A Refinement of Schwarz–Pick Lemma for Higher Derivatives

Abstract

In this paper, a Schwarz–Pick estimate of a holomorphic self map f of the unit disc D having the expansion $f\left(w\right)=c_0+c_n{(w-z)}^n+\dots$ in a neighborhood of some z in D is given. This result is a refinement of the Schwarz–Pick lemma, which improves a previous result of Shinji Yamashita.

1. Introduction

For the open unit disc D of the complex plane and the boundary $\partial D$ of D, the following Schwarz–Pick lemma(see [1], Lemma 1.2) is well-known.

Theorem 1.

Let $f:D⟶D$ be holomorphic and $z_0\in D.$ Then,

$$\left|\frac{f\left(z\right)-f\left(z_0\right)}{1-\overline{f\left(z_0\right)}f\left(z\right)}\right|\le \left|\frac{z-z_0}{1-\overline{z_0}z}\right|,z\in D,$$

(1)

and

$$\frac{|f^{\prime }\left(z_0\right)|}{1-{\left|f\right(z_0\left)\right|}^2}\le \frac{1}{1-|z_0{|}^2}.$$

(2)

Equality in (1) holds at some point $z\ne z_0$ or equality in (2) holds if and only if

$$f\left(z\right)=c\frac{z-a}{1-\overline{a}z},z\in D$$

(3)

for some $c\in \partial D$ and $a\in D$.

Among those interesting extensions of (2), there is a result of Shinji Yamashita(see [2], Theorem 1):

Theorem 2.

Let f be a function holomorphic and bounded, $\left|f\right|<1$, in D, and let $z\in D$. Suppose that

$$f\left(w\right)=c_0+c_n{(w-z)}^n+c_{n+1}{(w-z)}^{n+1}+\dots$$

in a neighborhood of z, where $n\ge 1$ depends on z and $c_n=0$ is possible. Then,

$$\frac{{(1-{\left|z\right|}^2)}^n\left|f^{\left(n\right)}\left(z\right)\right|}{n!(1-{\left|f\right(z\left)\right|}^2)}\le 1.$$

(4)

The inequality (4) is sharp in the sense that equality holds for the function

$$f\left(w\right)=e^{i\alpha }{\left(\frac{w-z}{1-\overline{z}w}\right)}^n(\alpha ;arealconstant)$$

of w.

For f holomorphic in D, $0\le r<1$, and $0\le p\le \infty$, as it is commonly used we denote $M_p(r,f)$ by the p-mean of f on $\partial D$, that is,

$$M_p(r,f)=\{\begin{array}{cc}exp\left({\int }_{-\pi }^{\pi }log\left|f\left(r{e}^{i\theta }\right)\right|\frac{d\theta }{2\pi }\right)\hfill & \mathrm{if}p=0,\hfill \\ {\left({\int }_{-\pi }^{\pi }{\left|f\right(r{e}^{i\theta }\left)\right|}^p\frac{d\theta }{2\pi }\right)}^{\frac{1}{p}}\hfill & \mathrm{if}0<p<\infty ,\hfill \\ \underset{\left|z\right|=r}{sup}\left|f\left(z\right)\right|\hfill & \mathrm{if}p=\infty .\hfill \end{array}$$

If f is holomorphic, then $M_p(r,f)$ is an increasing function of $p:0\le p\le \infty$ as well as an increasing function of $r:0\le r<1$ (see [3]).

For $a\in D$, let ${\phi }_a$ be defined by

$${\phi }_a\left(z\right)=\frac{z+a}{1+\overline{a}z},z\in D.$$

${\phi }_a$ satisfies ${\phi }_a\left({\phi }_{-a}\left(z\right)\right)=z$ for all $z\in D$. It is well-known that ${\phi }_a(\partial D)=\partial D$ and that the set of automorphisms, i.e., bijective biholomorphic mappings, of D consists of the mappings of the form $\alpha {\phi }_a\left(z\right)$, where $a\in D$ and $\left|\alpha \right|=1$.

Extending (2) in terms of $M_p(r,f)$, there is another result of Shinji Yamashita(see [4], Theorem 2):

Theorem 3.

Let f be a function holomorphic and bounded, $\left|f\right|<1$, in D and let $0\le p\le \infty .$ Then

$$\frac{(1-{\left|w\right|}^2)\left|f^{\prime }\left(w\right)\right|}{1-{\left|f\right(w\left)\right|}^2}\le \frac{1}{r}{M}_p(r,f_w)\le 1$$

(5)

for all $w\in D$ and $r:0<r<1$, where

$$f_w\left(z\right)=\frac{f(\frac{z+w}{1+\overline{w}z})-f\left(w\right)}{1-\overline{f\left(w\right)}f(\frac{z+w}{1+\overline{w}z})},z\in D.$$

If the equality $r^{-1}{M}_p(r,f_w)=1$ holds in (5) for $w\in D$ and $0<r<1$, then f is of the form (3).

Note that $n=1$ in (4) reduces to (2) and that (5) refines (2). As the same manner, it is expected that there might be a refinement of Theorem 2 which reduces to Theorem 3 when $n=1$. This is our objective of this note.

2. Result

The following is our corresponding result:

Theorem 4.

Let f be a function holomorphic and bounded, $\left|f\right|<1$, in D and let $z\in D$. If

$$f\left(w\right)=c_0+c_n{(w-z)}^n+c_{n+1}{(w-z)}^{n+1}+\cdots .$$

(6)

in a neighborhood of z, then

$$\left|\frac{f\left(w\right)-f\left(z\right)}{1-\overline{f\left(z\right)}f\left(w\right)}\right|\le {\left|\frac{w-z}{1-\overline{z}w}\right|}^n,w\in D,$$

(7)

and

$$\frac{{(1-{\left|z\right|}^2)}^n\left|f^{\left(n\right)}\left(z\right)\right|}{n!(1-{\left|f\right(z\left)\right|}^2)}\le \frac{1}{r^n}{M}_p(r,f_z)\le 1$$

(8)

for all $r:0<r<1$, and for all $p:$ $0\le p\le \infty$, where

$$f_z\left(w\right)=\frac{f\circ {\phi }_z\left(w\right)-f\left(z\right)}{1-\overline{f\left(z\right)}f\circ {\phi }_z\left(w\right)},w\in D.$$

(9)

Equality in (7) holds at some point $w\in D$, $w\ne z$ if and only if

$$f\left(w\right)=\frac{\alpha {\left({\phi }_{-z}\left(w\right)\right)}^n+c_0}{1+\alpha \overline{c_0}{\left({\phi }_{-z}\left(w\right)\right)}^n},w\in D$$

(10)

with $\left|\alpha \right|=1$.

Equality in the first inequality or in the second inequality of (8) holds for some $r:0<r<1$ and $p:0<p\le \infty$ if and only if f is of the form (10) (with $\left|\alpha \right|\le 1$ or $\left|\alpha \right|=1$, respectively).

Remark 1.

(1) The case $n=1$ of Theorem 4 should reduce to Theorem 1. Comparing (3) and (10), there should exist $z^{\prime }\in D$ and $\beta :\left|\beta \right|=1$ for which

$$\frac{\alpha {\phi }_{-z}\left(w\right)+c_0}{1+\alpha \overline{c_0}{\phi }_{-z}\left(w\right)}=\beta \frac{w-z^{\prime }}{1-\overline{z^{\prime }}w}$$

(11)

for all $w\in D$. This can be verified as follows:

Since any automorphism, i.e., bijective holomorphic mapping, of D is of the form of the right-hand side of (11), it suffices to show that the left-hand side of (11), denote $\mathrm{\Phi }\left(w\right)$, is an automorphism of D. That $\mathrm{\Phi }\left(w\right)$ is holomorphic and into D is obvious. We show $\mathrm{\Phi }\left(w\right)$ is bijective: If $\mathrm{\Phi }\left(w_1\right)=\mathrm{\Phi }\left(w_2\right)$, then ${\phi }_{-z}\left(w_1\right)={\phi }_{-z}\left(w_2\right)$, and the injectivity of ${\phi }_{-z}$ shows $w_1=w_2$. Thus, $\mathrm{\Phi }\left(w\right)$ is injective. Next, for any $\zeta \in D$, by the surjectivity of ${\phi }_{c_0}$, there exists $\eta \in D$ such that

$$\frac{\alpha \eta +c_0}{1+\alpha \overline{c_0}\eta }=\zeta .$$

For this η, there is $\xi \in D$ such that $\eta ={\phi }_{-z}\left(\xi \right)$, whence $\mathrm{\Phi }\left(w\right)$ is surjective.

(2) Fix $z\in D$ and self-map f of D. Then, applying Littlewood’s inequality (see [3,5,6]), it follows that

$$\left|\frac{f\left(w\right)-f\left(z\right)}{1-\overline{f\left(z\right)}f\left(w\right)}\right|\le \underset{j}{\mathrm{\Pi }}\left\{|z_j|:f\left(\frac{w+z_j}{1+{\overline{z}}_{j}w}\right)=f\left(z\right)\right\}=\underset{f\left(z_j\right)=f\left(z\right)}{\mathrm{\Pi }}\left|\frac{w-z_j}{1-\overline{z}jw}\right|,$$

(12)

with equality holding only if f is an inner function. Equation (7) follows directly from (12).

In addition, the inequality

$$\frac{{(1-{\left|z\right|}^2)}^n\left|f^{\left(n\right)}\left(z\right)\right|}{n!(1-{\left|f\right(z\left)\right|}^2)}\le 1$$

of (8) can be obtained as a one stroke limit from (7):

$$\begin{array}{cc}\hfill 1\ge \left|\frac{f\left(w\right)-f\left(z\right)}{1-\overline{f\left(z\right)}f\left(w\right)}\right|/{\left|\frac{w-z}{1-\overline{z}w}\right|}^n& =\left|\frac{f\left(w\right)-f\left(z\right)}{{(w-z)}^n}\right|\left|\frac{{(1-\overline{z}w)}^n}{1-\overline{f\left(z\right)}f\left(w\right)}\right|\hfill \\ & ⟶\frac{|f^{\left(n\right)}\left(z\right)|{(1-{\left|z\right|}^2)}^n}{n!(1-{\left|f\right(z\left)\right|}^2)}\hfill \end{array}$$

as $w\to z$ (by applying L’Hospital’s rule).

The point of Theorem 4 lies in its connection with $M_p(r,·)$ and in clarifying the condition of equality to make Yamashita type theorem complete.

After proving Theorem 4 in Section 3, applications of Theorem 4 to some coefficient problems will be given in Section 4.

3. Proof of Theorem 4

We may assume $c_n\ne 0.$ (7) can be expressed as

$$|{\phi }_{-f\left(z\right)}\circ f\left(w\right)|\le {|{\phi }_{-z}\left(w\right)|}^n,w\in D.$$

By (6), $f\left(w\right)-f\left(z\right)$ has a zero of order n at $w=z$ so that

$$\frac{{\phi }_{-f\left(z\right)}\circ f\left(w\right)}{{\phi }_{-z}{\left(w\right)}^n},w\in D$$

(13)

is holomorphic in D whose modulus at $w\in \partial D$ is not greater than 1, so that the maximum principle gives (7).

Next, to verify inequality (8), take $\delta >0$ such that (6) holds for $w:|w-z|<\delta .$ Then, by (6),

$$\begin{array}{cc}\hfill f\circ {\phi }_z\left(w\right)-f\left(z\right)=& c_n{({\phi }_z\left(w\right)-z)}^n+\cdots \hfill \\ \hfill =& c_n{\left(\frac{w}{1+\overline{z}w}\right)}^n{(1-{\left|z\right|}^2)}^n+O(w^{n+1})\hfill \end{array}$$

(14)

for $w:\left|w\right|<\frac{\delta }{1+\left|z\right|}.$ This is because

$$\left|w\right|<\frac{\delta }{1+\left|z\right|}\Rightarrow \left|w\right|<\frac{\delta |1+\overline{z}w|}{1-{\left|z\right|}^2}\Rightarrow |{\phi }_z\left(w\right)-z|<\delta .$$

Thus, $f_z\left(w\right)$ defined by (9) has a zero of order n at $w=0$. Hence,

$$h\left(w\right)=\frac{1}{w^n}{f}_z\left(w\right),w\in D,$$

(15)

is holomorphic in D. Since $h\left(0\right)\ne 0$ in a neighborhood of 0, $log\left|h\right|$ is harmonic in the neighborhood, hence there exists $r_0$ such that

$$\left|h\left(0\right)\right|=exp\left({\int }_{-\pi }^{\pi }log\left|h\right(r{e}^{i\theta }\left)\right|\frac{d\theta }{2\pi }\right)$$

(16)

for $r:r<r_0.$

On the other hand, by (15),

$$n!h\left(0\right)=\frac{d^n}{d{w}^n}\left(w^{n}h\left(w\right)\right){|}_{w=0}=\frac{d^n}{d{w}^n}{f}_z\left(w\right){|}_{w=0}.$$

(17)

In order to calculate the final term of (17), let’s put $F\left(w\right)=f\circ {\phi }_z\left(w\right)-f\left(z\right)$ and $G\left(w\right)=1-\overline{f\left(z\right)}f\circ {\phi }_z\left(w\right).$ Then,

$$\frac{d^n}{d{w}^n}{f}_z\left(w\right){|}_{w=0}=\sum _{j=0}^n\left(\genfrac{}{}{0pt}{}{n}{j}\right)F^{\left(j\right)}\left(0\right){\left(G^{-1}\right)}^{(n-j)}\left(0\right).$$

By (14),

$$F^{\left(j\right)}\left(0\right)=\{\begin{array}{cc}0,& \mathrm{if}j<n,\hfill \\ c_{n}n!{(1-{\left|z\right|}^2)}^n,& \mathrm{if}j=n,\hfill \end{array}$$

so that

$$\frac{d^n}{d{w}^n}{f}_z\left(w\right){|}_{w=0}=\frac{F^{\left(n\right)}\left(0\right)}{G\left(0\right)}=\frac{c_{n}n!{(1-{\left|z\right|}^2)}^n}{1-{\left|f\right(z\left)\right|}^2},$$

whence

$$h\left(0\right)=\frac{c_n{(1-{\left|z\right|}^2)}^n}{1-{\left|f\right(z\left)\right|}^2}.$$

(18)

Noting from (6) that $c_n=\frac{f^{\left(n\right)}\left(z\right)}{n!}$, we have, by (15), (16) and (18),

$$\frac{{(1-{\left|z\right|}^2)}^n\left|f^{\left(n\right)}\left(z\right)\right|}{n!(1-{\left|f\right(z\left)\right|}^2)}=exp\left({\int }_{-\pi }^{\pi }log\left|\frac{1}{r^n}{f}_z\left(r{e}^{i\theta }\right)\right|\frac{d\theta }{2\pi }\right)=\frac{1}{r^n}{M}_0(r,f_z)$$

(19)

for $r<r_0$.

Now, the first inequality of (8) follows from the fact that $M_p(r,h)$ is an increasing function of $p:0\le p\le \infty$ and also an increasing function of $r:0<r<1$.

In addition, since $M_p(r,h)\le M_{\infty }(r,h)$ and $\left|h\right|<1$ by the maximum principle, the second inequality of (8) follows.

We next check the conditions of equality. Elementary calculation shows that

$$\frac{{\phi }_{-f\left(z\right)}\circ f\left(w\right)}{{\phi }_{-z}{\left(w\right)}^n}=\alpha \iff f\left(w\right)=\frac{\alpha {\left({\phi }_{-z}\left(w\right)\right)}^n+c_0}{1+\alpha \overline{c_0}{\left({\phi }_{-z}\left(w\right)\right)}^n}\iff f_z\left(w\right)=\alpha w^n.$$

(20)

Thus, if equality in (7) holds, at some point $w\in D$, $w\ne z$; then, (13) is a constant function of modulus 1 by virtue of the maximum principle, which gives (10) with $\left|\alpha \right|=1$ by (20). To see that $f\left(w\right)$ of (10) with $\left|\alpha \right|=1$ gives the equality in (7) is straightforward also by (20).

If, for some p, $0<p\le \infty$, and for some $r:0<r<1$ the first inequality of (8) becomes equality, then, by (19), $M_0(r,h)=M_q(r,h)$ for $0\le q\le p$, so that $\left|h\right(r\zeta \left)\right|$=constant, a.e. $\zeta \in \partial D$. Since h is holomorphic and $\left|h\right(0\left)\right|=\left|h\right(\rho \zeta \left)\right|$=constant, a.e. $\zeta \in \partial D$ for $\rho \le r$, it follows that h is a constant function. Letting $h=\alpha$ with $\left|\alpha \right|\le 1$ and solving this, as in (20), gives (10).

Finally, suppose the second inequality of (8) becomes equal for some ${\rho }_0:0<{\rho }_0<1$ so that $M_p({\rho }_0,h)=\frac{1}{{\rho }^n}{M}_p({\rho }_0,f_z)=1$. Then, $M_p(\rho ,h)=1$ for $\rho :{\rho }_0\le \rho <1$. Since $log{M}_p(\rho ,h)$ is a convex function of $\rho$ (see [3]) and $log{M}_p(\rho ,h)=0$ for ${\rho }_0\le \rho <1,$ it follows that $log{M}_p(\rho ,h)\ge 0$ for $\rho \le {\rho }_0$ whence $log{M}_p(\rho ,h)=0$ for all $\rho :0<\rho <1.$ Thus,

$$h\left(0\right)=\underset{p\to 0}{lim}{M}_p(\rho ,f)=\underset{p\to 0}{lim}{e}^{log{M}_p(\rho ,h)}=1.$$

Since h maps D into D, this forces, by the maximum principle, that $h\left(w\right)$ is a constant, $h\left(w\right)=\alpha$, with $\left|\alpha \right|=1$. Hence, (20) gives (10).

Conversely, by (20), $f\left(w\right)$ of (10) with $\left|\alpha \right|\le 1$ makes h in (15) constant, so that the two inequalities in (8) become equalities.

4. Applications

Theorem 4 immediately gives the following estimate:

Corollary 1.

Let f be a function holomorphic and bounded, $\left|f\right|<1$, in D and let $z\in D$. If

$$f\left(w\right)=c_0+c_n{(w-z)}^n+c_{n+1}{(w-z)}^{n+1}+\cdots .$$

in a neighborhood of z, then

$$|c_n|\le \frac{1-{|c_0|}^2}{r^n{(1-{\left|z\right|}^2)}^n}{M}_p(r,f_z)\le \frac{1-{|c_0|}^2}{{(1-{\left|z\right|}^2)}^n}$$

(21)

for all $r:0<r<1$, and for all $p:$ $0\le p\le \infty$, where $f_z$ is defined by (9).

Equality in the first inequality or in the second inequality of (21) holds for some $r:0<r<1$ and $p:0<p\le \infty$ if and only if f is of the form (10) (with $\left|\alpha \right|\le 1$ or $\left|\alpha \right|=1$, respectively).

For the case $z=0$, we can also obtain

Corollary 2.

Let f be a function holomorphic and bounded, $\left|f\right|<1$, in D and let $z\in D$. If

$$f\left(w\right)=c_0+c_{1}w+c_2{w}^2+\cdots +c_n{w}^n+\cdots ,w\in D,$$

then

$$|c_n|\le (1-{|c_0|}^2)\frac{M_p(r,g_0)}{r^n}\le 1-{|c_0|}^2$$

(22)

for all $r:0<r<1$, and for all $p:$ $0\le p\le \infty$, where

$$g_0\left(w\right)=\frac{g\left(w\right)-g\left(0\right)}{1-\overline{g\left(0\right)}g\left(w\right)},w\in D$$

with

$$g\left(w\right)=\frac{1}{n}\sum _{k=1}^{n}f\left(e^{i2\pi k/n}w\right),w\in D.$$

Equality in the first inequality or in the second inequality of (22) holds for some $r:0<r<1$ and $p:0<p\le \infty$ if and only if g is of the form

$$g\left(w\right)=\frac{\alpha w^n+c_0}{1+\alpha \overline{c_0}{w}^n},w\in D$$

(23)

(with $\left|\alpha \right|\le 1$ or $\left|\alpha \right|=1$, respectively).

Proof of Corollary 2.

As was frequently used (see [7] for example), we make use of the facts that ${\sum }_{k=1}^n{e}^{i2\pi jk/n}=n$ if j is a multiple of n, and 0 if j is otherwise. Noting that

$$g\left(w\right)=\frac{1}{n}\sum _{k=1}^{n}f(e^{i2\pi k/n}w)=c_0+c_n{w}^n+c_{2n}{w}^{2n}+\cdots ,w\in D$$

is holomorphic and $\left|g\right|<1$ in D, by Corollary 1 with $z=0$, we have

$$|c_n|\le \frac{1-{|c_0|}^2}{r^n}{M}_p(r,g_0)\le 1-{|c_0|}^2$$

for all $r:0<r<1$, and for all $p:$ $0\le p\le \infty$.

In addition, by Corollary 1, equality in the first inequality or in the second inequality of (22) holds for some $r:0<r<1$ and $p:0<p\le \infty$ if and only if g is of the form (23) (with $\left|\alpha \right|\le 1$ or $\left|\alpha \right|=1$, respectively). □

5. Conclusions

With imperative applications to particular situations, various forms of Schwarz Lemma have been called for. In this paper, we presented Schwarz-Pick Lemma for higher derivatives in connection with p-mean $M_p(r,f)$(see Theorem 4). It refined a previous result of Shinji Yamashita and clarified the condition of equality. As an immediate consequence, the result could be applied to refine well-known estimates for n-th Taylor coefficient of holomorphic self maps of D (see Corollary 1 and 2). We are expecting its further extensions and applications.