AN IMPROVED LOWER BOUND FOR SCHWARZ LEMMA AT THE BOUNDARY

Abstract

In this paper, a boundary version of the Schwarz lemma for the holom- rophic function satisfying f(a) = b, |a| < 1, b ∈ ℂ and ℜf(z) > α, 0 ≤ α < |b| for |z| < 1 is invetigated. Also, we estimate a modulus of the angular derivative of f(z) function at the boundary point c with ℜf(c) = a. The sharpness of these inequalities is also proved.

1. INTRODUCTION

The classical Schwarz lemma gives information about the behavior of a holomorphic function on the unit disc D = {z : |z| <1} at the origin, subject only to the relatively mild hypotheses that the function map the unit disc to the disc and the origin to the origin. This lemma, named after Hermann Amandus Schwarz, is a result in complex analysis about holomorphic functions defined on the unit disc. In its must basic form, the familiar Schwarz lemma says this ([5], p.329):

Let D be the unit disc in the complex plane ℂ. Let f : D → D be a holomorphic function with f(0) = 0. Under these circumstances |f(z)| ≤ |z| for all z ∈ D, and |f′(0)| ≤ 1. In addition, if the equality |f(z)| = |z| holds for any z ≠ 0, or |f′(0)| = 1 then f is a rotation, that is, f(z) = zeiθ, θ real.

For historical background about the Schwarz lemma and its applications on the boundary of the unit disc, we refer to (see [1], [19]).

Let f(z) be holomorphic function in D, f(a) = b, |a| < 1, b ∈ ℂ and ℜf(z) > α, 0 ≤ α < |b| for |z| < 1.

Let

and

F(z) is holomorphic and ℜf(z) > 0 for |z| < 1 and hence

is holomorphic, |ϕ(z)| < 1 for |z| < 1 and ϕ(0) = 0. Thus, by Schwarz lemma, we obtain

The inequality in (1:1) is sharp with equality for the function

where −1 < a ≤ 0 and b is any integer ≥ 1.

It is an elementary consequence of Schwarz lemma that if f extends continuously to some boundary point to with |c| = 1, and if |f(c)| = 1 and f'(c) exists, then |f'(c)| ≥ 1. This result of Schwarz lemma and its generalization are described as Schwarz lemma at the boundary in the literature. This improvement was obtained in [20] by Helmut Unkelbach, and rediscovered by R. Osserman in [15] 60 years later.

In the last 15 years, there have been tremendous studies on Schwarz lemma at the boundary (see, [1], [3], [4], [6], [7], [9], [10], [15], [16], [17], [19] and references therein). Some of them are about the below boundary of modulus of the functions derivation at the points (contact points) which satisfies |f(c)| = 1 condition of the boundary of the unit circle.

In [15], R. Osserman offered the following boundary refinement of the classical Schwarz lemma. It is very much in the spirit of the sort of result.

Lemma 1.1. Let f : D → D be holomorphic function with f(0) = 0. Assume that there is a c ∈ ∂D so that f extends continuously to c, |f(c)| = 1 and f'(c) exists.

Then

Inequality (1.2) is sharp, with equality possible for each value of |f'(0)|.

Corollary 1.2. Under the hypotheses Lemma 1.1, we have

and

|f'(c)| > 1 unless f(z) = zeiθ, θ real.

Moreover, if f(z) = apzp + ap+1zp+1...., then

The equality in (1.4) occurs for the function f(z) = zp (z + γ) = (1 + γz) ; 0 ≤ γ ≤ 1.

Lemma 1.3 (Julia-Wolff lemma). Let f be a holomorphic function in D, f(0) = 0 and f(D) ⊂ D. If, in addition, the function f has an angular limit f(b) at b ∈ ∂D, |f(b)| = 1, then the angular derivative f'(b) exists and 1 ≤ |f'(b)| ≤ ∞ (see [18]).

D. M. Burns and S. G. Krantz [8] and D. Chelst [2] studied the uniqueness part of the Schwarz lemma. In M. Mateljević’s papers, for more general results and related estimates, see also ([11], [12], [13] and [14]).

Also, M. Jeong [7] showed some inequalities at a boundary point for different form of holomorphic functions and found the condition for equality and in [6] a holomorphic self map defined on the closed unit disc with fixed points only on the boundary of the unit disc.

 

2. MAIN RESULTS

Theorem 2.1. Let f(z) be holomorphic function in D, f(a) = b, |a| < 1, b ∈ ℂ and ℜf(z) > α, 0 ≤ α < |b| for |z| < 1. Assume that, for some c ∈ ∂D, f has an angular limit f (c) at c, ℜf (c) = α. Then

The inequality (2.1) is sharp, with equality for the function

where −1 < a ≤ 0 and b is any integer ≥ 1.

Proof. Consider the function

Then ϕ(z) is holomorphic function in the unit disc D, ϕ(0) = 0 and |ϕ(z)| < 1. In addition, for ,

and since ℜf (c) = α, we take

From (1:3), we obtain

Since , we have

Thus, we get

Now, we shall that the inequality (2.1) is sharp. Let

Then, we take

Since −1 < a ≤ 0 and b is any integer ≥ 1, (2.1) is satisfied with equality. □

Theorem 2.2. Let f(z) be holomorphic function in D, f(a) = b, |a| < 1, b ∈ ℂ and ℜf(z) > α, 0 ≤ α < |b| for |z| < 1. Assume that, for some c ∈ ∂D, f has an angular limit f (c) at c, ℜf (c) = α. Then

The equality in (2.2) occurs for the function

where −1 < a ≤ 0, b is any integer ≥ 1 and is an arbitrary number from [0, 1] (see (1.1)).

Proof. Let ϕ(z) be the same as in the proof of Theorem 2.1. From (1.2), for , we obtain

Since

we have

Therefore, we take the inequality (2:2).

To show that the inequality (2.2) is sharp, take the holomorphic function

Then

Since −1 < a ≤ 0 and b is any integer ≥ 1, (2.2) is satisfied with equality. □

If f(z) = b + ap (z − a)p + ap+1 (z − a)p+1 + ... is a holomorphic function in D f(a) = b, |a| < 1, b ∈ ℂ and ℜf(z) > α, 0 ≤ α < |b| for |z| < 1, then

Theorem 2.3. Let f(z) = b + ap (z − a)p + ap+1 (z − a)p+1 + ... is a holomorphic function in D f(a) = b, |a| < 1, b ∈ ℂ and ℜf(z) > α, 0 ≤ α < |b| for |z| < 1. Assume that, for some c ∈ ∂D, f has an angular limit f (c) at c, ℜf (c) = α. Then

Equality in (2.4) occurs for the function

where −1 < a ≤ 0, b is any integer ≥ 1 and is an arbitrary number from [0, 1] (see (2.3)).

Proof. Using the inequality (1.4) for the function ϕ(z), for , we obtain

Also, since

we may write

Thus, we take the inequality (2.4).

The equality (2.4) is obtained for the function

as show simple calculations.

In Theorem 2.3, the inequality (2.4) is obtained by adding the term ap of f(z) function. In the following theorem, the inequality (2.4) is obtained by adding ap and ap+1 that are consecutive terms of f(z) function.

Theorem 2.4. Let f(z) = b + ap (z − a)p + ap+1 (z − a)p+1 + ... , p ≥ 1 is a holomorphic function in D f(a) = b, |a| < 1, b ∈ ℂ and ℜf(z) > α, 0 ≤ α < |b| for |z| < 1. Assume that, for some c ∈ ∂D, f has an angular limit f (c) at c, ℜf (c) = α. Then

In addition, the equality in (2.5) occurs for the function

where −1 < a ≤ 0 and b is any integer ≥ 1.

Proof. Let ϕ(z) be the same as in the proof of Theorem 2.1. Also, let

We know that, from the maximum principle for each z ∈ D, we have |ϕ(z)| ≤ |zp|. Thus, k(z) is holomorphic function in D and |k(z)| < 1 for |z| < 1. In particular, we have

and

Moreover, we can see that, for ,

where ℏ(z) = zp.

The composite function

satisfies the assumptions of the Schwarz lemma on the boundary, whence we obtain

Because

and

we take

Therefore, we have

Now, we shall that the inequality (2.5) is sharp.

Consider the function

Then

and

Since −1 < a ≤ 0 and b is any integer ≥ 1, is satisfied with equality.

Also, because , (2.5) is satisfied with equality. □

If f(z) − b has no zeros different from z = a in Theorem 2.4, the inequality (2.5) can be further strengthened. This is given by the following Theorem.

Theorem 2.5. Let f(z) = b + ap (z − a)p + ap+1 (z − a)p+1 + ... p ≥ 1, ap > 0 is a holomorphic function in D f(a) = b, |a| < 1, b ∈ ℂ and ℜf(z) > α, 0 ≤ α < |b| for |z| < 1. and f(z) − b has no zeros in D except z = a. Assume that, for some c ∈ ∂D, f has an angular limit f (c) at c, ℜf (c) = α. Then

The equality in (2.7) occurs for the function

where −1 < a ≤ 0 and b is any integer ≥ 1.

Proof. Let ap > 0. Having in the mind inequality (2.6), we denote by ln k(z) the holomorphic branch of the logarithm normed by the condition

The auxiliary function

satisfies the assumptions of the Schwarz lemma on the boundary and so, for , we obtain

Since

and replacing arg2 k(z0) by zero, we take

Thus, we obtain the inequality (2.7).

The equality in (2.7) is obtained for the function

by simple calculations. □

Theorem 2.6. Under the same assumptions as in Theorem 2.5, we have

The equality in (2.8) occurs for the function

where −1 < a ≤ 0 and b is any integer ≥ 1.

Proof. Using the inequality (1.3) for the function Φ(z), we obtain

Replacing arg2 k(z0) by zero, we take

Therefore, we obtain the inequality (2.8) with an obvious equality case. □