1 Introduction and preliminaries
We denote by
$\mathbb {D}$
and
$\overline {\mathbb {D}}$
the open and closed unit disks in the complex plane
$\mathbb {C}$
. For
$c \in {\mathbb C}$
and
$\rho> 0$
, let
${\mathbb D}(c,\rho ) = \{ z \in {\mathbb C} : |z-c| < \rho \}$
and
$\overline {\mathbb D}(c,\rho ) = \{ z \in {\mathbb C} : |z-c| \leq \rho \}$
. Let
$\mathcal {H}$
be the class of analytic maps of
$\mathbb {D}$
into itself and
$\mathcal {H}_0=\{h\in \mathcal {H}:h(0)=0\}$
. In addition, denote by
$\mathcal {S}$
the Schur class, that is, the set of analytic functions from
$\mathbb {D}$
to
$\overline {\mathbb {D}}$
.
First, we recall the classical Schwarz lemma, which says that if
$f \in \mathcal {H}_0$
, then
$| f(z_0)| \le |z_0|$
for any nonzero
$z_0$
in
$\mathbb {D}$
and
$|f'(0)|\le 1$
, with equality if and only if
$ f(z)=e^{i \theta }z$
for some
$\theta \in \mathbb {R}$
. A number of sharpened and refined versions of the Schwarz lemma can be found in [Reference Beardon and Minda4–Reference Cho, Kim and Sugawa8].
The classical Schwarz–Pick lemma [Reference Pick17] states that
$$ \begin{align*}|f'(z)|\le \frac{1-|f(z)|^2}{1-|z|^2}\quad\mbox{for } f\in \mathcal{H} \mbox{ and } z\in \mathbb{D},\end{align*} $$
with equality if and only if
$f(z)=e^{i\theta }(z-a)/(1-\bar a z)$
for some
$\theta \in \mathbb {R}$
and
$a \in \mathbb {D}$
. If we define
$T_{a}$
by
$T_{a}(z):=(z+a) /(1+\bar {a} z)$
for
$z,a\in \mathbb D$
, then the Schwarz–Pick lemma can be slightly modified as the following lemma.
Theorem 1.1 (Schwarz–Pick lemma).
Let
$z_0,\ \gamma _0\in \mathbb {D}$
. Consider
$f\in \mathcal {H}$
with
$f(z_0)=\gamma _0$
. Then, the region of values of
$f'(z_0)$
, that is,
$\{ f'(z_0) : f \in \mathcal {H}, f(z_0) = \gamma _0\}$
, is the closed disk
$$ \begin{align*} \overline{\mathbb{D}}\bigg(0, \frac{1-|\gamma_0|^2}{1-|z_0|^2}\bigg) =\{f^{\prime}_{\alpha}(z_0):\alpha\in\overline{\mathbb{D}} \} \quad\mbox{where}\ f_{\alpha}(z) = T_{\gamma_0}(\alpha T_{-z_0}(z)), \end{align*} $$
and
$f(z)$
can be expressed as
$T_{\gamma _0}(T_{-z_0}(z) f^*(z))$
for some
$f^*\in \mathcal {S}$
. Further,
$f'(z_0)\in \partial \mathbb {D}(0, {(1-|\gamma _0|^2)}/{(1-|z_0|^2)})$
if and only if
$f(z)=f_{\alpha }(z)$
for some constant
$\alpha \in \partial \mathbb {D}$
.
There are many results on regions of variability (see, for example, [Reference Ali, Allu and Yanagihara1, Reference Hoshinaga and Yanagihara11, Reference Kaptanoğlu12, Reference Mercer15, Reference Roth and Schleißinger20, Reference Yanagihara23]). Among these is the Schwarz–Pick lemma of the second derivative, which describes the region of values of
$f"(z_0)$
for
$f\in \mathcal {H}$
and
$z_0\in \mathbb {D}$
with
$f(z_0)=\gamma _0$
.
Theorem 1.2 (Schwarz–Pick lemma of the second derivative; [Reference Chen6, Theorem 2.2]).
Let
$z_0, \gamma _0\in \mathbb {D}$
and
$ \gamma _1 \in \overline {\mathbb {D}}$
. Then,
$$ \begin{align*} \bigg\{ f"(z_0) : f \in \mathcal{H},f(z_0)=\gamma_0, f'(z_0)=\frac{1-|\gamma_0|^2}{1-|z_0|^2}\gamma_1\bigg\}= \overline{\mathbb{D}}(c_2,\rho_2), \end{align*} $$
where
$$ \begin{align} \left\{ \begin{aligned} c_2=c_2(z_0,\gamma_0,\gamma_1)&= \frac{2(1-|\gamma_0|^2)}{(1-|z_0|^2)^2} (\overline{z_0}-\overline{\gamma_0}\gamma_{1})\gamma_{1},\\ \rho_2=\rho_2(z_0,\gamma_0,\gamma_1)&=\frac{2(1-|\gamma_0|^2)(1-|\gamma_1|^2)} {(1-|z_0|^2)^2}. \end{aligned} \right. \end{align} $$
Further, for
$|\gamma _1|<1$
,
$f"(z_0)\in \partial \mathbb {D}(c_2, \rho _2)$
if and only if
$f(z)=T_{\gamma _0}(T_{-z_0}(z) T_{\gamma _1}(\alpha T_{-z_0}(z)))$
for some constant
$\alpha \in \partial \mathbb {D}$
.
The following interesting question arises from Theorems 1.1 and 1.2 for
$n\ge 1$
: is it possible to describe the variability region of
$f^{(n)}(z_0)$
for
$z_0\in \mathbb {D}$
and
$f\in \mathcal {H}$
? The answer is yes and the nth-order result is given in Theorem 2.5.
The nth derivative
$f^{(n)}(z_0)$
is related to the nth-order hyperbolic derivative
$H^nf(z_0)$
, which was first introduced by Rivard [Reference Rivard18]. For
$ z, w\in \overline {\mathbb {D}}$
, we define the complex pseudo-hyperbolic distance
$$ \begin{align*} [z,\,w]:= \begin{cases} \dfrac{z-w}{1-\overline{w}z} &\quad \text{if}~z\ne w, \\ 0 &\quad\text{if}~z=w. \end{cases} \end{align*} $$
Thus, we can define an operator
$\Delta _{z_0}$
by
$$ \begin{align*} \Delta_{z_0}f(z)= \begin{cases} \dfrac{[f(z),\,f(z_0)]}{[z,\,z_0]} &\quad \text{for}~z\ne z_0, \\[8pt] \dfrac{(1-|z_0|^2)f'(z_0)}{1-|f(z_0)|^2} &\quad\text{for}~z=z_0. \end{cases} \end{align*} $$
Following [Reference Baribeau, Rivard and Wegert2], we can iterate the process and construct the hyperbolic difference quotient of order j of the function f for distinct parameters
$z_1,\ldots ,z_{j}$
by
$$ \begin{align*} \Delta^jf(z;z_{j},\ldots,z_1)=(\Delta_{z_{j}}\circ\cdots\circ\Delta_{z_1})f(z). \end{align*} $$
We can rewrite this recursive definition as
$$ \begin{align*} \Delta^jf(z;z_j,\ldots,z_1)=\frac{[\Delta^{j-1}f(z;z_{j-1},\ldots,z_1),\Delta^{j-1}f(z_j;z_{j-1},\ldots,z_1)]}{[z,z_j]}. \end{align*} $$
Using a limiting process, we can define the hyperbolic divided quotient of order n of f with parameter z by
$$ \begin{align*} \Delta_z^n f(\zeta)=\Delta^n f(\zeta;z,\ldots,z):=\lim_{z_{n}\to z}\cdots\lim_{z_1\to z}\Delta^n f(\zeta;z_{n},\ldots,z_1)\quad(\zeta\neq z). \end{align*} $$
Therefore, we define the nth-order hyperbolic derivative of
$f\in \mathcal {H}$
at
$z\in \mathbb {D}$
by
$$ \begin{align*} H^n f(z):=\Delta_z^n f(z)=\Delta^n f(z;z,\ldots,z)=\lim_{\zeta\to z}\Delta^n f(\zeta;z,\ldots,z). \end{align*} $$
The usual hyperbolic derivative (see [Reference Beardon and Minda3]) coincides with the first-order hyperbolic derivative,
$$ \begin{align*} f^h(z)=\frac{(1-|z|^2)f'(z)}{1-|f(z)|^2}=H^{1} f(z). \end{align*} $$
To generalise Theorems 1.1 and 1.2, we consider the following problem.
Problem 1.3. For
$z_0\in \mathbb {D}$
and
$\gamma = (\gamma _0, \gamma _1 , \ldots , \gamma _{n-1}) \in {\mathbb D}^{n}$
, let
$$ \begin{align*} {\mathcal H} (\gamma) = \{f \in {\mathcal H} : f (z_0) = \gamma_0,H^1f (z_0) = \gamma_1,\ldots ,H^{n-1}f (z_0) = \gamma_{n-1} \}.\end{align*} $$
Determine the variability region
$\{ f^{(n)}(z_0) : f \in {\mathcal H} (\gamma ) \}$
.
Based on the Schwarz–Pick lemma, Dieudonné [Reference Dieudonné9] gave a description of the variability region of
$h'(z_0)$
for
$h\in \mathcal {H}_0$
and
$z_0\in \mathbb {D}$
(see also [Reference Beardon and Minda3, Reference Duren10]).
Theorem 1.4 (Dieudonné’s lemma).
Let
$z_0,w_0\in \mathbb {D}$
with
$|w_0|<|z_0|$
. Suppose that
$h\in \mathcal {H}_0$
and
$h(z_0) = w_0$
. Set
$\gamma _0=w_0/z_0$
,
$h_{\alpha }(z)=zT_{\gamma _0}(\alpha T_{-z_0}(z))$
. Then, the region of values of
$h'(z_0)$
, that is,
$\{ h'(z_0) : h \in \mathcal {H}_0, h(z_0) = w_0\}$
, is the closed disk
$$ \begin{align*} \overline{\mathbb{D}} ( c^{\prime}_1(z_0,w_0) , \rho^{\prime}_1(z_0,w_0) )=\{h^{\prime}_{\alpha}(z_0):\alpha\in \overline{\mathbb{D}}\}, \end{align*} $$
where
$$ \begin{align*}c^{\prime}_1(z_0,w_0)=\frac{w_0}{z_0}, \quad \rho^{\prime}_1(z_0,w_0)=\frac{|z_0|^2-|w_0|^2}{|z_0|(1-|w_0|^2)},\end{align*} $$
and
$h(z)$
can be expressed as
$z T_{\gamma _0}(T_{-z_0}(z) h^*(z))$
for some
$h^*\in \mathcal {S}$
. Further,
$h'(z_0) \in \partial \mathbb {D} ( c^{\prime }_1(z_0,w_0), \rho ^{\prime }_1(z_0,w_0)) $
if and only if
$h(z)=h_{\alpha }(z)$
for some constant
$\alpha \in \partial \mathbb {D}$
.
Rivard [Reference Rivard19] extended Dieudonné’s lemma to the second order.
Theorem 1.5 (Dieudonné’s lemma of the second derivative).
Let
$z_0, w_0\in \mathbb {D}$
with
$|w_0|<|z_0|$
,
$\gamma _1\in \overline {\mathbb {D}}$
and
$$ \begin{align*}w_1 =\frac{w_0}{z_0}+\frac{|z_0|^2-|w_0|^2}{\overline{z_0}(1-|w_0|^2)}\gamma_1.\end{align*} $$
Then,
$ \{ h"(z_0) : h \in \mathcal {H}_0, h(z_0) = w_0, h'(z_0)=w_1\}= \overline {\mathbb {D}}(c^{\prime }_2,\rho ^{\prime }_2), $
where
$$ \begin{align*} \left\{ \begin{aligned} c^{\prime}_2 &=c^{\prime}_2(z_0,w_0,\gamma_1)=\frac{2(|z_0|^2-|w_0|^2)}{|z_0|^2(1-|z_0|^2)^2} \gamma_1\bigg(1-\frac{z_0\overline{w_0}}{\overline{z_0}}\gamma_1\bigg),\\ \rho^{\prime}_2&=\rho^{\prime}_2(z_0,w_0,\gamma_1) =\frac{2(|z_0|^2-|w_0|^2)}{|z_0|(1-|z_0|^2)^2}(1-|\gamma_1|^2). \end{aligned} \right. \end{align*} $$
Further, for
$|\gamma _1|<1$
,
$h"(z_0)\in \partial \mathbb {D}(c^{\prime }_2,\rho ^{\prime }_2)$
if and only if
$h(z)=z T_{\gamma _0}\big (T_{-z_0}(z) T_{\gamma _1}(\alpha T_{-z_0}(z))\big )$
for some constant
$\alpha \in \partial \mathbb {D}$
, where
$\gamma _0=w_0/z_0$
.
Theorems 1.4 and 1.5 suggest an interesting question for
$n \ge 1$
: can we obtain an nth-order Dieudonné lemma for the region of values of
$h^{(n)}(z_0)$
for
$h\in \mathcal {H}_0$
and
$z_0\in \mathbb {D}$
? We give the nth-order result in Theorem 3.1.
The organisation of this paper is as follows. In Section 2, we obtain a useful formula for directly calculating the hyperbolic derivatives and use it to solve Problem 1.3. In Section 3, we apply the Schwarz–Pick lemma of the nth derivative to establish a Dieudonné lemma of the nth derivative. The connection between
$\mathcal {H}_0$
and
$\mathcal {H}$
is as follows. If
$h\in \mathcal {H}_0$
is not a rotation about the origin and
$f(z)=h(z)/z$
, then
$f\in \mathcal {H}$
. Since
$h^{(n)}(z)= zf^{(n)}(z)+nf^{(n-1)}(z)$
, we just need to determine the variability region of
$f^{(n)}(z_0)$
for
$f\in \mathcal {H}$
.
2 Formula for hyperbolic derivatives and the Schwarz–Pick lemma of the nth derivative
Since the recursive formula established by Rivard [Reference Rivard19] does not lend itself easily to simple calculations for hyperbolic derivatives, we will develop a method that enables us to calculate higher-order hyperbolic derivatives more efficiently.
For
$n=1,2,3,\ldots, $
we define a sequence of functions
$F_n$
of the n complex variables
$\gamma _1, \gamma _2, \ldots , \gamma _n$
inductively by
$$ \begin{align} \notag F_1(\gamma_1)&=\gamma_1, \quad F_2(\gamma_1,\gamma_2)=(1-|\gamma_1|^2)\gamma_2,\\ F_{n}(\gamma_1,\ldots, \gamma_{n}) &=(1-|\gamma_1|^2)F_{n-1}(\gamma_2,\ldots, \gamma_{n}) \notag\\ &\quad -\bar\gamma_1\sum_{k=2}^{n-1} F_{n-k}(\gamma_2,\ldots,\gamma_{n-k+1}) F_k(\gamma_1,\ldots, \gamma_k), \quad n\ge 3. \end{align} $$
By construction, the function
$F_n(\gamma _1,\ldots ,\gamma _n)$
is a polynomial in
$\gamma _1,\bar \gamma _1, \ldots ,$
$\gamma _{n-1},\bar \gamma _{n-1}, \gamma _n$
with integer coefficients. From (2.1),
$$ \begin{align*} F_3&=(1-|\gamma_1|^2)(1-|\gamma_2|^2)\gamma_3 -(1-|\gamma_1|^2)\bar\gamma_1\gamma_2^2, \\ F_4&=(1-|\gamma_1|^2)(1-|\gamma_2|^2)(1-|\gamma_3|^2)\gamma_4 \\ &\quad -(1-|\gamma_1|^2)(1-|\gamma_2|^2)\gamma_3(2\bar\gamma_1\gamma_2 +\bar\gamma_2\gamma_3)+(1-|\gamma_1|^2)\bar\gamma_1^2\gamma_2^3. \end{align*} $$
By induction, it is easy to verify Schur’s recurrence relation.
Lemma 2.1 [Reference Schur22].
For each
$n\ge 2,$
there exists a function
$G_n(\gamma _1,\ldots ,\gamma _{n-1})$
of
$n-1$
complex variables
$\gamma _1,\ldots ,\gamma _{n-1}$
such that
$$ \begin{align*} F_n(\gamma_1,\ldots,\gamma_n)=(1-|\gamma_1|^2)\cdots(1-|\gamma_{n-1}|^2)\gamma_n+G_n(\gamma_1,\ldots,\gamma_{n-1}). \end{align*} $$
Lemma 2.2 [Reference Li and Sugawa14].
Let
$g(z)=\sum _{n=1}^\infty a_n z^n$
be a function in
$\mathcal {H}_0$
with its hyperbolic derivatives
$H^jg(0)=\gamma _j (j=1,2,\ldots )$
. Then,
$a_n=F_n(\gamma _1,\gamma _2,\ldots , \gamma _n)$
for
$n=1, 2, 3, \ldots .$
For
$f\in \mathcal {H}$
, Peschl’s invariant derivatives
$D^n f(z)$
[Reference Peschl16] (see also [Reference Kim and Sugawa13, Reference Schippers21]) are defined by the Taylor series expansion
$$ \begin{align} z\mapsto g(z):=\frac{f(\tfrac{z+z_0}{1+\overline{z}_0 z})-f(z_0)}{1-\overline{f(z_0)}f(\tfrac{z+z_0}{1+\overline{z}_0 z})}=\sum_{n=1}^{\infty}\frac{D^n f(z_0)}{n!}z^n,\quad z, z_0\in \mathbb{D}. \end{align} $$
Then,
$D^n f(z_0)=g^{(n)}(0)$
and
$H^nf(z_0)=H^ng(0)$
for
$n=1,2,\ldots $
(see [Reference Cho, Kim and Sugawa8, Proposition 3.3]). The first few
$D^n f(z)$
are given by
$$ \begin{align*} D^1 f(z)&=\frac{(1-{\lvert z\rvert}^2)f'(z)}{1-{\lvert f(z)\rvert}^2},\\ D^2 f(z)&=\frac{(1-{\lvert z\rvert}^2)^2}{1-{\lvert f(z)\rvert}^2} \bigg[f"(z)-\frac{2\overline{z}f'(z)}{1-{\lvert z\rvert}^2} +\frac{2\overline{f(z)}f'(z)^2} {1-{\lvert f(z)\rvert}^2}\bigg],\\ D^3 f(z)&=\frac{(1-{\lvert z\rvert}^2)^3}{1-{\lvert f(z)\rvert}^2} \bigg[f"'(z)-\frac{6\overline{z}f"(z)}{1-{\lvert z\rvert}^2} +\frac{6\overline{f(z)}f'(z)f"(z)} {1-{\lvert f(z)\rvert}^2} +\frac{6\overline{z}^2f'(z)}{(1-{\lvert z\rvert}^2)^2}&\\ &\quad -\frac{12\overline{f(z)}f'(z)^2} {(1-{\lvert z\rvert}^2)(1-{\lvert f(z)\rvert}^2)} +\frac{6\overline{f(z)}^2f'(z)^3} {(1-{\lvert f(z)\rvert}^2)^2}\bigg]. \end{align*} $$
When we express g in (2.2) by the series expansion
$g(z)=\sum _{n=1}^\infty a_n z^n$
, we can obtain the formula for
$D^nf(z_0)$
,
$$ \begin{align} \frac{D^nf(z_0)}{n!}=(1-|\gamma_1|^2)\cdots(1-|\gamma_{n-1}|^2)\gamma_n +G_n(\gamma_1,\ldots,\gamma_{n-1}). \end{align} $$
Next, we obtain a formula for
$H^nf$
in terms of
$D^nf$
and the lower-order hyperbolic derivatives
$H^1f,\ldots ,H^{n-1}f$
, allowing us to calculate the hyperbolic derivatives in a more convenient way. From the preceding definitions,
$$ \begin{align*}H^1f(z)=D^1f(z)=f^h(z)=\frac{(1-|z|^2)f'(z)}{1-|f(z)|^2}.\end{align*} $$
A Blaschke product of degree
$n \in \mathbb {N}$
is a function of the form
$ B(z)=e^{i \theta }\prod _{j=1}^{n} [z,z_j]$
, with
$ z, z_j\in \mathbb {D}$
,
$\theta \in \mathbb {R}$
. The class of Blaschke products of degree n is denoted by
$\mathcal {B}_n$
.
Theorem 2.3. Let
$n\geq 2$
,
$f\in \mathcal {H}\setminus \bigcup _{j=0}^{j=n-1}\mathcal {B}_j$
and
$z_0\in \mathbb {D}$
. Then,
$$ \begin{align} H^nf(z_0)=\frac{D^nf(z_0)-n!G_n(\gamma_1,\ldots,\gamma_{n-1})} {n!(1-|\gamma_1|^2)\cdots(1-|\gamma_{n-1}|^2)} \quad\mbox{where } \gamma_j=H^jf(z_0). \end{align} $$
Using (2.4), one can easily obtain an explicit form for hyperbolic derivatives. We give the first few cases. Noting that
$G_2(\gamma _1)=0$
and
$G_3(\gamma _1,\gamma _2)= -(1-|\gamma _1|^2)\overline {\gamma _1}\gamma _2^2$
, we immediately obtain
$$ \begin{align*} H^2f(z_0)&= \frac{D^2f(z_0)}{2(1-|H^1f(z_0)|^2)},\\ H^3f(z_0)&=\frac{D^3f(z_0)+6(1-|H^1f(z_0)|^2)\overline{H^1f(z_0)}(H^2f(z_0))^2} {6(1-|H^1f(z_0)|^2)(1-|H^{2}f(z_0)|^2)}. \end{align*} $$
It is possible to obtain expressions for
$D^nf$
in terms of
$f^{(n)}$
and its lower-order derivatives, as well as the derivatives
$D^1f,\ldots ,D^nf$
.
Lemma 2.4 [Reference Kim and Sugawa13, Corollary 7.5].
Let
$f\in \mathcal {H}$
. Then,
$$ \begin{align*}D^nf(z)=\sum_{k=1}^n\alpha_{n,k} \frac{(\overline{z})^{n-k}(1-|z|^2)^kf^{(k)}(z)}{1-|f(z)|^2}-\sum_{k=2}^nk!(-\overline{f(z)})^{k-1}A_{n,k}(D^1f,\ldots,D^{n-k+1}f),\end{align*} $$
where
$$ \begin{align*}\alpha_{n,k}:= \left\{\begin{array}{@{}ll}(-1)^{n-k}\dfrac{n!(n-1)!}{k!(k-1)!(n-k)!}& \mbox{if } 1 \leq k \leq n, \\ 0&otherwise,\end{array}\right. \end{align*} $$
and
$A_{n, k}$
is the Bell polynomial given by
$$ \begin{align*} A_{n,k}(x_1,\ldots,x_{n-k+1})&:= \sum_{j\in I_k}{n! \over j_1!j_2!\cdots j_{n-k+1}!} \bigg({x_1\over 1!}\bigg)^{j_1}\bigg({x_2\over 2!}\bigg)^{j_2}\cdots\bigg({x_{n-k+1} \over (n-k+1)!}\bigg)^{j_{n-k+1}} \\ &\,= n! \sum_{j\in I_k} \prod_{i=1}^{n-k+1} \frac{x_i^{j_i}}{(i!)^{j_i} j_i!}, \end{align*} $$
where
$I_k$
consists of all multi-indexes
$(j_1,\ldots ,j_{n-k+1})$
such that
$$ \begin{align*} j_1,\ldots,j_{n-k+1}\ge 0, \quad j_1+j_2+\cdots+j_{n-k+1}=k, \quad j_1+2j_2+\cdots+(n-k+1)j_{n-k+1}=n. \end{align*} $$
We note here that the ordinary derivative
$f^{(n)}$
can also be expressed in terms of the invariant derivatives
$D^kf (k = 1, 2, \ldots , n)$
by
$$ \begin{align*}\frac{(1-|z|^2)^n}{1-|f(z)|^2}\frac{f^{(n)}(z)}{n!} =\sum_{k=1}^n\binom{n-1}{k-1}(\bar{z})^{n-k}\cdot b_k,\end{align*} $$
where
$$ \begin{align*}b_k=\sum_{l=1}^k\frac{l!}{k!}(-\overline{f(z)})^{l-1}A_{k,l}(D^1f,\ldots,D^{k-l+1}f).\end{align*} $$
Since
$\alpha _{n,n}=1$
, we have
$$ \begin{align*} D^nf(z)&=\frac{(1-|z|^2)^n f^{(n)}(z)}{1-|f(z)|^2}+s_{n-1}(z), \end{align*} $$
where
$$ \begin{align*} s_{n-1}(z)&= \sum_{k=1}^{n-1}\alpha_{n,k}\frac{(\overline{z})^{n-k}(1-|z|^2)^kf^{(k)}(z)}{1-|f(z)|^2}\\ &\quad-\sum_{k=2}^nk!(-\overline{f(z)})^{k-1}A_{n,k}(D^1f(z),\ldots,D^{n-k+1}f(z)). \end{align*} $$
Note that
$s_{n-1}(z)$
depends only on
$D^1f(z),\ldots ,D^{n-1}f(z)$
together with z and
$f(z)$
. By (2.4), it is easy to verify that
$s_{n-1}(z)$
depends only on
$H^1f(z),\ldots ,H^{n-1}f(z)$
together with z and
$f(z)$
. For instance,
$s_1(z)=-2H^1f(z)(\bar z-\overline {f(z)}H^1f(z))$
.
Together with (2.3), we have
$f^{(n)}(z_0)=c_n+\rho _n \gamma _n$
, where
$\gamma _n=H^nf(z_0)\in \overline {\mathbb {D}}$
and
$$ \begin{align} \left\{ \begin{aligned} c_n&=\frac{(1-|\gamma_0|^2)}{(1-|z_0|^2)^n} [n!G_n(\gamma_1,\ldots,\gamma_{n-1}) -s_{n-1}(z_0)],\\ \rho_n&=\frac{n!\prod_{k=0}^{n-1}(1-|\gamma_k|^2)}{(1-|z_0|^2)^n}. \end{aligned} \right. \end{align} $$
In addition, we set
$G_1=0$
and
$s_0(z_0)=0$
.
Theorem 2.5 (Schwarz–Pick lemma of the nth derivative).
Let
$n \in {\mathbb N}$
,
$z_0 \in {\mathbb D}$
and
$\gamma = (\gamma _0, \ldots , \gamma _{n-1} ) \in \overline {{\mathbb D}}^{n}$
. Suppose that
$f \in {\mathcal H}$
,
$f(z_0)=\gamma _0$
,
$H^1f(z_0)=\gamma _1, \ldots , H^{n-1}f(z_0)=\gamma _{n-1}$
.
-
(1) If
$|\gamma _0|<1, \ldots , |\gamma _{j-1}|<1$
and
$|\gamma _j|=1$
for some
$j=1, \ldots , n-1$
, then
$\gamma _{j+1}= \cdots = \gamma _{n-1}=0$
and
$f^{(n)}(z_0)=c_n$
. Furthermore, which is a Blaschke product of degree j.
$$ \begin{align*} f (z) = T_{\gamma_0} (T_{-z_0}(z) T_{\gamma_1}(\cdots T_{-z_0}(z) T_{\gamma_{j-1}}(\gamma_j T_{-z_0}(z)) \cdots )), \end{align*} $$
-
(2) If
$|\gamma _0|<1, \ldots , |\gamma _{n-1}|<1$
, then the region of values of
$f^{(n)}(z_0)$
is the closed disk where
$$ \begin{align*} \overline{\mathbb{D}}(c_n, \rho_n) =\{f^{(n)}_{\gamma,\varepsilon}(z_0):\varepsilon \in\overline{\mathbb{D}} \}, \end{align*} $$
$f_{\gamma ,\varepsilon } (z) = T_{\gamma _0} (T_{-z_0}(z) T_{\gamma _1}(\cdots T_{-z_0}(z) T_{\gamma _{n-1}}(\varepsilon T_{-z_0}(z)) \cdots ))$
, and
$f(z)$
can be expressed as for some
$$ \begin{align*} f (z) = T_{\gamma_0} (T_{-z_0}(z) T_{\gamma_1}(\cdots T_{-z_0}(z) T_{\gamma_{n-1}}(T_{-z_0}(z) f^* (z)) \cdots )) \end{align*} $$
$f^* \in \mathcal {S}$
. Further,
$f^{(n)}(z_0)\in \partial \mathbb {D}(c_n, \rho _n)$
if and only if
$f(z)=f_{\gamma ,\varepsilon }(z)$
for some constant
$\varepsilon \in \partial \mathbb {D}$
.
Theorems 1.1 and 1.2 are both simple corollaries of Theorem 2.5.
3 Dieudonné’s lemma of the nth derivative
In this section, we will apply Theorem 2.5 to formulate a Dieudonné lemma of the nth derivative
$h^{(n)}$
for
$h\in \mathcal {H}_0$
.
Let
$n \in {\mathbb N}$
,
$z_0 \in {\mathbb D}$
and
$\gamma = (\gamma _0, \ldots , \gamma _{n-1} ) \in {\mathbb D}^{n}$
. Suppose that
$f \in {\mathcal H}$
,
$f(z_0)=\gamma _0$
and
$H^1f(z_0)=\gamma _1, \ldots , H^{n-1}f(z_0)=\gamma _{n-1}$
. Then, we have
$h(z)=zf(z)\in \mathcal {H}_0$
and
$h^{(k)}(z)= kf^{(k-1)}(z)+zf^{(k)}(z)$
for
$k=1,\ldots ,n$
. In particular,
$h(z_0)=z_0\gamma _0$
,
$$ \begin{align*} h'(z_0)=f(z_0)+z_0f'(z_0) =\gamma_0+\frac{1-|\gamma_0|^2}{1-|z_0|^2}z_0\gamma_0 =\gamma_0+|z_0|\rho_1\frac{|z_0|}{\overline{z_0}}\gamma_0, \end{align*} $$
and, for
$k=2,\ldots ,n$
,
$$ \begin{align*} h^{(k)}(z_0)&=kf^{(k-1)}(z_0)+z_0f^{(k)}(z_0)\\ &=k(c_{k-1}+\rho_{k-1}\gamma_{k-1})+z_0(c_k+\rho_k\gamma_k)\\ &=[k(c_{k-1}+\rho_{k-1}\gamma_{k-1})+z_0c_k] +|z_0|\rho_k\frac{|z_0|}{\overline{z_0}}\gamma_k, \end{align*} $$
where
$\gamma _n=H^nf(z_0)\in \overline {\mathbb {D}}$
. Note that
$c_k$
and
$\rho _k$
are defined by (2.5). Therefore, we conclude that the region of values of
$h^{(n)}(z_0)$
is the closed disk
$\overline {\mathbb {D}}(c^{\prime }_n,\rho ^{\prime }_n)$
, where
$$ \begin{align*} \left\{ \begin{aligned} c^{\prime}_n&=n(c_{n-1}+\rho_{n-1}\gamma_{n-1})+z_0c_n,\\ \rho^{\prime}_n&=|z_0|\rho_n. \end{aligned} \right. \end{align*} $$
Theorem 3.1 (Dieudonné’s lemma of the nth derivative).
Let
$z_0, w_0\in \mathbb {D}$
with
$|w_0|<|z_0|$
and let
$\gamma _1,\ldots , \gamma _{n-1} \in \overline {\mathbb {D}}$
. For
$k=1,\ldots ,n-1$
, let
$w_k\in \overline {\mathbb {D}}(c^{\prime }_k,\rho ^{\prime }_k)$
be given by
$w_k=c^{\prime }_k+\rho ^{\prime }_k\gamma _k{|z_0|}/{\overline {z_0}}$
, where
$$ \begin{align*}c^{\prime}_1=\frac{w_0}{z_0}, \quad \rho^{\prime}_1=\frac{|z_0|^2-|w_0|^2}{|z_0|(1-|w_0|^2)}\end{align*} $$
and, for
$ k=2,\ldots ,n-1$
,
$$ \begin{align*} c^{\prime}_k=k(c_{k-1}+\rho_{k-1}\gamma_{k-1})+z_0c_k,\quad\rho^{\prime}_k=|z_0|\rho_k. \end{align*} $$
Suppose that
$h\in \mathcal {H}_0$
,
$h(z_0) = w_0, h'(z_0)=w_1,\ldots ,h^{(n-1)}(z_0)=w_{n-1}$
. Set
$\gamma _0=w_0/z_0$
.
-
(1) If
$|\gamma _1|<1, \ldots , |\gamma _{j-1}|<1$
,
$|\gamma _j|=1$
,
$\gamma _{j+1}= \cdots = \gamma _{n-1}=0$
for some
$j=1, \ldots , n-1$
, then
$h^{(n)}(z_0)=c^{\prime }_n$
and
$$ \begin{align*} h (z) = z T_{\gamma_0} (T_{-z_0}(z) T_{\gamma_1}(\cdots T_{-z_0}(z) T_{\gamma_{j-1}}(\gamma_j T_{-z_0}(z)) \cdots )). \end{align*} $$
-
(2) If
$|\gamma _1|<1$
,
$|\gamma _2|<1$
,…,
$|\gamma _{n-1}|<1$
, then the region of values of
$h^{(n)}(z_0)$
is the closed disk where
$$ \begin{align*}\overline{\mathbb{D}}(c^{\prime}_n, \rho^{\prime}_n)=\{h^{(n)}_{\gamma,\varepsilon}(z_0):\varepsilon \in\overline{\mathbb{D}}\},\end{align*} $$
$h_{\gamma ,\varepsilon }(z)=z T_{\gamma _0}(T_{-z_0}(z) T_{\gamma _1}(\cdots T_{-z_0}(z)T_{\gamma _{n-1}}(\varepsilon T_{-z_0}(z))\cdots ))$
, and
$h(z)$
can be expressed as for some
$$ \begin{align*} zT_{\gamma_0} (T_{-z_0}(z) T_{\gamma_1}(\cdots T_{-z_0}(z) T_{\gamma_{n-1}}(T_{-z_0}(z) h^* (z)) \cdots )) \end{align*} $$
$h^* \in \mathcal {S}$
. Further,
$h^{(n)}(z_0)\in \partial \mathbb {D}(c^{\prime }_n, \rho ^{\prime }_n)$
if and only if
$h(z)=h_{\gamma ,\varepsilon }(z)$
for some constant
$\varepsilon \in \partial \mathbb {D}$
.
Proof. We just need to consider the case
$\gamma _1,\ldots , \gamma _{n-1} \in \mathbb {D}$
for
$n\ge 2$
. Assume that
$f(z)=h(z)/z$
, with
$f\in \mathcal {H}$
. It is easily seen that
$f(z_0)=\gamma _0$
,
$H^1f(z_0)=\gamma _1, \ldots , H^{n-1}f(z_0)=\gamma _{n-1}$
. From the preceding discussion,
$h^{(n)}(z_0)\in \overline {\mathbb {D}}(c^{\prime }_n, \rho ^{\prime }_n)$
.
Next, we show that
$\overline {\mathbb {D}}(c^{\prime }_n,\rho ^{\prime }_n)$
is covered. Define the class
$\mathcal {H}_0 (z_0,w_0,\ldots ,w_{n-1})$
by
$$ \begin{align*}\mathcal{H}_0 (z_0,w_0,\ldots,w_{n-1}) = \{ h \in \mathcal{H}_0 : h(z_0) =w_0, h'(z_0)=w_1,\ldots, h^{(n-1)}(z_0)=w_{n-1}\}.\end{align*} $$
From the proof of Theorem 2.5 and the relations
$h^{(k)}(z)=kf^{(k-1)}(z)+zf^{(k)}(z)$
for
$k=1,\ldots ,n$
, it is easy to see that
$h_{\gamma ,\varepsilon }\in \mathcal {H}_0 (z_0,w_0,\ldots ,w_{n-1})$
and
$ h^{(n)}_{\gamma ,\varepsilon }(z_0) =c^{\prime }_n+\rho ^{\prime }_n\varepsilon {|z_0|}/{\overline {z_0}}$
. The closed disk
$\overline {\mathbb {D}}(c^{\prime }_n,\rho ^{\prime }_n)$
is covered since
$\varepsilon \in \overline {\mathbb {D}}$
is arbitrary.
Next,
$h^{(n)}(z_0)\in \partial \mathbb {D}(c^{\prime }_n, \rho ^{\prime }_n)$
if and only if
$h(z)=zf(z)$
, where f is a Blaschke product of degree n satisfying
$f(z_0)=\gamma _0$
,
$H^1f(z_0)=\gamma _1, \ldots , H^{n-1}f(z_0)=\gamma _{n-1}$
. From the proof of Theorem 2.5, we can easily check that
$$ \begin{align*}f(z)= T_{\gamma_0} (T_{-z_0}(z) T_{\gamma_1}(\cdots T_{-z_0}(z) T_{\gamma_{n-1}}(\varepsilon T_{-z_0}(z)) \cdots )), \quad \varepsilon\in \partial \mathbb{D}.\\[-37pt] \end{align*} $$
Theorems 1.4 and 1.5 can be directly derived from Theorem 3.1.
Acknowledgement
The author would like to express his deep gratitude to Prof. Toshiyuki Sugawa for his valuable comments and instructive suggestions.
