2 Definitions and preliminary results

In this section, we collect the required definitions and notation and prove some auxiliary results. The dual space of a topological linear space $X^{*}$ is the space of all continuous linear functionals on X; we denote this space by $X^{*}$ . If X and Y are topological linear spaces, then $X\hookrightarrow Y$ means that $X \subset Y$ and the inclusion map is continuous. We write $X=Y$ whenever $X\hookrightarrow Y$ and $Y\hookrightarrow X$ . Given two nonnegative functions f and g defined on the same set S, we write $f \sim g$ , if there exist positive constants $\gamma _1$ and $\gamma _2$ such that $\gamma _1\, g(s)\leqslant f(s) \leqslant \gamma _2\, g(s)$ for all $s\in S$ .

Let $\mathbb {D}$ be the open unit disk in the complex plane. The space of analytic functions on $\mathbb {D}$ will be denoted by $H(\mathbb {D})$ . As usual $\mathbb {T}$ denotes the unit circle, $\mathbb {T} = \{z\in \mathbb {C}; \, |z| = 1\}$ . We let m be the probability Lebesgue measure on $\mathbb {T}$ , and we denote by $L^0(\mathbb {T})$ the space of all m-measurable complex-valued almost everywhere finite functions on $\mathbb {T}$ . In what follows, we do not distinguish functions which are equal almost everywhere. The characteristic function of a measurable set E is denoted by $\chi _E$ .

If a function $f\in H(\mathbb {D})$ is such that $f^{*}(e^{i\theta }) := \lim _{r\to 1-}f(re^{i\theta })$ , then $f^{*}$ is called the radial limit function of f. For $p\in (0, \infty )$ , the Hardy space $H^p$ consists of all $f\in H(\mathbb {D})$ such that

$$\begin{align*}\|f\|_{H^p} := \sup_{r \in [0, 1)} \bigg(\frac{1}{2\pi} \int_0^{2\pi} |f(re^{i\theta})|^p\,d\theta\bigg)^{1/p} <\infty\,. \end{align*}$$

The Nevanlinna class, N, consists of those functions $f \in H(\mathbb {D})$ which satisfy

It is a well-known fact that functions in N have radial limits. Furthermore, by subharmonicity of $\log ^{+}|f|$ for any not identically $0$ holomorphic function f in $\mathbb {D}$ , we can change the supremum in the definition of N into the limit with r tending to $1-$ . We also have that is a topological linear group in sense Shapiro and Shields [Reference Shapiro and Shields17], and $(N,\rho )$ is a metric space, where $\rho $ is given by $\rho (f,g):= \rule[-1pt]{1pt}{9.5pt}\hskip2pt{f-g}\hskip 2pt\rule[-1pt]{1pt}{9.5pt}$ for all $f,g\in N$ . Note that $\rho $ does not generate a linear topology on N (see [Reference Shapiro and Shields17]). The largest subspace on N on which $\rho $ defines a linear topology is the Smirnov class, denoted by $N^{+}$ . Recall that the Smirnov class $N^{+}$ consists of all $f\in N$ for which the family $\{\log ^{+}|f_r|;\, r\in [0, 1)\}$ is uniformly integrable, where for every $f\in H(\mathbb {D})$ and $r \in [0, 1)$ , we let $f_r(z):= f(rz)$ for all $z\in \mathbb {T}$ . It is well-known that a function $f\in N^{+}$ if, and only if,

$$\begin{align*}\lim_{r\to 1-} \frac{1}{2\pi} \int_0^{2\pi} \log^{+} |f_r(e^{i\theta})| \,d\theta = \frac{1}{2\pi} \int_0^{2\pi} \log^{+}|f^{*}(e^{i\theta})|\,d\theta\,. \end{align*}$$

Yanagihara [Reference Yanagihara21] showed that is a Fréchet space. For more on spaces N and $N^{+}$ , see Duren’s book [Reference Duren1].

As usual by $A=A(\mathbb {D}),$ we denote the disk algebra, that is, the space of all functions holomorphic in $\mathbb {D}$ and continuous in $\overline {\mathbb {D}}$ . The following proper inclusions hold for $0<p<q\leqslant \infty $ :

$$\begin{align*}A \subset H^{q} \subset H^{p} \subset N^{+}\subset N\,. \end{align*}$$

Deep investigation of Nevanlinna and Hardy spaces and many interesting results obtained, among others, by Hardy, Littlewood, Riesz, and Nevanlinna were a motivation for some natural generalizations. Linking standard definitions of $H^p$ and N, Privalov defined spaces $N^p$ , $1<p<\infty $ (see [Reference Privalov14]). For given $p\in (1, \infty )$ , the space $N^p$ consists of all functions $f \in H(\mathbb {D})$ such that

$$ \begin{align*} \sup_{r\in [0, 1)} \frac{1}{2\pi} \int_{0}^{2\pi}\log^p(1+|f(re^{i\theta})|)\,d\theta <\infty\,. \end{align*} $$

For each $q>p>1$ , the following inclusions hold:

$$\begin{align*}N^q\subsetneq N^p,\quad \bigcup_{r>0}H^r\subsetneq\bigcap_{p>1}N^p, \quad\, \bigcup_{p>1}N^p\subsetneq N^{+}\subsetneq N\,. \end{align*}$$

We note that $f\in N^p$ if, and only if, $f \in N^{+}$ and $\log ^{+}|f^*|\in L^p(\mathbb {T})$ (see [Reference Stoll20]). If $f\in N^p$ then, for all $z\in \mathbb {D}$ , we have

$$\begin{align*}[\log^{+}|f(z)|]^p\leqslant \frac{1}{2\pi}\int_0^{2\pi}P(z, e^{i\theta})(\log^{+}|f^*(e^{i\theta})|)^{p}\,d\theta\,, \end{align*}$$

and

$$\begin{align*}\log^p(1+|f(z)|)\leqslant \frac{1}{2\pi}\int_0^{2\pi}P(z,e^{i\theta})\log^p(1+|f^*(e^{i\theta})|)\,d\theta\,, \end{align*}$$

where P is the Poisson kernel defined for all $z\in \mathbb {D}$ and $\theta \in [0, 2\pi )$ by

$$\begin{align*}P(z,e^{i\theta}) =\frac{1-|z|^2}{|e^{i\theta}-z|^2}\,. \end{align*}$$

Furthermore,

$$\begin{align*}\lim_{r\to 1-}\frac{1}{2\pi}\int_0^{2\pi}\log^p(1+|f(re^{i\theta})|)\,d\theta =\frac{1}{2\pi}\int_0^{2\pi}\log^p(1+|f^*(e^{i\theta})|)\,d\theta\,. \end{align*}$$

Observe that $N^{+}$ is a “natural limit” of $N^p$ as $p\to 1+$ . This observation leads to the definition of a topology on $N^p$ defined by the metric

$$\begin{align*}\rho_p(f,g) = \Big(\frac{1}{2\pi}\int_{-0}^{2\pi}\log^p(1+|f^{*}(e^{i\theta})-g^*(e^{i\theta})|)\, \mathit{d\theta}\Big)^{1/p}, \quad\, f, g\in N^p\,. \end{align*}$$

$(N^p,\rho _p)$ is an F-algebra. It is a generalization of Yanagihara’s concept for the class $N^{+}$ , which can be found in [Reference Yanagihara21]. We note that, for every $f\in N^p$ , we have

$$\begin{align*}\lim_{r\to 1-}\rho_p(f_r,f) = 0\,. \end{align*}$$

Our main motivation is to bring investigations on Privalov spaces to a more general setting, and find a general scheme for solving particular problems in the theory of abstract Nevanlinna classes. We will be mainly interested in problems concerning coefficient multipliers and the topological dual.

Before we will discuss in more details, these classes we point out that the study of many other important spaces of analytic functions is motivated by that fact, that for any not identically $0$ function $f\in H(\mathbb {D})$ and a convex nondecreasing function $\phi \colon \mathbb {R} \to [0, \infty )$ the function $\phi (\log |f|)$ is subharmonic (we let $\phi (-\infty ):= \lim _{t\to - \infty } \phi (t)=0$ and as usual it is understood that $\log 0 := - \infty $ ).

Following [Reference Rudin16], a function $\phi $ defined on $\mathbb {R}$ is said to be strongly convex if it is nonnegative, convex, and nondecreasing with $\phi (t)/t \to \infty $ as $t\to \infty $ . Let D be any domain in $\mathbb {C}$ . An analytic function on D is said to be in the Hardy–Orlicz class $H_\phi (D)$ if there exists a harmonic function h on D such that

$$\begin{align*}\phi(\log|f(z)|)\leqslant h(z), \quad\, z\in D\,. \end{align*}$$

The following result will be of essential importance for us. It refers to equivalent definitions of Hardy–Orlicz class $H_{\phi }(\mathbb {D})$ on the unit disk $\mathbb {D}$ ,

$$\begin{align*}H_{\phi}(\mathbb{D})= \bigg\{f\in N^{+}; \, \sup_{r\in [0, 1)} \int_0^{2\pi} \phi(\log|f(re^{i\theta})|)\,d\theta = \int_0^{2\pi} \phi(\log|f^{*}(e^{i\theta})|)\,d\theta \bigg\}\,. \end{align*}$$

For details and the background related to Hardy–Orlicz classes, we refer to [Reference Hasumi and Kataoka5], [Reference Leśniewicz8], [Reference Rosenblum and Rovnyak15], [Reference Stoll19].

The notation and assumptions that we set down here will be used throughout the paper. We will consider only strongly convex functions on $\mathbb {R}_{+}$ , that is, $\varphi \colon \mathbb {R}_{+} \to \mathbb {R}_{+}$ such that $\varphi $ is convex with $\phi (0)=0$ and $\varphi (t)/t\rightarrow \infty $ as $t\rightarrow \infty $ . The function $\varphi $ is said to satisfy the $\Delta _2$ -condition ( $\varphi \in (\Delta _2)$ for short), if there exists a constant $\gamma>0$ such that $\varphi (2t)\leqslant \gamma \varphi (t)$ for all $t>0$ . In what follows, we denote by $\Phi _2^{sc},$ the set of all continuously differentiable strongly convex functions that satisfy $\Delta _2$ -condition. Examples of functions in $\Phi _2^{sc}$ are $\varphi _p$ , $1\leqslant p<\infty $ , defined by $\varphi _{p}(t) = t^{p} \log (1 + t)$ for all $t\geqslant 0$ .

If $\varphi $ is a strongly convex function on $\mathbb {R}_{+}$ , we let $\phi (t):=0$ for all $t<0$ and $\phi (t):= \varphi (t)$ for all $t \geqslant 0$ . Then the Hardy–Orlicz class $H_{\phi }(\mathbb {D})$ is called the abstract Nevanlinna class and it is denoted by $N_\varphi $ . Following the above remarks,

$$\begin{align*}N_\varphi = \bigg\{f\in N^{+}; \,\, \sup_{r\in [0, 1)}\frac{1}{2\pi}\int_0^{2\pi} \varphi(\log^{+} |f(re^{i\theta})|)\,d\theta <\infty\bigg\}\,. \end{align*}$$

It follows from [Reference Rudin16] that, for all strongly convex functions $\varphi \colon \mathbb {R} \to \mathbb {R}_{+}$ , one has

$$\begin{align*}H_{\varphi} \subset N^{+} \qquad \text{and} \qquad N^{+} = \bigcup H_{\phi}\,, \end{align*}$$

where the sum is taken over all strongly convex functions $\phi \colon \mathbb {R} \to \mathbb {R}_{+}$ . Furthermore, $f\in N_\varphi $ if and only if $\varphi (\log ^{+} |f^{*}|)\in L^1(\mathbb {T})$ and, for all $z\in \mathbb {D},$

$$ \begin{align*} \varphi(\log^{+} |f(z)|))\leqslant F(z):=\frac{1}{2\pi}\int_0^{2\pi}P(z,e^{i\theta})\varphi(\log^{+} |f^*(e^{i\theta})|)\,d\theta\,. \end{align*} $$

Due to the inequalities

$$ \begin{align*} \log^+|t|\leqslant\log(1+|t|)\leqslant\log 2 + \log^{+} |t|, \quad\, t\in \mathbb{R}\,, \end{align*} $$

$\log ^{+}|f|$ can be exchanged with $\log (1+|f|)$ in the definition of N and $N^{+}$ . Furthermore, the $\Delta _2$ -condition for $\varphi $ yields

$$ \begin{align*} \varphi(\log(1+|z|))& \leqslant \varphi(\log 2+\log^+|z|) \leqslant \varphi\Big(2\Big(\frac{1}{2}\log 2+\frac{1}{2}\log^+|z|\Big)\Big)\\ & \leqslant \frac{\gamma}{2} \big(\varphi(\log 2)+\varphi(\log^+|z|)\big), \quad\, z\in\mathbb{C}\,. \end{align*} $$

We conclude that, in the case $\varphi \in (\Delta _2)$ , the above inequality shows that, for every $f\in N_\varphi $ , we have

$$ \begin{align*} \sup_{r\in [0, 1)}\int_0^{2\pi}\varphi(\log(1+|f(re^{i\theta})|))\,d\theta<\infty\,. \end{align*} $$

Now, it is easy to see that the above condition allows us to define $N_{\varphi }$ in a new way. Moreover, from Theorem [Reference Stoll19, Th. 2] and the fact that $z\mapsto \log (1+|z|)$ is subharmonic in $\mathbb {D}$ , we deduce that, for all $f\in N_\varphi $ , one has

$$ \begin{align*} \varphi(\log(1+|f(z)|)) \leqslant \frac{1}{2\pi}\int_0^{2\pi}P(z, e^{i\theta})\varphi(\log(1+|f^{*}(e^{i\theta})|))\,d\theta, \quad\, z\in \mathbb{D}\,, \end{align*} $$

and from the properties of subharmonic functions, Fatou’s Lemma and Tonelli’s Theorem, we easily get

$$ \begin{align*} \frac{1}{2\pi}\int_0^{2\pi}\varphi(\log(1+|f^*(e^{i\theta})|))\,d\theta & = \lim_{r\rightarrow 1-}\frac{1}{2\pi}\int_0^{2\pi}\varphi(\log(1+|f(re^{i\theta})|))\,d\theta \\ & = \sup_{r\in [0, 1)}\int_0^{2\pi}\varphi(\log(1+|f(re^{i\theta})|))\,d\theta\,. \end{align*} $$

It is now clear that we can define $N_\varphi $ as the space of all functions $f\in N^{+}$ for which

$$\begin{align*}\|f\|_\varphi :=\frac{1}{2\pi}\int_0^{2\pi}\varphi(\log(1+|f^{*}(e^{i\theta})|))\,d\theta<\infty\,. \end{align*}$$

We note that, from the above inequality, we can derive the following useful estimate valid for all functions f in $N_\varphi $ :

$$ \begin{align*} \varphi(\log(1+|f(z)|))\leqslant \frac{1+|z|}{1-|z|}\|f\|_\varphi, \quad\, z\in \mathbb{D}\,. \end{align*} $$

In particular, this implies that, for every $f\in N_\varphi $ , we have

$$ \begin{align*} \sup_{|z|\leqslant r} |f(z)| \leqslant\exp\bigg(\varphi^{-1}\bigg(\frac{2\|f\|_\varphi}{1-r}\bigg)\bigg)-1, \quad\, 0\leqslant r<1\,. \end{align*} $$

Now, we define a topological structure on spaces $N_\varphi $ . Our goal is to define a metric for $N_\varphi $ , for all functions $\varphi $ satisfying the conditions from the previous section. Let us begin with the following definition. Let X be a linear space. Following [Reference Kalton, Peck and Roberts6], the functional $\|\cdot \|\colon X\to [0,\infty )$ is called a $\Delta $ -norm on X, if it fulfills the following conditions for all x, $y\in X$ :

1. (i) $\|x\|>0$ for $x\neq 0$ .

2. (ii) $\|\lambda x\|\leqslant \|x\|$ for all $x\in X$ and $|\lambda |\leqslant 1$ .

3. (iii) There exists a constant $C>0$ such that $\|x+y\| \leqslant C\max \{\|x\|,\|y\|\}$ for all $x, y\in X$ .

4. (iv) $\lim _{\lambda \to 0}\|\lambda \,x\|=0$ for all $x\in X$ .

If $\|\cdot \|$ is a $\Delta $ -norm on X, it defines a metrizable topology $\tau $ on X for which

$$\begin{align*}U_n=\{x\in X;\, \|x\|<1/n\}, \quad\, n\in\mathbb{N}\end{align*}$$

forms a countable base of 0 neighborhoods. Hence, a sequence $\{x_n\}\subset X$ is convergent in $(X,\tau )$ to some element $x\in X$ if, and only if, $\|x_n-x\|\to 0$ . We note that from [Reference Kalton, Peck and Roberts6, Th. 1.2], it follows that the functional $\|\cdot \|^{*}$ defined by

$$ \begin{align*} \|x\|^{*} = \inf \bigg\{\sum_{j=1}^n\|x_j\|^{\delta}_{\varphi}; \,\,\, \sum_{j=1}^{n} x_j=x, \,\, n\in \mathbb{N} \bigg\} \end{align*} $$

is an F-norm on $N_\varphi $ satisfying condition

$$ \begin{align*} 4^{-1} \|x\|^\delta \leqslant \|x\|^{*} \leqslant \|x\|^{\delta}, \quad\, x \in X\,, \end{align*} $$

where $\delta>0$ is chosen such that $C^\delta =2$ .

We observe that $\|\cdot \|_\varphi $ defines a $\Delta $ -norm on $N_\varphi $ . Conditions (i) and (ii) are obvious and (iii) follows by convexity and the $\Delta _2$ -condition (with constant $\gamma $ ) of $\varphi $ ,

$$ \begin{align*} \|f+g\|_\varphi & \leqslant\frac{1}{2\pi}\int_0^{2\pi}\varphi(\log(1+|f^*(e^{i\theta})|)+\log(1+|g^*(e^{i\theta})|))\,d\theta\\ &\leqslant\frac{\gamma}{2}(\|f\|_\varphi+\|g\|_\varphi)\leqslant \gamma \max\big\{\|f\|_\varphi,\|g\|_\varphi\big\}\,, \end{align*} $$

for all $f,g\in N_\varphi $ . Condition (iv) follows from the Lebesgue Dominated Convergence Theorem. The above mentioned remarks yield that the functional $\|\cdot \|^*_\varphi \colon N_\varphi \to [0,\infty )$ given by the formula

$$ \begin{align*} \|f\|^{*}_\varphi = \inf\bigg\{\sum_{j=1}^n\|f_j\|^\delta_\varphi; \,\,\, \sum_{j=1}^nf_j=f, \,\, n \in \mathbb{N}\bigg\} \end{align*} $$

is an F-norm on $N_\varphi $ satisfying the condition

$$ \begin{align*} 4^{-1} \|f\|_{\varphi}^{\delta} \leqslant\|f\|^{*}_{\varphi} \leqslant \|f\|_{\varphi}^{\delta}, \quad\, f\in N_{\varphi}\,, \end{align*} $$

where $\delta>0$ is chosen such that $\gamma ^{\delta }=2$ .

It follows that $\rho _\varphi (f,g)=\|f-g\|^{*}_\varphi $ is a metric on $N_\varphi $ . We can now proceed to the problem of completeness of $N_\varphi $ . From the above inequalities, it is also clear that topologies on $N_\varphi $ generated by $\|f-g\|_\varphi $ and $\rho _\varphi (f,g)$ are equivalent. This observation will simplify our approach.

Using methods from [Reference Stoll20] in some generalized way, we will prove the following theorem.

Theorem 2.1. Let $\varphi \in \Phi _2^{sc}$ . The space $N_\varphi $ , with the topology given by the metric $\rho _\varphi $ , is an $F-$ algebra $($ topological vector space with the topology given by complete, translation invariant metric $)$ . Furthermore, if $f\in N_\varphi $ , then

$$\begin{align*}\lim_{r \rightarrow 1}\rho_\varphi(f_r, f)=0\,. \end{align*}$$

Before giving the proof of the above theorem, we need the following lemma.

Lemma 2.2. If $\varphi \in \Phi ^{sc}_2$ and $f\in N_\varphi $ , then

$$\begin{align*}\lim_{r\rightarrow 1}\frac{1}{2\pi}\int_0^{2\pi}\varphi(\log^+|f_r(e^{i\theta})-f^{*}(e^{i\theta})|)\,d\theta =0\,. \end{align*}$$

Proof. From [Reference Stoll20, Th. 4], it is sufficient to show that, if $\varphi \in \Phi ^{sc}_2$ , then the inequality

$$\begin{align*}\varphi(\log^{+}(a+b))\leqslant c_1 [\varphi(\log^{+}a) + \varphi(\log^{+}b) + c_2] \end{align*}$$

is valid for some $c_1$ , $c_2>0$ and all a, $b\geqslant 0$ . In fact, by $\varphi \in (\Delta _2)$ , we get that

$$ \begin{align*} \varphi(\log^{+}(a+b)) & \leqslant \varphi(\log(1+a+b)) \leqslant \gamma [\varphi(\log(1+a)+\varphi(\log(1+b)] \\ &\leqslant \gamma^2[(\varphi(\log^{+}a) + \varphi(\log^{+}b) + 2\varphi(\log2))]\,.\\[-37pt] \end{align*} $$

Proof of Theorem 2.1

The fact that $N_\varphi $ is an algebra follows from the inequality

$$\begin{align*}\log^+|fg|\leqslant\log^{+} |f| + \log^{+}|g|, \quad\, f,g\in N_{\varphi}\,. \end{align*}$$

We show now that $(N_\varphi , \|\cdot \|_\varphi ^{*})$ is a complete metric space. Let $\{f_n\}$ be a Cauchy sequence in $N_\varphi $ . Thus, there exists $C>0$ such $\|f_n\|_\varphi \leqslant C$ for each $n\in \mathbb {N}$ . Applying the estimate

$$\begin{align*}\sup_{|z|\leqslant r} |f(z)| \leqslant\exp\bigg(\varphi^{-1}\bigg(\frac{2\|f\|_\varphi}{1-r}\bigg)\bigg)-1, \quad\, 0\leqslant r<1\,, \end{align*}$$

we conclude that the sequence $\{f_n\}$ converges uniformly on any compact subset of $\mathbb {D}$ to some function f, and so from the Weierstrass theorem, $f\in H(\mathbb {D})$ . Hence

$$\begin{align*}\frac{1}{2\pi}\int_0^{2\pi}\varphi(\log(1+|f(re^{i\theta})|))dt=\lim_{n\rightarrow\infty}\frac{1}{2\pi} \int_0^{2\pi}\varphi(\log(1+|f_n(re^{i\theta})|))\,d\theta\leqslant C\,, \end{align*}$$

for every $0<r<1$ , so $f\in N_\varphi $ (see [Reference Hasumi and Kataoka5]). We also observe that by $f \in N^{+}$ one has

$$ \begin{align*} \frac{1}{2\pi}&\int_0^{2\pi}\varphi(\log(1+|f_n(re^{i\theta})-f(re^{i\theta})|))\,d\theta \\ &\leqslant \lim_{m\rightarrow\infty}\frac{1}{2\pi}\int_0^{2\pi}\varphi(\log(1+|f_n(re^{i\theta}) - f_m(re^{i\theta})|))\,d\theta \leqslant \lim_{m\rightarrow \infty} \|f_n-f_m\|_{\varphi}\,. \end{align*} $$

Combining with $\lim _{r\to 1-}\frac {1}{2\pi }\int _0^{2\pi }\varphi (\log (1+|f(re^{i\theta })|))\,d\theta =\frac {1}{2\pi }\int _0^{2\pi } \varphi (\log (1+|f^*(e^{i\theta })|))\,d\theta $ , we get that

$$\begin{align*}\|f_n-f\|_\varphi \leqslant \lim_{m\rightarrow\infty}\|f_n-f_m\|_\varphi\,. \end{align*}$$

This yields that $\rho _\varphi (f_n, f) \to 0$ as $n\to \infty $ . Thus, it remains to prove that

$$\begin{align*}\lim_{r \to 1-}\rho_\varphi(f_r,f)=0\,. \end{align*}$$

Since $f_r(\xi ) \to f^{*}(\xi )\ m$ -a.e. on $\mathbb {T}$ as $r\rightarrow 1-$ , $\varphi (\log (1+|f_r(\xi ) - f^{*}(\xi )|)) \to 0\ m$ -a.e. as $r\rightarrow 1-$ . Take a sequence $\{r_n\}$ with $r_n\rightarrow 1-$ as $n\to \infty $ and fix $\varepsilon>0$ . From Egorov’s theorem, there exists a set $E\in \mathbb {T}$ such that, $m(\mathbb {T}\backslash E)<\varepsilon $ and $\lim _{n\to \infty } \varphi (\log (1 + |f_{r_n}(e^{i\theta }) - f^{*}(e^{i\theta })|)) = 0$ uniformly on E. Clearly,

$$\begin{align*}\varphi(\log(1+|f_{r_n}(e^{i\theta}) - f^{*}(e^{i\theta})|)) \leqslant \gamma [\varphi(\log2) + \varphi(\log^{+} (|f_{r_n}(e^{i\theta}) - f^{*}(e^{i\theta})|))]\,. \end{align*}$$

In consequence, we obtain

$$ \begin{align*} &\lim_{n\rightarrow\infty} \frac{1}{2\pi}\int_0^{2\pi}\varphi(\log(1+|f_{r_n}(e^{i\theta})-f^{*}(e^{i\theta})|))\,d\theta\\ &=\lim_{n\rightarrow\infty}\frac{1}{2\pi}\Big(\int_E+\int_{\mathbb{T}\backslash E}\varphi(\log(1+|f_{r_n}(e^{i\theta}) -f^{*}(e^{i\theta})|))\,d\theta \Big)\\ &=\lim_{n\rightarrow\infty}\frac{1}{2\pi}\int_{\mathbb{T}\backslash E}\varphi(\log(1+|f_{r_n}(e^{i\theta}) - f^{*}(e^{i\theta})|))\,d\theta \\ & \leqslant \frac{\gamma}{2\pi}\Big[\varphi(\log 2)m(\mathbb{T}\backslash E) + \lim_{n\rightarrow\infty}\int_0^{2\pi}\varphi(\log^+|f_{r_n}(e^{i\theta}) - f^*(e^{i\theta})|)dt\Big]\,. \end{align*} $$

This completes the proof by Lemma 2.2.

From now on, we will be using some standard notation, namely,

$$\begin{align*}M_p(r,f)&:=\frac{1}{2\pi}\int_0^{2\pi}|f(re^{i\theta})|^{p}\,d\theta, \quad\, 0<p<\infty\,,\\M_\infty(r,f)&:= \sup_{\theta \in[0,2\pi)}|f(re^{i\theta})|\,. \end{align*}$$

By $F_\varphi $ , we denote the space of all functions $f\in H(\mathbb {D})$ for which

$$ \begin{align*} \lim_{r\rightarrow 1-}(1-r)\varphi(\log^{+}M_{\infty}(r,f)) = 0\,. \end{align*} $$

The next result shows the relationship between Nevanlinna’s abstract classes and $F_\varphi $ spaces.

Proof. Let $f\in N_\varphi $ . Take $\varepsilon>0$ and a function $g\in L^\infty (\mathbb {T})$ such that $0\leqslant g \leqslant \varphi (\log ^{+}|f^{*}|) m$ -a.e. on $\mathbb {T}$ and

$$\begin{align*}\frac{1}{2\pi}\int_0^{2\pi}[\varphi(\log^{+}|f^{*}(e^{i\theta})|) - g(e^{i\theta})]\,d\theta \leqslant \varepsilon\,. \end{align*}$$

We recall that the following estimate holds:

$$ \begin{align*} \varphi(\log^+(|f(z)|))\leqslant F(z):= \frac{1}{2\pi}\int_0^{2\pi}P(z,e^{i\theta})\varphi(\log^+(|f^*(e^{i\theta})|)\,d\theta\,. \end{align*} $$

Since for all $\theta $ , $t\in [0, 2\pi )$ , $0\leqslant r<1$ , $P(re^{i\theta }, e^{it})\leqslant \frac {1+r}{1-r}$ , we conclude that, for all $\theta \in [0, 2\pi ),$

$$ \begin{align*} |F(re^{i\theta})|& \leqslant \frac{1}{2\pi}\int_0^{2\pi}P(re^{i\theta},e^{it})[\varphi(\log^+|f^*(e^{it})|)- g(e^{it})+ g(e^{it})]\,dt\\ & \leqslant \frac{1+r}{1-r}\varepsilon + \|g\|_\infty\,. \end{align*} $$

Clearly, this implies that $(1-r)M_\infty (r,F)\leqslant (1+r)\varepsilon +\|g\|_\infty (1-r)$ . Hence

$$\begin{align*}\lim_{r \to 1-}(1-r)M_\infty(r,F)=0\,. \end{align*}$$

This combined with the above estimates yields

$$ \begin{align*} \lim_{r \rightarrow 1-}(1-r) \varphi(\log^{+} M_\infty(r,f))=0, \end{align*} $$

and this completes the proof.

Let $\varphi \in \Phi _2^{sc}$ . Then, for all $t\geqslant 0$ , $\overline {\varphi }(t):=\sup_{s>0}\frac {\varphi (st)}{\varphi (s)}$ . Note that, if a function $\varphi $ satisfies the $\Delta _2$ -condition, then $\overline {\varphi }(t)<\infty $ for all $t\geqslant 0$ and so, for all $t,s>0$ , we have $\varphi (st)\leqslant \overline {\varphi }(t)\varphi (s)$ .

By $F_\varphi ^{\text {int}}$ , we denote the space of all functions $f\in H(\mathbb {D})$ , such that, for all $c>0$ , we have

(2.1) $$ \begin{align} \|f\|_{\varphi,c}:=\int_0^1 M_\infty(r,f)\exp\Big[-c\,\varphi^{-1}\Big(\frac{1}{1-r}\Big)\Big]\,dr<\infty\,. \end{align} $$

Observe that $\{\|\cdot \|_{\varphi ,c}; \, c>0\}$ is a family of norms on $F_\varphi ^{int}$ , thus, it means that we can define a locally convex topology on this space.

Proof. Fix $f\in F_\varphi $ . Then, the function $g\colon [0,1)\to \mathbb {R}$ given by

$$\begin{align*}g(r):=(1-r)\varphi(\log^{+} M_\infty(r,f)), \quad\, r\in [0,1)\,, \end{align*}$$

is bounded on $[0,1)$ and continuous on $(0,1)$ . Clearly, $g(r)\to 0$ as $r\rightarrow 1-$ and the inequality

$$\begin{align*}M_\infty(r,f)\leqslant \exp\Big[\varphi^{-1}\Big(\frac{g(r)}{1-r}\Big)\Big] \end{align*}$$

is valid for all $r\in (0,1)$ . Thus, for all $c>0$ , we have

$$ \begin{align*} \int_0^1M_\infty(r,f)\exp&\Big[-c\,\varphi^{-1}\Big(\frac{1}{1-r}\Big)\Big]dr \\ &\leqslant\int_0^1 \exp\Big[\big(\overline{\varphi^{-1}}(g(r))-c\big)\varphi^{-1}\Big(\frac{1}{1-r}\Big)\Big]\,dr\,. \end{align*} $$

From the fact that $\varphi ^{-1}\in (\Delta _2)$ , it follows that $\overline {\varphi ^{-1}}(t)\to 0$ as $t\to 0+$ . Because $g(r)\to 0$ , so there exists $r_0\in (0,1)$ , such that $\overline {\varphi ^{-1}}(g(r))\leqslant c/2$ for $r\in [r_0,1)$ . In consequence,

$$ \begin{align*} \int_0^1\exp\Big[\big(\overline{\varphi^{-1}}(g(r))&-c\big)\varphi^{-1}\Big(\frac{1}{1-r}\Big)\Big]\,dr \\ &\leqslant\int_0^{r_0}\exp\Big[\big(\overline{\varphi^{-1}}(g(r))-c\big)\varphi^{-1}\Big(\frac{1}{1-r}\Big)\Big]\,dr\\ &+ \int_{r_0}^1\exp\Big[-\frac{c}{2}\varphi^{-1}\Big(\frac{1}{1-r}\Big)\Big]\,dr<\infty\,, \end{align*} $$

because both integrated functions on the right-hand side of inequality are bounded on the appropriate intervals.

From Theorem 2.3 and Proposition 2.4, we get the following corollary, which will become crucial in the sequel.

3 Multipliers form $N_\varphi $ to the Hardy spaces $H^p$

In this section, we will be interested in describing coefficient multipliers between $N_\varphi $ and the classical Hardy spaces $H^p$ . Let us recall that a complex sequence $\{\lambda _n\}:=\{\lambda _n\}_{n=0}^\infty $ is a multiplier from space X to Y of holomorphic functions in $\mathbb {D}$ if, for every $f\in X$ with power series $f(z)=\sum _{n=0}^\infty \widehat {f}(n) z^n$ for all $z\in \mathbb {D}$ , we have that the function $g(z)=\sum _{n=0}^\infty \lambda _n \widehat {f}(n) z^n$ belongs to Y. It is clear that to every multiplier $\{\lambda _n\}$ , we can associate a linear operator $\Lambda \colon X\to Y$ given by the formula:

$$\begin{align*}\Lambda f(z):= \sum_{n=0}^\infty \lambda_n \widehat{f}(n) z^{n}, \quad\, z\in \mathbb{D}\,, \end{align*}$$

for every $f\in X$ with power series $f(z)=\sum _{n=0}^\infty \widehat {f}(n) z^n$ for all $z\in \mathbb {D}$ .

We will use the following standard fact which easily follows from the Closed Graph Theorem.

It can be easily seen from the previously established inequality

(*) $$\begin{align} \sup_{|z|\leqslant r} |f(z)| \leqslant\exp\bigg(\varphi^{-1}\bigg(\frac{2\|f\|_\varphi}{1-r}\bigg)\bigg)-1, \quad\, r\in [0, 1) \end{align} $$

that one has $N_\varphi \hookrightarrow H(\mathbb {D})$ . Since $H^p \hookrightarrow H(\mathbb {D})$ for all $0<p\leqslant \infty $ , it follows that $\Lambda \colon N_\varphi \to H^p$ is indeed a continuous operator.

Our first goal in this section is to describe a necessary condition for a sequence $\{\lambda _n\}\in \boldsymbol {\omega }$ to be a multiplier from $N_\varphi $ to $H^p$ . We will start with some basic definitions and technical facts from the theory of Orlicz spaces.

For a function $\varphi \in \Phi _2^{sc}$ , the lower (resp. upper) Matuszewska–Orlicz index $p(\varphi )$ (resp. $q(\varphi )$ ) is given by the formula (see [Reference Matuszewska and Orlicz9, Reference Matuszewska and Orlicz10])

$$\begin{align*}p(\varphi)=\sup_{0<s<1}\frac{\log\overline{\varphi}(s)}{\log s} \qquad \bigg(\text{resp.} \, \, q(\varphi) =\inf_{s>1}\frac{\log\overline{\varphi}(s)}{\log s}\bigg)\,. \end{align*}$$

Since $\overline {\varphi }\colon [0,\infty )\to [0,\infty )$ is nondecreasing and submultiplicative, the following are true (see [Reference Krein, Petunin and Semenov7, pp. 53–54]):

$$\begin{align*}p(\varphi)=\lim_{s\to 0+}\frac{\log\overline{\varphi}(s)}{\log s}, \qquad q(\varphi)=\lim_{s\to\infty}\frac{\log\overline{\varphi}(s)}{\log s}\,. \end{align*}$$

Using the convexity of $\varphi $ , we get that $p(\varphi )\geqslant 1$ , while from $\varphi \in (\Delta _2)$ , we have $q(\varphi )<\infty $ . Hence $1\leqslant p(\varphi )\leqslant q(\varphi )<\infty $ . From the above formulas, we obtain that for every $\varepsilon>0$ there exists $C\geqslant 1$ , such that

$$\begin{align*}\max\big\{s^{p(\varphi)},s^{q(\varphi)}\big\}\leqslant \overline{\varphi}(s) \leqslant C\max\big\{s^{p(\varphi)-\varepsilon},s^{q(\varphi)+\varepsilon}\big\}, \quad\ s>0\,. \end{align*}$$

Let us recall that a subset E of a linear topological space X is called bounded, if it is absorbed by any zero neighborhood V, that is, if there exists $\alpha _0>0$ , such that $\alpha E\subset V$ for all $\alpha \in \mathbb {C}$ , for which $|\alpha | \leqslant \alpha _0$ .

Lemma 3.2. Let $\varphi \in \Phi _2^{sc}$ be a function such that $p(\varphi )>1$ , and let $\{r_n\} \subset [1/2,1)$ be a sequence such that $r_n\uparrow 1$ as $n\to \infty $ . Furthermore, let $\{\beta _n\}\subset (0,1)$ be a sequence satisfying the condition

$$\begin{align*}\lim_{n\to\infty}(1-r_n)\varphi\Big(\frac{\beta_n}{1-r_n}\Big)= 0\,. \end{align*}$$

Define a sequence $\{f_n\}$ by the following formula:

$$\begin{align*}f_n(z)=\exp\Big(\beta_n\frac{1+r_nz}{1-r_nz}\Big), \quad\, z\in\mathbb{D}\,. \end{align*}$$

Then, $\{f_{n};\, n\in \mathbb {N}\}$ is a bounded subset of $N_\varphi $ .

Proof. We first show that

$$ \begin{align*} C_{\varphi}:=\int_{-\infty}^{\infty}\overline{\varphi}\Big(\big(1+ 2 t^2/\pi \big)^{-1/2}\Big)\,dt <\infty\,. \end{align*} $$

Since $p(\varphi )>1$ , so for all $\varepsilon \in (0,p(\varphi )-1)$ , there exists constant $C=C(\varepsilon )>0$ , such that

$$\begin{align*}\overline{\varphi}(s)\leqslant Cs^{p(\varphi)-\varepsilon}, \quad\, s\in(0,1]\,. \end{align*}$$

Hence, for $p:=p(\varphi )-\varepsilon $ , we have $p>1$ and so

$$\begin{align*}C_{\varphi} = \int_{-\infty}^{\infty}\overline{\varphi}\Big(\big(1+ 2t^2/\pi \big)^{-1/2}\Big)\, dt\leqslant \int_{-\infty}^{\infty}\big(1+ 2 t^2/\pi \big)^{-p/2}\,dt < \infty\,. \end{align*}$$

Let $\{\varepsilon _k\}$ and $\{\delta _k\}$ be nonnegative sequences satisfying conditions $\varepsilon _k\downarrow 0$ and $\delta _k\downarrow 0$ , when $k\to \infty $ , and additionally

$$ \begin{align*} \frac{1-r^2_k}{1-2r\cos\theta+r_k^2}\leqslant 1 \end{align*} $$

for $|\theta |\geqslant \varepsilon _k$ and $r\geqslant r_k$ , $k\in \mathbb {N}$ . For a given zero neighborhood,

$$\begin{align*}V: = \big\{g\in N_\varphi; \, \rho_\varphi(g,0)<\eta\big\} \end{align*}$$

in $N_\varphi $ choose $m\in \mathbb {N}$ , such that

$$ \begin{align*} \varphi(\log(1+\delta_m))+\frac{\gamma}{2\pi}\varphi(\log 2)\varepsilon_m+\frac{\gamma^2}{4\pi} C_{\varphi}\,(1-r_m)\varphi\Big(\frac{\beta_m}{1-r_m}\Big)<\eta^{1/\delta}\,, \end{align*} $$

where $\gamma $ is a $\Delta _2$ constant for $\varphi $ and $\delta =\frac {\log 2}{\log \gamma }$ . Furthermore, suppose that $\alpha _0$ , $0<\alpha _0<1$ , is chosen so that

$$\begin{align*}\alpha_0\exp\frac{1+r_m}{1-r_m }\leqslant \delta_{m}\,, \end{align*}$$

and note that, in particular, $\alpha _0 e < \delta _m$ .

Observe that for all $\theta \in [0,2\pi ]$ and each $k\in \mathbb {N}$ , we have

$$\begin{align*}f_{k}^{*}(e^{i\theta})=\lim_{r \to 1-}\exp\bigg(\beta_k\frac{1+r_kre^{i\theta}}{1-r_kre^{i\theta}}\bigg) =\exp\bigg(\beta_k\frac{1+r_ke^{i\theta}}{1-r_ke^{i\theta}}\bigg)\,, \end{align*}$$

and hence (by $\beta _k<1$ )

$$ \begin{align*} |f_{k}^{*}(e^{i\theta})|& = \exp \bigg(\beta_k\text{Re}\,\frac{1+r_ke^{i\theta}}{1-r_ke^{i\theta}}\bigg)\\ &= \exp\bigg(\beta_k\frac{1-r_k^2}{1-2r_k\cos \theta + r_k^2}\bigg) \leqslant \exp\bigg(\beta_k\frac{1+r_k}{1-r_k}\bigg)\,. \end{align*} $$

Because the function $[0,1)\ni x\mapsto \frac {1+x}{1-x}$ is increasing and $r_n\uparrow 1$ , we get that

$$\begin{align*}|\alpha_0 f_k^*(e^{i\theta})|\leqslant |\alpha_0|\exp\frac{1+r_k}{1-r_k}\leqslant\delta_{m}, \quad\, k \leqslant m\,. \end{align*}$$

In consequence, the above estimates yields that, for all $|\alpha |\leqslant \alpha _0$ and each $k\leqslant m$ , we have

$$ \begin{align*} \|\alpha f_k\|_\varphi=\frac{1}{2\pi}\int_{-\pi}^{\pi}\varphi(\log(1+|\alpha f_k^*(e^{i\theta})|))\,d\theta \leqslant \varphi(\log(1+\delta_m))<\eta^{1/\delta}\,. \end{align*} $$

This shows that $\rho _\varphi (\alpha f_k,0) = \|\alpha f_k\|_\varphi ^\delta <\eta $ . We conclude that $\alpha f_k\in V$ for each $k\leqslant m$ and all $|\alpha |\leqslant \alpha _0$ .

We can now proceed to the case $k>m$ . For all $|\alpha |\leqslant \alpha _0$ , we have

$$ \begin{align*} \|\alpha f_k\|_\varphi & =\frac{1}{2\pi}\int_{-\pi}^{\pi}\varphi(\log(1+|\alpha f_k^*(e^{i\theta})|))\,d\theta =\frac{1}{2\pi}\int_{|\theta|\geqslant \varepsilon_k}\varphi(\log(1+|\alpha f_k^*(e^{i\theta})|))\,d\theta \\[3pt] & \quad +\frac{1}{2\pi}\int_{|\theta|< \varepsilon_k} \varphi(\log(1+|\alpha f_k^*(e^{i\theta})|))\,d\theta\,. \end{align*} $$

From the above shown inequality, one has

$$ \begin{align*} \frac{1}{2\pi}\int_{|\theta|\geqslant \varepsilon_k} \varphi(\log(1+|\alpha f_k^*(e^{i\theta})|))\,d\theta\leqslant \varphi(\log(1+\alpha_0 e)\leqslant\varphi(\log(1+\delta_m)). \end{align*} $$

Clearly,

$$ \begin{align*} \frac{1}{2\pi}\int_{|\theta|< \varepsilon_k}\varphi(\log(1 & + |\alpha f_k^*(e^{i\theta})|))\,d\theta \\ &\leqslant \frac{\gamma}{2\pi}\varphi(\log 2)\varepsilon_k +\frac{\gamma}{4\pi}\int_{|\theta|< \varepsilon_k}\varphi(\log^{+}|\alpha f_{k}^{*}(e^{i\theta})|)\,d\theta\,, \end{align*} $$

where $\gamma $ comes from the $\Delta _2$ -condition for $\varphi $ .

To estimate the second term in the sum above note that $|\sin (\theta /2)| \geqslant |\theta |/\pi $ for all $\theta \in [-\pi , \pi ]$ and $r_k/(1-r_k)^2 \geqslant 2$ for each $r_k \in [1/2, 1)$ . Hence

$$ \begin{align*} 1- 2 r_k \cos \theta + r_k^2 & = (1- r_k)^2 + 2 r_k(1 - \cos \theta) \\[3pt] & = (1 - r_k)^2 + 4r_k \sin^2 \Big(\frac{\theta}{2}\Big) \\[3pt] & \geqslant (1- r_k)^2\,\bigg(1 + \frac{2\theta^2}{\pi(1 - r_k)^2}\bigg)\,. \end{align*} $$

Combining this estimate with

$$ \begin{align*} |f_{k}^{*}(e^{i\theta})| \leqslant \exp\bigg(\frac{2\beta_k}{|1- r_k e^{i\theta}|}\bigg) = \exp\Biggl(\frac{2\beta_k}{\sqrt{1-2r_k\cos \theta + r_k^2}}\Biggr)\,, \end{align*} $$

we obtain by $|\alpha | \leqslant \alpha _0 <1$ and $\varphi \in (\Delta _2)$ with constant $\gamma $ ,

$$\begin{align*}\frac{\gamma}{4\pi}\int_{|\theta|< \varepsilon_k}\varphi(\log^+|\alpha f_{k}^{*}(e^{i\theta})|)d\theta \leqslant \frac{\gamma^2}{4\pi}\int_{-\pi}^{\pi}\varphi\Biggl(\frac{\beta_k}{(1-r_k)\sqrt{1+\frac{2\theta^2}{\pi(1-r_k)^2}}}\Biggr)\,d\theta\,. \end{align*}$$

Since $\varphi (st)\leqslant \varphi (s)\overline {\varphi }(t)$ for all $s,t>0$ , so substituting $t=\frac {\theta }{1-r_k}$ yields

$$ \begin{align*} \frac{\gamma}{4\pi}\int_{|\theta|< \varepsilon_k}&\varphi(\log^{+} |\alpha f_k^{*}(e^{i\theta})|)\,d\theta \\ & \leqslant\frac{\gamma^2}{4\pi}(1-r_k)\varphi\Big(\frac{\beta_k}{1-r_k}\Big){\int_{-\infty}^{\infty}\overline{\varphi}\Big(\big(1+ 2t^2/\pi \big)^{-1/2}\Big)\,dt}\,. \end{align*} $$

In consequence, for all $k>m$ and $|\alpha |\leqslant \alpha _0$ ,

$$ \begin{align*} &\frac{1}{2\pi}\int_{|\theta|< \varepsilon_k}\varphi(\log(1+|\alpha f_k^*(e^{i\theta}))|)\,d\theta \\ &\leqslant \frac{\gamma}{2\pi}\varphi(\log 2)\varepsilon_k + \frac{\gamma^2}{4\pi}(1-r_k)\varphi\Big(\frac{\beta_k}{1-r_k}\Big){\int_{-\infty}^{\infty}\overline{\varphi}\Big(\big(1+ 2t^2/\pi \big)^{-1/2}\Big)dt}\\&\leqslant \frac{\gamma}{2\pi}\varphi(\log 2)\varepsilon_k + C_{\varphi}\, \frac{\gamma^2}{4\pi}(1-r_k)\varphi\Big(\frac{\beta_k}{1-r_k}\Big)\,. \end{align*} $$

Combining with the estimates above, we obtain the desired statement.

From now on, for a given $\varphi \in \Phi _2^{sc}$ , we define $h_\varphi \colon (0, \infty ) \to (0, \infty )$ by

$$\begin{align*}h_{\varphi}(y):=\varphi^{-1}\Big(\frac{1}{y}\Big), \quad\, y>0, \end{align*}$$

and for given $t>0$ let $\widetilde {y}(t)$ be a solution of the equation $h_{\varphi }(y)=ty$ , with variable being $y>0$ . Furthermore, let $\omega _{\varphi }(t)=\omega (t):=h_\varphi (\widetilde {y}(t))$ for all $t>0$ . Throughout the rest of the paper, the function $\omega $ will be called associated with $\varphi $ .

In what follows, we will need the following properties of $\varphi \in \Phi _2^{sc}$ . For the sake of completeness, we include some technical details.

Remarks. (1) Since $\varphi ^{-1}$ is a nondecreasing function, $h_\varphi $ is nonincreasing. Clearly, $\lim _{y\to \infty }h(y)=0$ and $\lim _{y\to 0+}h(y)=\infty $ . Hence $\lim _{t\to \infty }\widetilde {y}(t)=0$ .

(2) From definition of $\omega _\varphi $ , we get that $\omega (t)=\varphi ^{-1}\big (\frac {1}{\widetilde {y}(t)}\big )$ , for all $t>0$ . Since $\varphi ^{-1}\big (\frac {1}{\widetilde {y}(t)}\big )=h(\widetilde {y}(t))=t\widetilde {y}(t)$ ,

$$\begin{align*}\varphi(\omega(t))\omega(t)=\frac{1}{\widetilde{y}(t)}\varphi^{-1}\Big(\frac{1}{\widetilde{y}(t)}\Big)=t, \quad\, t>0\,. \end{align*}$$

In consequence, $\varphi (t)=\frac {\omega ^{-1}(t)}{t}$ for all $t> 0$ . Clearly, $\varphi \in \Phi _2^{sc}$ implies that $\omega $ is continuous, nondecreasing, and nonnegative function. It is easy to check that $\omega ^{-1}\in (\Delta _2)$ and $t^2=o(\omega ^{-1}(t))$ as $t\to \infty $ .

(3) If $\varphi \in \Phi _2^{sc}$ and $\varphi (1)\neq 1$ , then the function $\phi \in \Phi _2^{sc}$ given by $\phi (t):=\frac {\varphi (t)}{\varphi (1)}$ for all $t\geqslant 0$ , satisfies the condition $\phi (1)=1$ . Obviously, $N_\phi =N_\varphi $ and the topologies on both spaces are equivalent. Since $\phi ^{-1}(t)=\varphi ^{-1}(t\varphi (1))$ , for all $t\geqslant 0$ , we have

$$\begin{align*}h_\phi(y)=\phi^{-1}\Big(\frac{1}{y}\Big)=\varphi^{-1}\Big(\frac{\varphi(1)}{y}\Big) =h_{\varphi}\Big(\frac{y}{\varphi(1)}\Big), \quad\, y>0\,. \end{align*}$$

It follows that

$$\begin{align*}\min\big\{1,\varphi(1)\big\}h_\varphi(y)\leqslant h_\phi(y)\leqslant \max\big\{1,\varphi(1)\big\}h_\varphi(y), \quad\, y>0, \end{align*}$$

and the associated functions are equivalent.

We also will need the following lemma.

Proof. (a). Since $\varphi (t)=\frac {\omega ^{-1}(t)}{t}$ for $t> 0$ , $\varphi (x)=o(\omega ^{-1}(x))$ as $x\to \infty $ . We claim that $\varphi ^{-1}(x\varphi (x))/x\to \infty $ as $x\to \infty $ . If we suppose the opposite, then there would exist constant $C>0$ and sequence $\{x_n\}$ such that $x_n\to \infty $ as $n\to \infty $ and, for large enough n,

$$\begin{align*}\varphi^{-1}(x_n\varphi(x_n)){x_n}\leqslant C x_{n}\,. \end{align*}$$

Since $\varphi \in (\Delta _2)$ , there exists a constant $\gamma>0$ and a positive integer k such that $C \leqslant 2^k$ and $\varphi (C t)\leqslant \gamma ^k \varphi (t)$ for all $t>0$ . Hence $x_n\varphi (x_n)\leqslant \gamma ^k\varphi (x_n)$ yields that $x_n\leqslant \gamma ^k$ for each $n\in \mathbb {N}$ . This contradiction shows that the claim is true. Combining this fact with $\omega ^{-1}(x)=x\varphi (x)\to \infty $ as $x\to \infty $ , we get that

$$\begin{align*} \lim_{x\to \infty} \frac{\varphi^{-1}(x)}{\omega(x)} = \lim_{x\to \infty} \frac{\varphi^{-1}(\omega^{-1}(x))}{x} =\lim_{x\to \infty} \frac{\varphi^{-1}(x\varphi(x))}{x} = \infty\,. \end{align*}$$

In consequence, $\omega (x)=o(\varphi ^{-1}(x))$ as $x\to \infty $ , which proves (a).

(b). Assume by a contradiction that there exists a constant $\gamma> 0$ , such that for every $x_0>0,$ there exists $x>x_0$ with $\omega (x)\leqslant \gamma \log x$ . We claim that is not true for large enough $x>0$ . Otherwise, we would get that, for large enough $x> e$ ,

$$\begin{align*}x = \varphi(\omega(x)) \omega(x) \leqslant \gamma \varphi(\gamma \log x)\log x\,. \end{align*}$$

Since $\varphi \in (\Delta _2)$ , it follows that for large enough $x>e$ , we get that $x \leqslant c \varphi (\log x)\log x$ for some $c>0$ . It is easy to see that $\varphi \in (\Delta _2)$ yields that there are constants $C>0$ and $\beta \geqslant 1$ such that $\varphi (x) \leqslant C x^\beta $ for all $x>0$ . Combining these estimates yields that $\sup_{x>e} x/\log ^{\beta +1} x <\infty $ . Since $x/\log ^{\beta +1} x \to \infty $ as $x\to \infty $ , we get a contradiction. This completes the proof of (b).

To finish observe that, from (a) and (b), we get that $\log x=o(\varphi ^{-1}(x))$ as $x\to \infty $ , which is equivalent to

$$\begin{align*}\log\frac{1}{x}=o(\log h(x)), \quad\, x\to 0+, \end{align*}$$

and so the statement (c) follows.

In what follows, we will use the following lemma proved by Yanagihara [Reference Yanagihara21].

Lemma 3.4. Let $0<c\leqslant 1$ , and let $\{a_n(c)\}$ be defined by

$$\begin{align*}\exp\Big(\frac{c}{2}\frac{1+z}{1-z}\Big)=\sum_{n=0}^\infty a_n(c)z^n, \quad\, z \in \mathbb{D}\,. \end{align*}$$

Then $a_n(c)$ satisfies the following asymptotic formula:

$$\begin{align*}\log|a_n(c)|\geqslant\sqrt{cn}+O(\log n) + O(\log c), \quad\, n\in\mathbb{N}\,. \end{align*}$$

The following corollary is a immediate consequence of Lemma 3.4.

Corollary 3.5. If $\{c_k\}\subset (0,1]$ is a sequence such that $\lim _{k\to \infty } c_kk=\infty $ , then the Taylor coefficients of the function given by the formula

$$\begin{align*}\exp\Big(\frac{c_k}{2}\frac{1+z}{1-z}\Big)=\sum_{n=0}^\infty a_n(c_k)z^n, \quad\, z \in \mathbb{D}\,, \end{align*}$$

satisfy the condition

$$\begin{align*}\log|a_k(c_k)|\geqslant\sqrt{c_kk}\,(1+o(1)),\quad\, k\in\mathbb{N}\,. \end{align*}$$

The following lemma is a generalization of Yanagihara’s Lemma $2.1$ from [Reference Yanagihara21], proved there for the special case $\{x_k\}=\{\sqrt {k}\}$ .

Lemma 3.6. Let $\{x_k\}\in \boldsymbol {\omega }(\mathbb {N}_0)$ be a positive and increasing sequence such that $x_k \to \infty $ as $k\to \infty $ . Suppose that the complex sequence $\{\lambda _k\}$ satisfies the condition $\lambda _k=O(\exp (-c_kx_k))$ for every positive sequence $\{c_k\}\in \boldsymbol {\omega }$ , such that, $c_k\downarrow 0$ . Then, there exists $c>0$ such that

$$\begin{align*}\lambda_k = O(\exp(-cx_k))\,. \end{align*}$$

Proof. Clearly, $\lambda _k=O(\exp (-c_kx_k))$ yields that $\lambda _k\to 0$ and $\limsup _{k\to \infty }\frac {\log |\lambda _k|}{x_k} \leqslant 0$ . It is enough to show that $\limsup _{k\to \infty }\frac {\log |\lambda _k|}{x_k}<0$ . Suppose that $\limsup _{k\to \infty }\frac {\log |\lambda _k|}{x_k}=0$ . Then, by passing to a subsequence, if necessary, we can assume that $b_k \downarrow 0$ as $k\to \infty $ , where

$$\begin{align*}b_k:=-\frac{1}{x_k}\log|\lambda_k| \downarrow 0, \quad\, k\in \mathbb{N}_0\,. \end{align*}$$

From our hypothesis, it follows that for every positive sequence $c=\{c_k\}\in \boldsymbol {\omega }$ with $c_k\downarrow 0$ , there exists a constant $A(c)>0$ such that $|\lambda _k|\leqslant A(c)\exp (-c_kx_k)$ and so, $c_k x_k + \log |\lambda _k|\leqslant \log A(c)$ for each $k\in {\mathbb {N}_0}$ . From the definition of $\{b_k\}$ , we have $\log |\lambda _k|=-b_kx_k$ . Thus, we conclude that

$$\begin{align*}(c_k-b_k)x_k \leqslant \log A(c), \quad\, k\geqslant 0\,. \end{align*}$$

Now, observe that the sequence $c'=\{c_{k}'\}$ given by $c_{k}'=\max \big \{2b_k,\frac {1}{\sqrt {x_k}}\big \}$ , for each $k\geqslant 0$ , is decreasing and $c_k'\downarrow 0$ as $k\to \infty $ (by $b_k\downarrow 0$ and $x_k\uparrow \infty $ as $k\to \infty $ ). Combining the above facts, now gives that there exists a constant $A(c')>0$ such that, for each $k\geqslant 0$ , we have $(c_k'-b_k)x_k\leqslant \log A(c')$ . Thus, we conclude that

$$\begin{align*}\sqrt{x_k}\leqslant 2 \log A(c'), \quad\, k\geqslant 0\,. \end{align*}$$

Since $x_k\to \infty $ as $k\to \infty $ , we get a contradiction. This gives the desired conclusion.

We are now ready to prove the necessary condition for a sequence $\{\lambda _k\}$ to be a multiplier from $N_\varphi $ to $H^p$ .

Theorem 3.7. Let $\varphi \in \Phi _2^{sc}$ be a function satisfying condition $p(\varphi )>1$ , and let $\omega $ be a function associated with $\varphi $ . If $\{\lambda _k\}\in \boldsymbol {\omega }(\mathbb {N}_0)$ is a multiplier from $N_\varphi $ to $H^p$ for some $p\in (0,\infty ]$ , then for some constant $c>0$ , we have

$$ \begin{align*} \lambda_k=O(\exp[-c\,\omega(k)])\,. \end{align*} $$

Proof. From Remark (3), without loss of generality, we can assume $\varphi (1)=1$ . By Lemma 3.6, it suffices to show that, for any positive sequence $\{c_k\}\in \boldsymbol {\omega }$ such that $c_k\downarrow 0$ , we have

$$ \begin{align*} \lambda_k=O(\exp[-c_k\omega(k)])\,. \end{align*} $$

Observe that, taking a sequence $\{c_k'\}\in \boldsymbol {\omega }$ of the form

$$\begin{align*}c_k'=\min\Big\{\frac{1}{2}, \max\Big\{\frac{1}{\omega(k)},c_n\Big\}\Big\}, \quad\, k\in\mathbb{N}\,, \end{align*}$$

we have $c_k'\downarrow 0$ . Furthermore, if we show that

$$\begin{align*}\lambda_k=O(\exp[-c_k'\omega(k)])\,, \end{align*}$$

then, for a given sequence $\{c_k\}$ as above, the formula $\lambda _k=O(\exp [-c_k\omega (k)])$ is satisfied. This means that without loss of generality, we can assume that

$$\begin{align*}\frac{1}{\omega(k)}\leqslant c_k\leqslant \frac{1}{2}, \quad\, k\geqslant k_{0}\,, \end{align*}$$

where $k_0$ is the smallest nonnegative integer such that $\omega (k_0)\geqslant 2$ . Choose one of such sequences $\{c_k\}$ and let $\{\beta _k\}$ be the sequence given by

$$\begin{align*}\beta_k :=c_k(1-r_k)\varphi^{-1}\Big(\frac{1}{1-r_k}\Big), \quad\, k\in{\mathbb{N}_0}\,, \end{align*}$$

where $\{r_k\} \subset (0, 1)$ is a sequence such that $r_k\uparrow 1$ as $k\to \infty $ . The concavity of $\varphi ^{-1}$ implies that the function $t\mapsto \varphi ^{-1}(t)/t$ is nonincreasing on $(0,\infty )$ . Hence $(1-r_k)\varphi ^{-1}\big (\frac {1}{1-r_k}\big )\leqslant \varphi ^{-1}(1)=1$ and so $\beta _k\leqslant 1/2$ for each $k\in {\mathbb {N}_0}$ . From the convexity of $\varphi $ , we get

$$\begin{align*}(1-r_k)\varphi\Big(\frac{\beta_k}{1-r_k}\Big) =(1-r_k)\varphi\Big(c_k\varphi^{-1}\Big(\frac{1}{1-r_k}\Big)\Big)\leqslant c_k\to 0\,, \end{align*}$$

as $k\to \infty $ . In consequence, the sequence $\{\beta _k\}$ satisfies all of the assumptions of Lemma 3.2. Now, observe that the sequence $\{f_k\}$ (generated by $\{\beta _k\}$ given above) defined in Lemma 3.2 is topologically bounded in $N_\varphi $ . Thus, from Lemma 3.1, we conclude that the sequence $\{\Lambda f_k\}$ is bounded in $H^p$ . Furthermore,

$$\begin{align*}\Lambda f_k(z)=\sum_{n=0}^{\infty}\lambda_n\widehat{f}_k(n)r_k^n z^n, \quad\, z \in \mathbb{D}\,, \end{align*}$$

where $\widehat {f}_k(n)$ is the nth Taylor coefficient of $\exp \big (\beta _k\frac {1+z}{1-z}\big )$ , $z\in \mathbb {D}$ . Since $\{\Lambda f_k\}$ is a bounded sequence in $H^p$ there exists a constant K, such that, for all $k\in {\mathbb {N}_0}$ , the inequality $\|\Lambda f_k\|_{H^p} \leqslant K$ holds. From [Reference Duren1, Th. 6.4], it follows that there exists $C_p>0$ such that, for each $k\in \mathbb {N}$ and $n\in {\mathbb {N}_0}$ , we have

(**) $$\begin{align} |\lambda_nf_k(n)r_k^n|\leqslant \begin{cases} K C_{p} n^{-1+\frac{1}{p}}, & \text{if} \quad\, 0<p<1\,,\\ K C_{p}, & \text{if}\quad\, 1\leqslant p\leqslant\infty\,. \end{cases} \end{align} $$

On the other hand, using notation from Lemma 3.4, we get that

$$\begin{align*}f_k(z)=\sum_{n=0}^\infty a_n(2\beta_k)(r_kz)^n, \quad\, z \in \mathbb{D}\,. \end{align*}$$

Now, observe that $\omega (1)=1$ (because $\varphi (t)=\omega ^{-1}(t)/t$ for all $t>0$ ). From the assumption $c_k\omega (k)\geqslant 1$ for $k\geqslant k_0$ , we obtain

$$ \begin{align*} \frac{1}{k}\leqslant c_k\widetilde{y}(k)=\frac{c_k\omega(k)}{k}\leqslant c_k\leqslant\frac{1}{2}, \quad k\geqslant k_0\,. \end{align*} $$

Since $c_k\downarrow 0$ and $\widetilde {y}(k)=\frac {\omega (k)}{k}\downarrow 0$ as $k\to \infty $ , the sequence $\{r_k\}$ given by $r_k=1-c_k\widetilde {y}(k)$ for each $k\in {\mathbb {N}_0}$ , satisfies $r_k\in (0,1)$ and $r_k\uparrow 1$ as $k\to \infty $ . The function $t\mapsto \varphi ^{-1}(t)/t$ is nonincreasing, which gives that the mapping $t\mapsto x\varphi ^{-1}(1/x)$ is nondecreasing on $(0,\infty )$ . Using this observation and the above equality, we get that

$$\begin{align*}\beta_k\omega(k)=c_k\,\omega(k)\,\Big[c_k\widetilde{y}(k)\varphi^{-1}\Big(\frac{1}{c_k\widetilde{y}(k)}\Big)\Big] \geqslant \frac{\varphi^{-1}(k)}{k}\,, \end{align*}$$

which is

$$\begin{align*}k\beta_k\geqslant\frac{\varphi^{-1}(k)}{\omega(k)}, \quad k\geqslant k_{0}\,. \end{align*}$$

From Lemma 3.3(a), we know that $2k\beta _k\to \infty $ as $k\to \infty $ . Hence, from Corollary 3.5, it follows that

$$\begin{align*}|a_k(2\beta_k)|\geqslant\exp\big[\sqrt{2\beta_kk}(1+o(1))\big]\,. \end{align*}$$

Furthermore, for each $k\in {\mathbb {N}_0}$ , we have (by $k\widetilde {y}(k)=\omega (k)$ for $k\in {\mathbb {N}_0}$ )

$$ \begin{align*} \beta_{k}\,k& = c_{k}^2\,k\,\widetilde{y}(k) \varphi^{-1}\Big(\frac{1}{c_k\widetilde{y}(k)}\Big) = c_{k}^2\,\omega(k) \varphi^{-1}\Big(\frac{1}{c_k\widetilde{y}(k)}\Big)\\ &\geqslant c_{k}^2\,\omega(k) \varphi^{-1}\Big(\frac{1}{\widetilde{y}(k)}\Big)=c_k^2\omega^2(k)\,. \end{align*} $$

Thus

$$\begin{align*}|a_k(2\beta_k)|\geqslant\exp\big[\sqrt{2}c_k\omega(k)(1+o(1))\big]. \end{align*}$$

Since $\lim _{x\to 0+}\big (1-x\big )^{-1/x}=e$ and $c_k\widetilde {y}(k)\to 0$ as $k\to \infty $ , we obtain

$$\begin{align*}(1-c_k\widetilde{y}(k))^{-k}=\Big[(1-c_k\widetilde{y}(k))^{-\frac{1}{c_k\widetilde{y}(k)}}\Big]^{kc_k\widetilde{y}(k)} =\exp[c_k\omega(k)(1+o(1))]\,. \end{align*}$$

Now, we apply $(**)$ for $1\leqslant p\leqslant \infty $ to get that, for some $\gamma>0$ and for each $k\in \mathbb {N}$ , the following estimates hold:

$$ \begin{align*} |\lambda_k| & \leqslant \gamma (1-c_k\widetilde{y}(k))^{-k}\exp\big[-\sqrt{2\beta_kk}(1+o(1))\big] \\ & \leqslant \gamma \exp[c_k\omega(k){(1+o(1))}]\exp[-\sqrt{2}c_k\omega(k)(1+o(1))] \\ & \leqslant \gamma \exp[-(\sqrt{2}-1)c_k\omega(k)(1+o(1))]\,. \end{align*} $$

Clearly, this gives the desired conclusion in the case $1\leqslant p<\infty $ . When $0<p<1$ , we have

$$\begin{align*}|\lambda_k|k^{\frac{1}{p}-1}\leqslant C\exp[-(\sqrt{2}-1)c_k\omega(k)(1+o(1))]\,. \end{align*}$$

Using Lemma 3.6, we can find constants $\alpha>0$ and $\gamma _p>0$

$$\begin{align*}|\lambda_k|k^{1-\frac{1}{p}}\leqslant \gamma_p \exp(-\alpha \omega(k)), \quad\, k \in \mathbb{N}\,, \end{align*}$$

that is,

$$\begin{align*}|\lambda_k|\leqslant \gamma_p \exp\Big(\Big(\frac{1}{p}-1\Big)\log k\Big)\exp(-\alpha \omega(k)), \quad\, k \in \mathbb{N}\,. \end{align*}$$

From Lemma 3.3(b), it follows that

$$\begin{align*}\lim_{k\to\infty}\frac{\log k}{\omega(k)}=0\,. \end{align*}$$

Thus, there exists $k_0\in \mathbb {N}$ , such that

$$\begin{align*}\Big(\frac{1}{p}-1\Big)\log k\leqslant \frac{a}{2}\omega(k),\quad k\geqslant k_{0}\,. \end{align*}$$

Combining the above, we conclude that there exists $\widetilde {C}>0$ such that for $c=\alpha /2$ , we have

$$\begin{align*}|\lambda_k|\leqslant \widetilde{C}\exp(-c\,\omega(k)),\quad k\in{\mathbb{N}_0}\,. \end{align*}$$

This completes the proof.

Now, we proceed to the investigation of sufficient conditions for the multipliers. We start with the definition of $\mathcal {H}(\omega )$ spaces given by Zayed in [Reference Zayed23]. These are the spaces of analytic functions generated by some function $\omega $ . The original assumptions on $\omega $ come from theory of Beurling distributions, but we will present a weaker set of conditions for which the results in [Reference Zayed23] are true. Let $\omega \colon \mathbb {R}_{+} \to \mathbb {R}_{+}$ be a function satisfying the following conditions:

1. (1) $\omega $ is concave and continuously differentiable function with $\omega (0)=0$ .

2. (2) $\omega (x)=o(x)$ as $x\to \infty $ .

3. (3) $\log (x)=o(\omega (x))$ as $x\to \infty $ .

4. (4) $\omega (ax)\leqslant \delta (a)\omega (x)$ , where $a\geqslant 1$ and $\delta (a)=o(a)$ as $a\to \infty $ .

We denote the set of all functions satisfying (1)–(4) by $\boldsymbol {\Omega }$ . Observe, for every $\omega \in \boldsymbol {\Omega }$ , the function $x\mapsto \omega (x)/x$ is decreasing on $(0, \infty )$ . Thus, for every $r\in [0, 1),$ there exists a unique solution x denoted by $\widetilde {x}(1-r)$ of the equation $\omega (x)/x=1-r$ . In what follows, we let

$$\begin{align*}\Omega\Big(\frac{1}{1-r}\Big):=\omega(\widetilde{x}(1-r)), \quad\, r\in [0, 1)\,. \end{align*}$$

For any $\omega \in \boldsymbol {\Omega }$ , we define $H(\omega )$ to be the space of all $f\in H(\mathbb {D})$ , which for any $c>0$ satisfies

$$ \begin{align*} \|f\|_{c}: = \int_{0}^{1} M_\infty(r,f) \exp\Big[-c\,\Omega\Big(\frac{1}{1-r}\Big)\Big]\,dr<\infty\,. \end{align*} $$

Clearly, for any $c>0$ , $\|\cdot \|_c$ defines a norm in $H(\omega )$ . Furthermore, it is obvious that, any $\varphi \in \Phi _2^{sc}$ replacing $\Omega $ with $\varphi ^{-1}(x)$ gives the space $F^{int}_\varphi $ . Of course, not every function $\varphi \in \Phi _2^{sc}$ can be represented as $\Omega $ for some $\omega $ satisfying conditions (1)–(4). In view of this observation, in what follows the following conditions on $\omega $ will be imposed.

1. (i) $\omega $ is concave and continuously differentiable function with $\omega (0)=0$ .

2. (ii) $\omega ^{-1}$ is $2$ –strongly convex, that is, $x^2=o(\omega ^{-1}(x))$ as $x\to \infty $ .

3. (iii) $\omega ^{-1}\in (\Delta _2)$ .

4. (iv) $\omega (ax)\leqslant \delta (a)\omega (x)$ , where $a\geqslant 1$ and $\delta (a)=o(a)$ as $a\to \infty $ .

If the function $\omega $ satisfies all of the above conditions, we write $\omega \in {\boldsymbol {\widetilde {\Omega }}}$ . It can be seen that, if $\omega \in \boldsymbol {\widetilde {\Omega }}$ , then $\omega \in \boldsymbol {\Omega }$ , so every function $\omega \in \boldsymbol {\widetilde {\Omega }}$ generates some space $H(\omega )$ . Now, for any $\omega \in {\boldsymbol {\widetilde {\Omega }}}$ , define $\varphi _\omega (x)\sim \frac {\omega ^{-1}(x)}{x}$ . It is equivalent to $\varphi _{\omega }^{-1}(\omega _{*}(x))\sim \omega (x)$ , where $\omega _{*}(x):=\frac {x}{\omega (x)}$ , for all $x>0$ . Hence,

$$\begin{align*}\varphi_\omega^{-1}\Big(\frac{1}{1-r}\Big)\sim\omega\Big(\omega_{*}^{-1}\Big(\frac{1}{1-r}\Big)\Big), \quad\, r\in[0,1)\,. \end{align*}$$

Since $\omega _{*}^{-1}\big (\frac {1}{1-r}\big )$ is the solution of equation $\frac {\omega (x)}{x}=1-r$ with respect to x, so

$$\begin{align*}\varphi_\omega^{-1}(x) \sim \Omega(x)\,. \end{align*}$$

It is also easy to see that, if $\omega \in {\boldsymbol {\widetilde {\Omega }}}$ , then $\varphi _\omega \in \Phi _2^{sc}$ . We denote the set of all functions $\varphi _\omega $ , where $\omega \in \boldsymbol {\widetilde {\Omega }}$ , by $\Phi _2^{sc}(\boldsymbol {\widetilde {\Omega }})$ . From previous observations, we get the following corollary.

The equality $F^{int}_{\varphi }=H(\omega )$ is an important step to give sufficient condition for the multipliers. In case of $F^{int}_{\varphi }$ , $\varphi \in \Phi _2^{sc}$ , there is no equivalent way to define them using growth of Taylor coefficients. But, when we impose some stricter conditions on $\varphi $ , that is, $\varphi \in \Phi _2^{sc}(\boldsymbol {\widetilde {\Omega }})$ ( $F^{int}_{\varphi }=H(\omega )$ ), it is possible to do that, as the following theorem shows. Its proof can be found in [Reference Zayed23].

Theorem 3.9. Let $f\in H(\mathbb {D})$ with $f(z)=\sum _{n=0}^{\infty }\widehat {f}(n)z^n$ for all $z\in \mathbb {D}$ . Then $f\in H(\omega )$ if and only if,

$$ \begin{align*} \widehat{f}(n)=O(\exp[o(\omega(n))]), \quad\, n\to\infty\,. \end{align*} $$

Now, we are ready to show that the condition $\lambda _k=O(\exp [-c\,\omega (k)])$ is sufficient for a sequence to be a multiplier from $N_\varphi $ to $H^p$ , where $\varphi \in \Phi _2^{sc}(\boldsymbol {\widetilde {\Omega }})$ .

Proof. Suppose that $\lambda _n=O(\exp (-c\,\omega (n)))$ for some $c>0$ . If $f\in N_\varphi $ with $\varphi \in \Phi _2^{sc}(\boldsymbol {\widetilde {\Omega }})$ , then from Theorems 2.3 and 3.9, we get that there exists $C_1>0$ , such that

$$\begin{align*}|\widehat{f}(n)| \leqslant C_1\exp[c_n(\omega(n))], \quad\, n\geqslant 0\,, \end{align*}$$

and $c_n\downarrow 0$ as $n\to \infty $ . Let $n_0$ be such that $c_k<c/2$ for each $n\geqslant n_0$ . Then, we have

$$\begin{align*}|\lambda_n\widehat{f}(n)|\leqslant C_2\exp[-c\,\omega(k)/2], \quad\, n\geqslant n_0 \end{align*}$$

for some constant $C_2>0$ . Since $\log (x)=o(\omega (x))$ as $x\to \infty $ ,

$$\begin{align*}\sum_{n=0}^{\infty}\exp[-c\,\omega(k)/2]<\infty\,. \end{align*}$$

This means that $\Lambda f$ is a continuous function on $\overline {\mathbb {D}}$ , and hence, $\Lambda f\in H^p$ for every $p>0$ .

From Theorems 3.7 and 3.10, we obtain the following corollary.

4 Continuous linear functionals on $N_{\varphi }$

We now proceed to the problem of finding the topological dual for the spaces $N_\varphi $ . We will start with a necessary condition true for all $\varphi \in \Phi _2^{sc}(\boldsymbol {\widetilde {\Omega }})$ . To prove the sufficient condition, we will have to extend slightly the set of assumptions for $\varphi $ , but as it will be seen, we will do it in rather natural way. We will need the following lemma.

Theorem 4.2. Let $\varphi \in \Phi _2^{sc}(\boldsymbol {\widetilde {\Omega }})$ satisfy $p(\varphi )>1$ . For any continuous linear functional $L \in (N_\varphi )^*$ , there exists a unique function $g\in A(\mathbb {D})$ with $g(z)=\sum _{n=0}^\infty b_nz^n$ for all $z\in \mathbb {D}$ , for which there exists $c>0$ such that $b_n=O(\exp [-c\,\omega (n)])$ and

$$ \begin{align*} L(f)=\lim_{r\to 1-}\frac{1}{2\pi}\int_0^{2\pi}f(re^{i\theta})g(e^{-i\theta})\,d\theta =\sum_{n=0}^\infty \widehat{f}(n)b_n, \quad\, f \in N_{\varphi}\,, \end{align*} $$

where $f(z)=\sum _{n=0}^\infty \widehat {f}(n)z^n$ for all $z\in \mathbb {D}$ .

Proof. Let $L \in (N_\varphi )^{*}$ and $b_n=L(z^n)$ for each $n\in {\mathbb {N}_0}$ . Since $\{z^{n}; \, n\in {\mathbb {N}_0}\}$ is a bounded set in $N_\varphi $ , it follows that $\{b_n\}$ is a bounded sequence. Hence, the function g given by $g(z):=\sum _{n=0}^\infty b_nz^n$ for all $z\in \mathbb {D}$ is well-defined and analytic in $\mathbb {D}$ . Let $h(\xi ) := L(f_\xi )$ , where $f_\xi (z)=f(\xi z)$ for all $\xi , z \in \mathbb {D}$ . It can be seen that for all $\xi \in \mathbb {D}$ , we get (from the continuity of L) that

$$ \begin{align*} h(\xi)=L(f_\xi)=\lim_{N\to\infty} L\Big(\sum_{n=0}^N\widehat{f}(n)\xi^nz^n\Big)=\sum_{n=0}^\infty\widehat{f}(n)b_n\xi^n\,. \end{align*} $$

From Lemma 4.1, $\{f_\xi ;\, \xi \in \mathbb {D}\}$ is a bounded set in $N_\varphi $ , so h is a bounded, analytic function in $\mathbb {D}$ . Hence, $\{b_n\}$ is a multiplier from $N_\varphi $ to $H^\infty $ , and therefore $b_n = O(\exp [-c\,\omega (n)])$ for some $c>0$ . Because $\varphi \in \Phi _2^{sc}(\boldsymbol {\widetilde {\Omega }})$ , we know that $N_\varphi \subset F_\varphi = H(\omega )$ , so by Theorem 3.9, $\widehat {f}(n)=O(\exp [o(\omega (n))])$ as $n\to \infty $ . This shows that the series $\sum _{n=0}^\infty \widehat {f}(n)b_n\xi ^n$ in the above formula converges for every $\xi \in \mathbb {D}$ . Now, from the fact that $f_r\to f$ in $N_\varphi $ as $r\to 1-$ and Abel’s theorem, we conclude that

$$\begin{align*}L(f)=\lim_{r\to 1-} \,L(f_r)=\lim_{r\to 1-}\sum_{n=0}^\infty\widehat{f}(n)b_n r^n =\sum_{n=0}^\infty \widehat{f}(n)b_{n}\,. \end{align*}$$

The proof of equality of the integral and the series is standard.

Now, we will proceed to the sufficient condition. From Remark (2), we know that every function $\varphi \in \Phi _2^{sc}$ can be represented in the form $\varphi (x)= \frac {\omega ^{-1}(x)}{x}$ for all $x>0$ , where $\omega $ satisfies conditions (i)–(iv) from the definition of a set $\boldsymbol {\widetilde {\Omega }}$ . If we suppose that $\varphi \in \Phi _2^{sc}$ is supermultiplicative (i.e., there exists $\gamma>0$ such that $\varphi (st)\geqslant \gamma \varphi (s) \varphi (t)$ for all $s, t>0$ ), then $\omega \in \boldsymbol {\widetilde {\Omega }}$ and hence $F^{int}_{\varphi }=H(\omega )$ . If we were to impose the condition that all the function $\omega \in \boldsymbol {\widetilde {\Omega }}$ is submultiplicative (i.e., there exists $\gamma>0$ such that $\omega (st) \leqslant \gamma \omega (s)\omega (t)$ for all $s, t>0$ ), then from the above-mentioned representation of the function $\varphi \in \Phi _2^{sc}$ , it follows that $\varphi $ is supermultiplicative. Thus, the previously defined set $\Phi _{2}^{sc}(\boldsymbol {\widetilde {\Omega }}) = \{\varphi \in \Phi _2^{sc};\, \varphi \,\, \text {is supermultiplicative}\}$ . Here, it is important to notice that we assumed something strictly weaker for $\omega $ (i.e., condition (iv) in $\boldsymbol {\widetilde {\Omega }}$ ) and thus, the right-hand side set is strictly smaller than $\Phi _2^{sc}(\boldsymbol {\widetilde {\Omega }})$ . We will use the following result.

Proof. Suppose that $\{f_n\}\subset N_\varphi $ and $\|f_n\|_\varphi \to 0$ as $n\to \infty $ . From the inequality $(*)$ in §2, it follows that, for every $r\in [0,1)$ and each $n\in \mathbb {N}$ , we have

$$ \begin{align*} M_\infty(r,f_n)\leqslant\exp\Big(\varphi^{-1}\Big(\frac{2\|f_n\|_\varphi}{1-r}\Big)\Big)-1\,. \end{align*} $$

Hence, $\lim _{n\to \infty } M_\infty (r,f_n) =0$ for every $r\in [0, 1)$ .

Since $\varphi ^{-1}$ is submultiplicative, the above inequality yields

$$\begin{align*}M_\infty(r,f_n)\leqslant\exp\Big(\varphi^{-1}\Big(\frac{1}{1-r}\Big)\varphi^{-1}\Big(2\|f_n\|_\varphi\Big)\Big), \quad\, r\in[0,1)\,. \end{align*}$$

Choose $c>0$ and take $N\in \mathbb {N}$ , such that, for $n\geqslant N$ , the inequality $\varphi ^{-1}(2\|f_n\|_\varphi )\leqslant c$ holds. Then we get that, for each $n\geqslant N$ ,

$$\begin{align*}\exp\Big(-c\,\varphi^{-1}\Big(\frac{1}{1-r}\Big)\Big)M_\infty(r,f_n)\leqslant 1, \quad\, r\in[0,1)\,. \end{align*}$$

From Lebesgue’s Dominated Convergence Theorem, we obtain

$$\begin{align*}\|f_n\|_{\varphi,c}=\int_{0}^{1} M_\infty(r,f_n) \exp\Big(-c\,\varphi^{-1}\Big(\frac{1}{1-r}\Big)\Big)\,dr\to 0\,, \end{align*}$$

and so we are done.

As we observed earlier, for supermultiplicative $\varphi \in \Phi _2^{sc}$ we know that $N_\varphi \subset F^{int}_\varphi =H(\omega )$ . The following theorem can be found in [Reference Zayed23].

Theorem 4.4. For every functional $L\in H(\omega )^*$ , there exists a unique function $g\in A(\mathbb {D})$ with $g(z)=\sum _{n=0}^\infty b_nz^n$ for all $z\in \mathbb {D}$ , where $b_n=O(\exp [-c\,\omega (n)]),$ for some $c>0,$ and such that

$$ \begin{align*} L(f)= \lim_{r\to 1-}\frac{1}{2\pi}\int_0^{2\pi}f(re^{i\theta})g(e^{-i\theta})\,d\theta =\sum_{n=0}^\infty \widehat{f}(n)b_n, \quad\, f\in H(\omega)\,, \end{align*} $$

where $f(z)=\sum _{n=0}^\infty \widehat {f}(n)z^n$ for all $z\in \mathbb {D}$ . Conversely, every function $g\in A(\mathbb {D})$ with $g(z)=\sum _{n=0}^\infty b_nz^n$ for all $z\in \mathbb {D}$ , where $b_n=O(\exp [-c\,\omega (n)])$ for some $c>0$ , defines a linear continuous functional on $H(\omega )$ via the above formula.

An immediate consequence of Theorems 4.3 and 4.4 is the following result.

Theorem 4.5. Let $\varphi \in \Phi _2^{sc}$ be a supermultiplicative function. Every function $g\in A(\mathbb {D})$ with power series $g(z)=\sum _{n=0}^\infty b_nz^n$ for all $z\in \mathbb {D}$ and $b_n=O(\exp [-c\,\omega (n)])$ for some $c>0$ , defines a linear continuous functional L on $N_\varphi $ by the formula:

$$ \begin{align*} L(f)=\lim_{r\to 1-}\frac{1}{2\pi}\int_0^{2\pi}f(re^{i\theta})g(e^{-i\theta})\,d\theta =\sum_{n=0}^\infty \widehat{f}(n)b_n, \quad\, f \in N_{\varphi}\,, \end{align*} $$

where $f(z)=\sum _{n=0}^\infty \widehat {f}(n)z^n$ for all $z\in \mathbb {D}$ .

We conclude with a remark concerning Theorems $6$ , $8,$ and $9$ in this section that the class of analytic functions g in $\mathbb {D}$ satisfying the condition $\widehat {g}(n) = O(\exp [-c\,\omega (n)])$ is known as a Gevrey-type space (the classical Gevery spaces arise when $\omega $ is a power function). Notice that in [Reference Dyakonov3], Dyakonov studies the problem, for which inner functions $\vartheta $ and functions f in Hardy space $H^2$ , the image $H_{\bar {\vartheta }}f$ belongs to some class X of functions in $L^2$ defined by smoothness conditions. Here, $H_{\bar {\vartheta }}$ is the Hankel operator generated by the conjugate $\bar {\vartheta }$ of $\vartheta $ . In particular, the author considers this problem when X is the Gevrey space. This space also appears as a duality result stated in Lemma $3$ which is interesting on its own.