This completes the proof.

We now proceed with the proof of Theorem 2.

Proof

(i) Since $f = h + \overline {g} \in \mathcal {S}_{H_q}^0(\mathcal {S})$ , that is, there exists some $\epsilon $ with $\vert \epsilon \vert = 1$ and a function $\psi \in \mathcal {S}$ such that $ h + \epsilon g =\xi (z)\,\psi (\eta (z)) $ , where the functions $\xi $ and $\eta $ are defined in (1.4). Assume further that $\xi $ extends continuously to $\overline {\mathbb {D}}$ , is absolutely continuous on $\mathbb {T}$ , and that $ \eta ' \in H^{\infty }(\mathbb {D}). $ Also, we have $h' = \phi _\epsilon '/(1 + \epsilon w ), $ where $ w(z) = g'(z)/h'(z) $ is the dilatation of $ f $ satisfying $ w(0) = 0 $ and $ \vert w(z)\vert < 1 $ . Applying Lemmas D and 1, we obtain
$$ \begin{align*} \lim_{r \to 1^-} M_p^p(r, f) &\leq C \int_0^1 (1 - s)^{p-1} M_p^p(s, h') \, ds \nonumber \\ &= C \int_0^1 (1 - s)^{p-1} \left( \frac{1}{2\pi} \int_0^{2\pi} \left \vert \frac{\phi_\epsilon'(se^{it})}{1 + \epsilon w(se^{it})} \right \vert^p dt \right) ds \nonumber \\ &\leq \frac{C_p C}{2\pi} \int_0^1 \frac{1}{(1 - s)^{3p}} \, ds. \nonumber\end{align*} $$

The last integral is finite for $ p < 1/3 $ . Therefore, $f \in h^p$ for $ p < 1/3 $ .
(ii) Since f is sense-preserving, it suffices to show that
$$ \begin{align*}\int_{\mathbb{D}} \vert h'(z)\vert ^p \, dxdy < +\infty \quad \text{for all } p < 1/2. \end{align*} $$

From the representation of $h'$ , it follows that
$$ \begin{align*} \int_{\mathbb{D}} \vert h'(z) \vert^p \, dxdy &= \int_0^1 \int_0^{2\pi} \left \vert \frac{\phi_\epsilon'(re^{it})}{1 + \epsilon w(re^{it})} \right \vert^p \, r\,dt\,dr \nonumber \\ &\leq \int_0^1 \frac{C_p}{(1 - r)^{4p - 1}}\,dr \quad \text{(by Lemma 1}) \nonumber. \end{align*} $$

The last integral is finite if $p<1/2$ . This completes the proof.

3.3 Proof of Theorem 3

The proof of Theorem 3 relies on the following lemmas. First, we establish growth and distortion theorem for $\varphi \in \mathcal {S}^{(m)}$ .

Lemma 2 Let $\varphi \in \mathcal {S}^{(m)}$ and $\vert z \vert = r < 1$ , the following inequalities hold:

(3.5) $$ \begin{align} \frac{r}{\left(1 + r^m\right)^{2/m}} &\leq \vert \varphi(z) \vert \leq \frac{r}{\left(1 - r^m\right)^{2/m}}, \end{align} $$

(3.6) $$ \begin{align} \frac{1 - r^m}{\left(1 + r^m\right)^{1+\frac{2}{m}}} &\leq \vert \varphi'(z) \vert \leq \frac{1 + r^m}{\left(1 - r^m\right)^{1+\frac{2}{m}}}. \end{align} $$

Moreover, both the upper and lower bounds are sharp for the functions $z/(1-z^m)^{2/m}$ and $z/(1+z^m)^{2/m}$ , respectively.

Proof By the definition of the m-fold symmetric class $\mathcal {S}^{(m)}$ , for each $\varphi \in \mathcal {S}^{(m)}$ , there exists a function $f \in \mathcal {S}$ such that $\varphi (z) = [f(z^m)]^{1/m}$ . Applying the classical growth theorem for $\mathcal {S}$ [Reference Duren9, Theorem 2.6], we have

$$ \begin{align*}\frac{|w|}{(1 + |w|)^2} \leq |f(w)| \leq \frac{|w|}{(1 - |w|)^2}, \quad w \in \mathbb{D}. \end{align*} $$

Setting $w = z^m$ and taking the m-th root yields the growth estimates for $\varphi \in \mathcal {S}^{(m)}$ :

(3.7) $$ \begin{align} \frac{|z|}{(1 + |z|^m)^{2/m}} \leq |\varphi(z)| \leq \frac{|z|}{(1 - |z|^m)^{2/m}}. \end{align} $$

To derive the distortion bounds, we logarithmic differentiate the relation $\varphi (z)^m = f(z^m)$ to obtain

$$ \begin{align*}\frac{\varphi'(z)}{\varphi(z)} = \frac{z^{m-1} f'(z^m)}{f(z^m)} \implies \frac{z \varphi'(z)}{\varphi(z)} = \frac{z^m f'(z^m)}{f(z^m)}. \end{align*} $$

Using the well-known distortion bounds for the logarithmic derivative of functions in $\mathcal {S}$ [Reference Duren9, Theorem 2.7], we have

(3.8) $$ \begin{align} \frac{1 - r^m}{1 + r^m} \leq \left| \frac{z \varphi'(z)}{\varphi(z)} \right| \leq \frac{1 + r^m}{1 - r^m}, \quad |z|=r < 1. \end{align} $$

Finally, combining (3.8) with the growth bounds in (3.7), we establish the sharp distortion inequalities for $\mathcal {S}^{(m)}$ :

$$ \begin{align*}\frac{1 - r^m}{(1 + r^m)^{1+\frac{2}{m}}} \leq |\varphi'(z)| \leq \frac{1 + r^m}{(1 - r^m)^{1+\frac{2}{m}}}. \end{align*} $$

This completes the proof.

Lemma 3 Let $\varphi \in \mathcal {S}^{(m)}$ with $m \geq 2$ . Then, for every exponent p satisfying

(3.9) $$ \begin{align} \frac{2m}{m+4}<p<2, \end{align} $$

there exists a positive constant $C_{p,m},$ depending only on p and m, such that

$$ \begin{align*}M_p^p(r,\varphi') \leq \frac{C_{p,m}}{(1-r)^{\left(1+\frac{2}{m}\right)p-1}}, \qquad 0<r<1. \end{align*} $$ upper M Subscript p Superscript p Baseline left parenthesis r comma phi Superscript prime Baseline right parenthesis less than or equals StartFraction upper C Subscript p comma m Baseline Over left parenthesis 1 minus r right parenthesis Superscript left parenthesis 1 plus StartFraction 2 Over m EndFraction right parenthesis p minus 1 Baseline EndFraction comma 0 less than r less than 1 period

Proof Set

$$ \begin{align*}r_0:=2^{-1/m}. \end{align*} $$

By the distortion theorem for the class $\mathcal {S}^{(m)}$ (see Lemma 2), there exists a constant $C_m>0$ , depending only on m, such that

$$ \begin{align*}|\varphi'(z)|\le C_{m}, \qquad |z|\le r_0. \end{align*} $$

Thus, without loss of generality, it suffices to prove the desired estimate for $ r_0< r<1. $ For such r, if we set $ s:=r^m, $ then $ \frac 12< s<1 $ .

By the m-th root transformation, there exists a function $f\in \mathcal {S}$ such that

$$ \begin{align*}\varphi(z)^m=f(z^m), \qquad z\in\mathbb D. \end{align*} $$

Differentiating this identity, we obtain

$$ \begin{align*}m\varphi(z)^{m-1}\varphi'(z)=mz^{m-1}f'(z^m), \end{align*} $$

and hence

$$ \begin{align*}\varphi'(z)=\frac{z^{m-1}f'(z^m)}{\varphi(z)^{m-1}}. \end{align*} $$ StartLayout 1st Row phi prime left parenthesis z right parenthesis equals StartFraction z Superscript m minus 1 Baseline f prime left parenthesis z Superscript m Baseline right parenthesis Over phi left parenthesis z right parenthesis Superscript m minus 1 Baseline EndFraction period EndLayout

Also,

$$ \begin{align*}|\varphi'(re^{i\theta})|^p = r^{(m-1)p}|f'(r^me^{im\theta})|^p|f(r^me^{im\theta})|^{-\frac{(m-1)p}{m}}. \end{align*} $$

Now

$$ \begin{align*}-\frac{(m-1)p}{m}=-p+\frac{p}{m}, \end{align*} $$

$$ \begin{align*}|\varphi'(re^{i\theta})|^p = r^{(m-1)p} \left|\frac{f'(r^me^{im\theta})}{f(r^me^{im\theta})}\right|^p{} |f(r^me^{im\theta})|^{\frac{p}{m}}. \end{align*} $$

Therefore,

$$ \begin{align*} 2\pi M_p^p(r,\varphi') &= \int_0^{2\pi}|\varphi'(re^{i\theta})|^p\,d\theta \\ &= r^{(m-1)p}\int_0^{2\pi} \left|\frac{f'(r^me^{im\theta})}{f(r^me^{im\theta})}\right|^p{} |f(r^me^{im\theta})|^{\frac{p}{m}}\,d\theta \\ &= \frac{r^{(m-1)p}}{m}\int_0^{2\pi m} \left|\frac{f'(se^{it})}{f(se^{it})}\right|^p{} |f(se^{it})|^{\frac{p}{m}}\,dt \\ &= r^{(m-1)p}\int_0^{2\pi} \left|\frac{f'(se^{it})}{f(se^{it})}\right|^p{} |f(se^{it})|^{\frac{p}{m}}\,dt. \end{align*} $$

StartLayout 1st Row 1st Column 2 pi upper M Subscript p Superscript p Baseline left parenthesis r comma phi Superscript prime Baseline right parenthesis 2nd Column equals integral Subscript 0 Superscript 2 pi Baseline StartAbsoluteValue phi prime left parenthesis r e Superscript i theta Baseline right parenthesis EndAbsoluteValue Superscript p Baseline d theta 2nd Row 1st Column Blank 2nd Column equals r Superscript left parenthesis m minus 1 right parenthesis p Baseline integral Subscript 0 Superscript 2 pi Baseline StartAbsoluteValue StartFraction f prime left parenthesis r Superscript m Baseline e Superscript i m theta Baseline right parenthesis Over f left parenthesis r Superscript m Baseline e Superscript i m theta Baseline right parenthesis EndFraction EndAbsoluteValue Superscript p Baseline StartAbsoluteValue f left parenthesis r Superscript m Baseline e Superscript i m theta Baseline right parenthesis EndAbsoluteValue Superscript StartFraction p Over m EndFraction Baseline d theta 3rd Row 1st Column Blank 2nd Column equals StartFraction r Superscript left parenthesis m minus 1 right parenthesis p Baseline Over m EndFraction integral Subscript 0 Superscript 2 pi m Baseline StartAbsoluteValue StartFraction f prime left parenthesis s e Superscript i t Baseline right parenthesis Over f left parenthesis s e Superscript i t Baseline right parenthesis EndFraction EndAbsoluteValue Superscript p Baseline StartAbsoluteValue f left parenthesis s e Superscript i t Baseline right parenthesis EndAbsoluteValue Superscript StartFraction p Over m EndFraction Baseline d t 4th Row 1st Column Blank 2nd Column equals r Superscript left parenthesis m minus 1 right parenthesis p Baseline integral Subscript 0 Superscript 2 pi Baseline StartAbsoluteValue StartFraction f prime left parenthesis s e Superscript i t Baseline right parenthesis Over f left parenthesis s e Superscript i t Baseline right parenthesis EndFraction EndAbsoluteValue Superscript p Baseline StartAbsoluteValue f left parenthesis s e Superscript i t Baseline right parenthesis EndAbsoluteValue Superscript StartFraction p Over m EndFraction Baseline d t period EndLayout

Write $\frac {p}{m}=\alpha +\beta $ , where $\alpha ,\beta>0$ . Then

$$ \begin{align*}2\pi M_p^p(r,\varphi') = r^{(m-1)p}\int_0^{2\pi} \left|\frac{f'(se^{it})}{f(se^{it})}\right|^p{} |f(se^{it})|^{\alpha+\beta}\,dt. \end{align*} $$

Write

$$ \begin{align*}A(t):= \left|\frac{f'(se^{it})}{f(se^{it})}\right|^p{} |f(se^{it})|^\alpha, \qquad B(t):=|f(se^{it})|^\beta. \end{align*} $$

Then

$$ \begin{align*}A(t)B(t)= \left|\frac{f'(se^{it})}{f(se^{it})}\right|^p{} |f(se^{it})|^{\alpha+\beta}. \end{align*} $$

Applying Hölder’s inequality, we get

$$ \begin{align*} 2\pi M_p^p(r,\varphi') &\le r^{(m-1)p} \left(\int_0^{2\pi}A(t)^{2/p}\,dt\right)^{p/2} \left(\int_0^{2\pi}B(t)^{2/(2-p)}\,dt\right)^{(2-p)/2}. \end{align*} $$

Hence,

$$ \begin{align*} 2\pi M_p^p(r,\varphi') &\le r^{(m-1)p} \left( \int_0^{2\pi} |f(se^{it})|^{\frac{2\alpha}{p}-2}|f'(se^{it})|^2\,dt \right)^{p/2} \\ &\qquad\times \left( \int_0^{2\pi}|f(se^{it})|^{\frac{2\beta}{2-p}}\,dt \right)^{(2-p)/2}. \end{align*} $$

Denote

$$ \begin{align*}I_3(s):= \left( \int_0^{2\pi} |f(se^{it})|^{\frac{2\alpha}{p}-2}|f'(se^{it})|^2\,dt \right)^{p/2}, \end{align*} $$

and

$$ \begin{align*}I_4(s):= \left( \int_0^{2\pi}|f(se^{it})|^{\frac{2\beta}{2-p}}\,dt \right)^{(2-p)/2}. \end{align*} $$

Then

$$ \begin{align*}2\pi M_p^p(r,\varphi')\le r^{(m-1)p}I_3(s)I_4(s). \end{align*} $$

We next estimate $I_3(s)$ and $I_4(s)$ .

Estimate of $I_3(s)$ . Since $\alpha>0$ . Put

$$ \begin{align*}\delta:=\frac{2\alpha}{p}>0. \end{align*} $$ delta colon equals StartFraction 2 alpha Over p EndFraction greater than 0 period

Then

$$ \begin{align*}I_3(s)= \left( \int_0^{2\pi}|f(se^{it})|^{\delta-2}|f'(se^{it})|^2\,dt \right)^{p/2}. \end{align*} $$

Since $f\in \mathcal {S}$ and $\frac 12< s<1$ , it follows from [Reference Pommerenke22, Lemma 5.2] that

$$ \begin{align*}\int_0^{2\pi}|f(se^{it})|^{\delta-2}|f'(se^{it})|^2\,dt \le C_\delta \frac{M_\infty(s,f)^\delta}{1-s}. \end{align*} $$

Therefore,

$$ \begin{align*}I_3(s)\le \left( C_\delta \frac{M_\infty(s,f)^\delta}{1-s} \right)^{p/2}. \end{align*} $$ upper I 3 left parenthesis s right parenthesis less than or equals left parenthesis upper C Subscript delta Baseline StartFraction upper M Subscript infinity Baseline left parenthesis s comma f right parenthesis Superscript delta Baseline Over 1 minus s EndFraction right parenthesis Superscript p divided by 2 Baseline period

Since $f \in \mathcal {S}$ , the growth theorem (see [Reference Duren9, Theorem 2.6]) yields

$$ \begin{align*}M_\infty(s,f)\le \frac{s}{(1-s)^2}\le \frac{1}{(1-s)^2}. \end{align*} $$

Substituting this estimate, we get

$$ \begin{align*} I_3(s) &\le C^{\prime}_\delta\left( \frac{1}{(1-s)^{2\delta}}\cdot\frac{1}{1-s} \right)^{p/2} = C^{\prime}_\delta(1-s)^{-\left(\delta p+\frac{p}{2}\right)}. \end{align*} $$

Since $\delta =\dfrac {2\alpha }{p}$ delta equals StartFraction 2 alpha Over p EndFraction , so $\delta p=2\alpha $ , we get

$$ \begin{align*}I_3(s)\le C^{\prime}_\delta(1-s)^{-\left(2\alpha+\frac{p}{2}\right)}. \end{align*} $$

Estimate of $I_4(s)$ . Put

$$ \begin{align*}\lambda:=\frac{2\beta}{2-p}.\end{align*} $$ lamda colon equals StartFraction 2 beta Over 2 minus p EndFraction period

Theorem I yields

$$ \begin{align*}\int_0^{2\pi}|f(se^{it})|^\lambda\,dt \le \frac{C_\lambda'}{(1-s)^{2\lambda-1}},~\qquad \lambda>\frac{1}{2}. \end{align*} $$ integral Subscript 0 Superscript 2 pi Baseline StartAbsoluteValue f left parenthesis s e Superscript i t Baseline right parenthesis EndAbsoluteValue Superscript lamda Baseline d t less than or equals StartFraction upper C prime Subscript lamda Baseline Over left parenthesis 1 minus s right parenthesis Superscript 2 lamda minus 1 Baseline EndFraction comma lamda greater than one half period

Moreover,

$$ \begin{align*}\lambda>\frac12 \iff \beta>\frac{2-p}{4}. \end{align*} $$

Therefore,

$$ \begin{align*} I_4(s) &= \left( \int_0^{2\pi}|f(se^{it})|^\lambda\,dt \right)^{(2-p)/2} \le C^{\prime\prime}_{\lambda}(1-s)^{-(2\lambda-1)(2-p)/2}= C^{\prime\prime}_{\lambda}(1-s)^{-\left(2\beta-1+\frac{p}{2}\right)}, \end{align*} $$

provided that

$$ \begin{align*}\frac{2-p}{4}<\beta<\frac{p}{m}.\end{align*} $$

Such a choice of $\beta $ is possible because p satisfies (3.9).

Combining the estimates for $I_3(s)$ and $I_4(s)$ , we obtain

$$ \begin{align*} 2\pi M_p^p(r,\varphi') &\le r^{(m-1)p} C^{\prime}_\delta C^{\prime\prime}_{\lambda}(1-s)^{-\left(2\alpha+\frac{p}{2}\right)} (1-s)^{-\left(2\beta-1+\frac{p}{2}\right)} \\ &\le C^{\prime}_{p,m}(1-s)^{-\left(\left(1+\frac{2}{m}\right)p-1\right)}\qquad (\text{since } \alpha+\beta=p/m)\\ &= C^{\prime}_{p,m}(1-r^m)^{-\left(\left(1+\frac{2}{m}\right)p-1\right)}\qquad (\text{since}~ s=r^m)\\ &\le C^{\prime\prime}_{p,m}(1-r)^{-\left(\left(1+\frac{2}{m}\right)p-1\right)}. \end{align*} $$

Hence,

$$ \begin{align*}M_p^p(r,\varphi') \le \frac{C_{p,m}}{(1-r)^{\left(1+\frac{2}{m}\right)p-1}}, \qquad 0< r<1.\\[-42pt] \end{align*} $$ upper M Subscript p Superscript p Baseline left parenthesis r comma phi Superscript prime Baseline right parenthesis less than or equals StartFraction upper C Subscript p comma m Baseline Over left parenthesis 1 minus r right parenthesis Superscript left parenthesis 1 plus StartFraction 2 Over m EndFraction right parenthesis p minus 1 Baseline EndFraction comma 0 less than r less than 1 period

Lemma 4 Let $\phi $ be an m-fold symmetric analytic function on $\mathbb D$ with $\phi (0)=0$ , and suppose that $ \phi \prec _q \varphi $ , where $\varphi \in \mathcal S^{(m)}$ with $m\ge 2$ and quasi-subordination relation is defined by (1.4). Assume further that $\xi $ extends continuously to $\overline {\mathbb D}$ , is absolutely continuous on $\mathbb T$ , and that $ \eta ' \in H^\infty (\mathbb D). $ Let $\omega $ be a Schwarz function. Then, for every

(3.10) $$ \begin{align} \frac{2m}{3m+4}<p<1 \end{align} $$

and $0<r<1$ , there exists a constant $C^{\prime }_{p,m}>0$ , depending only on p, m, $\|\xi ^{\prime }\|_{H^1}$ , and $\|\eta ^{\prime }\|_\infty $ , such that

$$ \begin{align*}\int_0^{2\pi} \left| \frac{\phi'(re^{it})}{1-\omega(re^{it})} \right|^p\,dt \le \frac{C^{\prime}_{p,m}}{(1-r)^{\left(2+\frac{2}{m}\right)p-1}}. \end{align*} $$

Proof Since

$$ \begin{align*}\phi(z) = \xi(z)\varphi(\eta(z)). \end{align*} $$

As $\frac {2m}{3m+4}<p<1$ , Lemma F yields

$$ \begin{align*}|\phi'(z)|^p \le |\xi'(z)\varphi(\eta(z))|^p + |\xi(z)\varphi'(\eta(z))\eta'(z)|^p.\end{align*} $$

Set

Then

$$ \begin{align*}I(r) \le I_5(r)+I_6(r), \end{align*} $$

where

$$ \begin{align*}I_5(r):= \int_0^{2\pi} \left| \frac{\xi'(re^{it})\varphi(\eta(re^{it}))}{1-\omega(re^{it})} \right|^pdt \end{align*} $$

and

$$ \begin{align*}I_6(r):= \int_0^{2\pi} \left| \frac{\xi(re^{it})\varphi'(\eta(re^{it}))\eta'(re^{it})}{1-\omega(re^{it})} \right|^pdt. \end{align*} $$

We estimate $I_5(r)$ and $I_6(r)$ separately.

Estimate of $I_{5}(r)$ . By Hölder inequality with exponents $1/p$ and $1/(1-p)$ , we obtain

$$ \begin{align*}I_{5}(r)\leq \left(\int_{0}^{2\pi} |\xi'(re^{it})|\,dt\right)^{p} \left(\int_{0}^{2\pi} \left| \frac{\varphi(\eta(re^{it}))}{1-\omega(re^{it})} \right|^{\frac{p}{1-p}} \,dt\right)^{1-p}. \end{align*} $$

upper I 5 left parenthesis r right parenthesis less than or equals left parenthesis integral Subscript 0 Superscript 2 pi Baseline StartAbsoluteValue xi prime left parenthesis r e Superscript i t Baseline right parenthesis EndAbsoluteValue d t right parenthesis Superscript p Baseline left parenthesis integral Subscript 0 Superscript 2 pi Baseline StartAbsoluteValue StartFraction phi left parenthesis eta left parenthesis r e Superscript i t Baseline right parenthesis right parenthesis Over 1 minus omega left parenthesis r e Superscript i t Baseline right parenthesis EndFraction EndAbsoluteValue Superscript StartFraction p Over 1 minus p EndFraction Baseline d t right parenthesis Superscript 1 minus p Baseline period

Since $\xi $ extends continuously to $\overline {\mathbb {D}}$ and is absolutely continuous on $\mathbb {T}$ , it follows from [Reference Mashreghi18, Theorem 8.3] that

(3.11) $$ \begin{align} \left(\int_{0}^{2\pi} |\xi'(re^{it})|\,dt\right)^{p} \leq C_p , \end{align} $$ left parenthesis integral Subscript 0 Superscript 2 pi Baseline StartAbsoluteValue xi prime left parenthesis r e Superscript i t Baseline right parenthesis EndAbsoluteValue d t right parenthesis Superscript p Baseline less than or equals upper C Subscript p Baseline comma

where $C_p$ is a constant depending only on p and $\|\xi ^{\prime }\|_{H^1}$ . Hence,

$$ \begin{align*}I_5(r) \lesssim \left( \int_0^{2\pi} \left\vert \frac{\varphi(\eta(re^{it}))}{1-\omega(re^{it})} \right\vert^q dt \right)^{1-p}, \qquad q:=\frac{p}{1-p}. \end{align*} $$

Since $\varphi \in \mathcal {S}^{(m)}$ , there exists a function $F \in \mathcal {S}$ such that

$$ \begin{align*}\varphi(z)^m = F(z^m), ~ z \in \mathbb{D}. \end{align*} $$

Using Lemmas A–C, together with Theorem E, we obtain

$$ \begin{align*} \left( \log \left| \frac{\varphi(\eta(re^{it}))}{1-\omega(re^{it})} \right| \right)^* &\leq (\log |\varphi(\eta(re^{it}))|)^* + \left( \log \left| \frac{1}{1-\omega(re^{it})} \right| \right)^* \\[1ex] &\leq (\log |\varphi(re^{it})|)^* + \left( \log \left| \frac{1}{1-re^{it}} \right| \right)^* \\[1ex] &= \left( \log |F(r^me^{imt})|^{\frac{1}{m}} \right)^* + \left( \log \left| \frac{1}{1-re^{it}} \right| \right)^* \\[1ex] &\leq \left( \log |k(r^me^{imt})|^{\frac{1}{m}} \right)^* + \left( \log \left| \frac{1}{1-re^{it}} \right| \right)^* \\[1ex] &= \left( \log \frac{r}{|1-r^me^{imt}|^{\frac{2}{m}}\,} \right)^* + \left( \log \left| \frac{1}{1-re^{it}} \right| \right)^*\\[1ex] &\leq \left( \log r \right)^*+\left( \log \frac{1}{|1-r^me^{imt}|^{\frac{2}{m}}\,} \right)^* + \left( \log \left| \frac{1}{1-re^{it}} \right| \right)^*. \end{align*} $$

StartLayout 1st Row 1st Column left parenthesis log StartAbsoluteValue StartFraction phi left parenthesis eta left parenthesis r e Superscript i t Baseline right parenthesis right parenthesis Over 1 minus omega left parenthesis r e Superscript i t Baseline right parenthesis EndFraction EndAbsoluteValue right parenthesis Superscript asterisk 2nd Column less than or equals left parenthesis log StartAbsoluteValue phi left parenthesis eta left parenthesis r e Superscript i t Baseline right parenthesis right parenthesis EndAbsoluteValue right parenthesis Superscript asterisk Baseline plus left parenthesis log StartAbsoluteValue StartFraction 1 Over 1 minus omega left parenthesis r e Superscript i t Baseline right parenthesis EndFraction EndAbsoluteValue right parenthesis Superscript asterisk Baseline 2nd Row 1st Column Blank 2nd Column less than or equals left parenthesis log StartAbsoluteValue phi left parenthesis r e Superscript i t Baseline right parenthesis EndAbsoluteValue right parenthesis Superscript asterisk Baseline plus left parenthesis log StartAbsoluteValue StartFraction 1 Over 1 minus r e Superscript i t Baseline EndFraction EndAbsoluteValue right parenthesis Superscript asterisk Baseline 3rd Row 1st Column Blank 2nd Column equals left parenthesis log StartAbsoluteValue upper F left parenthesis r Superscript m Baseline e Superscript i m t Baseline right parenthesis EndAbsoluteValue Superscript StartFraction 1 Over m EndFraction Baseline right parenthesis Superscript asterisk Baseline plus left parenthesis log StartAbsoluteValue StartFraction 1 Over 1 minus r e Superscript i t Baseline EndFraction EndAbsoluteValue right parenthesis Superscript asterisk Baseline 4th Row 1st Column Blank 2nd Column less than or equals left parenthesis log StartAbsoluteValue k left parenthesis r Superscript m Baseline e Superscript i m t Baseline right parenthesis EndAbsoluteValue Superscript StartFraction 1 Over m EndFraction Baseline right parenthesis Superscript asterisk Baseline plus left parenthesis log StartAbsoluteValue StartFraction 1 Over 1 minus r e Superscript i t Baseline EndFraction EndAbsoluteValue right parenthesis Superscript asterisk Baseline 5th Row 1st Column Blank 2nd Column equals left parenthesis log StartFraction r Over StartAbsoluteValue 1 minus r Superscript m Baseline e Superscript i m t Baseline EndAbsoluteValue Superscript StartFraction 2 Over m EndFraction Baseline EndFraction right parenthesis Superscript asterisk Baseline plus left parenthesis log StartAbsoluteValue StartFraction 1 Over 1 minus r e Superscript i t Baseline EndFraction EndAbsoluteValue right parenthesis Superscript asterisk Baseline 6th Row 1st Column Blank 2nd Column less than or equals left parenthesis log r right parenthesis Superscript asterisk Baseline plus left parenthesis log StartFraction 1 Over StartAbsoluteValue 1 minus r Superscript m Baseline e Superscript i m t Baseline EndAbsoluteValue Superscript StartFraction 2 Over m EndFraction Baseline EndFraction right parenthesis Superscript asterisk Baseline plus left parenthesis log StartAbsoluteValue StartFraction 1 Over 1 minus r e Superscript i t Baseline EndFraction EndAbsoluteValue right parenthesis Superscript asterisk Baseline period EndLayout

Now, since

$$ \begin{align*}\frac{1}{(1-z^m)^{2/m}}\prec \frac{1}{(1-z)^{2/m}}. \end{align*} $$

Therefore, Lemma C gives

$$ \begin{align*}\left( \log \frac{1}{\vert 1-r^me^{imt}\vert^{2/m}} \right)^* \le \left( \log \frac{1}{\vert 1-re^{it}\vert^{2/m}} \right)^*. \end{align*} $$

Moreover, for each fixed $r\in (0,1)$ , the functions

$$ \begin{align*}\frac{1}{\vert 1-re^{it}\vert^{2/m}} \qquad\text{and}\qquad \frac{1}{\vert 1-re^{it}\vert} \end{align*} $$

are symmetric on $[-\pi ,\pi ]$ and non-increasing on $[0,\pi ]$ . Hence, by the equality case in Lemma E,

$$ \begin{align*}\left( \log \left\vert \frac{\varphi(\eta(re^{it}))}{1-\omega(re^{it})} \right\vert \right)^* \le \left( \log \frac{r}{\vert 1-re^{it}\vert^{1+\frac{2}{m}}} \right)^*. \end{align*} $$

Choosing $\Phi (x)=e^{qx}$ in Lemma A, we get

$$ \begin{align*}\int_{0}^{2\pi} \left| \frac{\varphi(\eta(re^{it}))}{1-\omega(re^{it})} \right|^{q} dt \leq \int_{0}^{2\pi} \frac{r^{q}} {|1-re^{it}|^{(1+\frac{2}{m})q}} \,dt. \end{align*} $$

Since

$$ \begin{align*}p>\frac{2m}{3m+4}, \end{align*} $$

we have

$$ \begin{align*}\left(1+\frac{2}{m}\right)\frac{p}{1-p}>1, \end{align*} $$ left parenthesis 1 plus StartFraction 2 Over m EndFraction right parenthesis StartFraction p Over 1 minus p EndFraction greater than 1 comma

that is,

$$ \begin{align*}\left(1+\frac{2}{m}\right)q>1. \end{align*} $$

Hence, Lemma E yields

$$ \begin{align*}\int_0^{2\pi} \frac{r^{q}}{\vert 1-re^{it}\vert^{(1+\frac{2}{m})q}}\,dt \lesssim \frac{1}{(1-r)^{(1+\frac{2}{m})q-1}}. \end{align*} $$

Consequently,

$$ \begin{align*}I_5(r) \lesssim \frac{1}{(1-r)^{\left(\left(1+\frac{2}{m}\right)q-1\right)(1-p)}} = \frac{1}{(1-r)^{(2+\frac{2}{m})p-1}}. \end{align*} $$

Estimate of $I_6(r)$ . Since $ |\xi (z)| \leq 1$ and $\eta ' \in H^\infty (\mathbb {D})$ , we have

$$ \begin{align*}I_6(r) \leq \|\eta^{\prime}\|_\infty^p \int_0^{2\pi} \left| \frac{\varphi'(\eta(re^{it}))}{1 - \omega(re^{it})} \right|^p \, dt. \end{align*} $$

Applying Hölder’s inequality with conjugate exponents $\alpha $ and $\beta $ , we obtain

$$ \begin{align*} I_6(r)\lesssim \left( \int_0^{2\pi} \left| \varphi'\big(\eta(re^{it})\big) \right|^{p \alpha}{} \, dt \right)^{1/\alpha} \left( \int_0^{2\pi} \left| 1 - \omega(re^{it}) \right|^{-p \beta}{} \, dt \right)^{1/\beta}. \end{align*} $$

Put $a=p\alpha $ and $ b=p\beta $ . Then

Thus,

$$ \begin{align*} I_6(r)\lesssim \left( \int_0^{2\pi} \left| \varphi'\big(\eta(re^{it})\big) \right|^{a}{} \, dt \right)^{p/a} \left( \int_0^{2\pi} \left| 1 - \omega(re^{it}) \right|^{-b}{} \, dt \right)^{p/b}.\end{align*} $$

Since $\eta $ is Schwarz function, we have $\varphi '\circ \eta \prec \varphi '$ , and thus Theorem D gives

$$ \begin{align*}\int_{0}^{2\pi} |\varphi'(\eta(re^{it}))|^{a}\,dt \leq \int_{0}^{2\pi} |\varphi'(re^{it})|^{a}\,dt. \end{align*} $$

Now Lemma 3 implies

$$ \begin{align*}\int_{0}^{2\pi} |\varphi'(re^{it})|^{a}\,dt \lesssim \frac{1}{(1-r)^{\left(1+\frac{2}{m}\right)a-1}}, ~\quad a>\frac{2m}{m+4}. \end{align*} $$

Therefore,

$$ \begin{align*}\left( \int_{0}^{2\pi} \left|\varphi'(\eta(re^{it}))\right|{}^{a}\,dt \right)^{p/a}\lesssim \frac{1}{(1-r)^{\frac{\left(\left(1+\frac{2}{m}\right)a-1\right)p}{a}}}. \end{align*} $$

left parenthesis integral Subscript 0 Superscript 2 pi Baseline StartAbsoluteValue phi prime left parenthesis eta left parenthesis r e Superscript i t Baseline right parenthesis right parenthesis EndAbsoluteValue Superscript a Baseline d t right parenthesis Superscript p divided by a Baseline less than or equivalent to StartFraction 1 Over left parenthesis 1 minus r right parenthesis Superscript StartStartFraction left parenthesis left parenthesis 1 plus StartFraction 2 Over m EndFraction right parenthesis a minus 1 right parenthesis p OverOver a EndEndFraction Baseline EndFraction period

For the second factor, an application of Lemma E, together with Theorem D, yields

$$ \begin{align*}\left( \int_{0}^{2\pi} |1-\omega(re^{it})|^{-b}\,dt \right)^{p/b} \lesssim \frac{1}{(1-r)^{\frac{(b-1)p}{b}}},~\qquad ~b>1. \end{align*} $$

Moreover,

$$ \begin{align*}b>1 \iff a<\frac{p}{1-p}. \end{align*} $$

Combining the above estimates, we obtain

(3.12) $$ \begin{align} I_{6}(r)\lesssim \frac{1}{(1-r)^{\frac{\left(\left(1+\frac{2}{m}\right)a-1\right)p}{a}}} \cdot \frac{1}{(1-r)^{\frac{(b-1)p}{b}}}, \end{align} $$

provided that

$$ \begin{align*}\max\left\{ \frac{2m}{m+4},\, p \right\} < a < \frac{p}{1-p}. \end{align*} $$ max left brace StartFraction 2 m Over m plus 4 EndFraction comma p right brace less than a less than StartFraction p Over 1 minus p EndFraction period

Such a choice of a is possible because p satisfies (3.10).

Finally, since

$$ \begin{align*}\frac{\left(\left(1+\frac{2}{m}\right)a-1\right)p}{a}+\frac{(b-1)p}{b} = \left(1+\frac{2}{m}\right)p-\frac{p}{a}+p-\frac{p}{b} = \left(2+\frac{2}{m}\right)p-1, \end{align*} $$

we get

$$ \begin{align*}I_6(r)\lesssim \frac{1}{(1-r)^{\left(2+\frac{2}{m}\right)p-1}}. \end{align*} $$ upper I 6 left parenthesis r right parenthesis less than or equivalent to StartFraction 1 Over left parenthesis 1 minus r right parenthesis Superscript left parenthesis 2 plus StartFraction 2 Over m EndFraction right parenthesis p minus 1 Baseline EndFraction period

Combining the estimates for $I_5(r)$ and $I_6(r)$ , we conclude that

$$ \begin{align*}I(r)\le I_5(r)+I_6(r) \le \frac{C^{\prime}_{p,m}}{(1-r)^{\left(2+\frac{2}{m}\right)p-1}}. \end{align*} $$

This completes the proof.

The proof of Theorem 3 follows by applying Lemmas D, 3, and 4. The proof follows an argument analogous to that of Theorem 2.

3.4 Proof of Theorem 4

The inequality

$$\begin{align*}\big|\,|a_n| - |b_n|\,\big| \le |a_n + \epsilon b_n| \le n, \qquad n \ge 2, \end{align*}$$

follows by arguments analogous to those used in the proof of Theorem 1. Since the dilatation $w(z)=q^2(z)$ , this implies $\vert q(z)\vert <1$ and $q(0)=0$ . By the Schwarz lemma, it follows that $ \vert w(z)\vert \leq \vert z\vert ^2 $ for all $z \in \mathbb {D}$ . So, the Taylor coefficients of $w(z)/(1 + \epsilon w(z))$ and $\phi _\epsilon '(z)$ are dominated in modulus by those of $z^2/(1-z^2)$ and the derivative of the Koebe function $ k(z) = z/(1 - z)^2$ , respectively. In accordance with the terminology from [Reference Goodman11, Chapter 7, p. 82], we may express this as

$$ \begin{align*} g^{\prime}(z)=\dfrac{w(z)}{1 + \epsilon w(z)} \, \phi_{\epsilon}'(z) \ll \frac{z^2}{1 - z^2} \cdot \frac{1 + z}{(1 - z)^3} = \frac{1}{6} \sum_{n=2}^\infty (n - 1)n(n + 1)z^{n},\end{align*} $$

which implies $\vert b_n\vert \leq (n - 1)(n - 2)/6,~ \text {for all } n \geq 2.$ Similarly, we have

$$\begin{align*}h^{\prime}(z)=\frac {\phi_{\epsilon}'(z)}{1 + \epsilon w(z)} \ll \frac{1}{1-z^2} \cdot \frac{1 + z}{(1 - z)^3} = 1 + \frac{1}{6}\sum_{n=2}^{\infty} (n + 1)(n +2)(n+3)z^{n}.\end{align*}$$

This implies $\vert a_n \vert \leq (n + 1)(n + 2)/6, ~\text {for all } n \geq 2.$ The proof of growth estimate follows the same technique as that of Theorem 1, and therefore the details are omitted here.

Acknowledgements

The authors declare that they have no competing interests relevant to the content of this article.

Footnotes

The first author acknowledges the Council of Scientific and Industrial Research (CSIR), India, for financial support in the form of a Senior Research Fellowship. The second author received partial funding through the Core Research Grant (CRG/2022/008920) awarded by the Anusandhan National Research Foundation (ANRF), India.

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Article contents

Coefficients and integral mean estimates for harmonic mappings via quasi-subordination

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MSC classification

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1 Introduction and main results

Conjecture A (Clunie and Sheil-Small)

1.1 Quasi-subordination

1.2 Hardy spaces

1.3 Harmonic mappings associated with m-fold symmetric functions

1.4 Harmonic mappings and minimal surfaces

Theorem A [Reference Duren10, p. 177]

2 Preliminaries and auxiliary results

Theorem B [Reference Hayman12, p. 249]

Theorem C [Reference Clunie and Sheil-Small4]

Lemma A [Reference Duren9, p. 218]

Lemma B [Reference Duren9, p. 230]

Lemma C [Reference Duren9, p. 230]

Theorem D (Littlewood’s subordination theorem) [Reference Duren9]

Lemma D [Reference Das and Kaliraj5]

Theorem E [Reference Duren9, p. 215]

Lemma E [Reference Pommerenke23, p. 176]

Lemma F [Reference Duren8, p. 57]

3 Proofs of main results

3.1 Proof of Theorem 1

3.2 Proof of Theorem 2

3.3 Proof of Theorem 3

3.4 Proof of Theorem 4

Acknowledgements

Footnotes

References

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