A SHARP SCHWARZ LEMMA AT THE BOUNDARY

Abstract

In this paper, a boundary version of Schwarz lemma is investigated. For the function holomorphic f(z) = a + c_pz^p + c_p+₁z^p+1 + ... defined in the unit disc satisfying |f(z) − 1| < 1, where 0 < a < 2, we estimate a module of angular derivative at the boundary point b, f(b) = 2, by taking into account their first nonzero two Maclaurin coefficients. The sharpness of these estimates is also proved.

1. INTRODUCTION

In recent years, a boundary version of Schwarz lemma was investigated in D. M. Burns and S. G. Krantz [8], R. Osserman [11], V. N. Dubinin [3, 4], M. Jeong [6, 7], H. P. Boas [1], X. Tang, T. Liu and J. Lu [10], D. Chelst [2], X. Tang and T. Liu [9] and other’s studies.

The classical Schwarz lemma states that an holomorphic function f mapping the unit disc D = {z : |z| < 1} into itself, with f(0) = 0, satisfies the inequality |f(z)| ≤ |z| for any point z ∈ D and |f'(0)| ≤ 1. Equality in these inequalities (in the first one, for z ≠ 0) occurs only if f(z) = λz, |λ| = 1 [5, p.329].

Let f(z) = a + cpzp + cp+1zp+1 + ..., cp ≠ 0, be a holomorphic function in the unit disc D, and |f(z) − 1| < 1 for |z| < 1, where 0 < a < 2. Consider the functions

and

Then h(z) and φ(z) are holomorphic functions on D with |h(z)| < 1 and |φ(z)| < 1 for |z| < 1 and φ(0) = 0. Therefore, from the Schwarz lemma, we obtain

and

The equality in (1.1) for some nonzero z ∈ D or in (1.2) holds if and only if

where θ is a real number (see [6]).

It is an elementary consequence of Schwarz lemma that if f extends continuously to some boundary point b with |b| = 1, and if |f(b)| = 1 and f'(b) exists, then |f'(b)| ≥ 1, which is known as the Schwarz lemma on the boundary.

R. Osserman [11] considered the case that only one boundary fixed point of f is given and obtained a sharp estimate based on the values of the function. He has first showed that

and

under the assuumption f(0) = 0 where f is a holomorphic function mapping the unit disc into itself and b is a boundary point to which f extends continuously and |f(b)| = 1. In addition, the equality in (1.3) holds if and only if f is of the form

where θ is a real number and α ∈ D satisfies argα = arg b. Also, the equality in (1.4) holds if and only if f(z) = zeiθ, where θ is a real number.

If, in addition, the function f has an angular limit f(b) at b ∈ ∂D, |f(b)| = 1, then by the Julia-Wolff lemma the angular derivative f'(b) exists and 1 ≤ |f'(b)| ≤ ∞ (see [14]).

Inequality (1.4) and its generalizations have important applications in geometric theory of functions (see, e.g., [5], [14]). Therefore, the interest to such type results is not vanished recently (see, e.g., [1], [3], [4], [6], [7], [9], [10], [11], [13] and references therein).

Vladimir N. Dubinin [2] has continued this line of research and has made a refinement on the boundary Schwarz lemma under the assumption that f(z) = cpzp + cp+1zp+1 + ..., with a zero set {ak}.

X. Tang, T. Liu and J. Lu [10] established a new type of the classical boundary Schwarz lemma for holomorphic self-mappings of the unit polydisk Dn in ℂn . They extended the classical Schwarz lemma at the boundary to high dimensions.

Also, M. Jeong [6] showed some inequalities at a boundary point for different form of holomorphic functions and found the condition for equality.

 

2. MAIN RESULTS

In this section, we can obtain more general results on the angular derivatives of holomorphic function on the unit disc at boundary by taking into account cp, cp+1 and zeros of f(z)−a if we know the first and the second coefficient in the expansion of the function f(z) = a + cpzp + cp+1zp+1 + .... We obtain a sharp lower bound of |f'(b)| at the point b, where |b| = 1.

Theorem 2.1. Let f(z) = a+cpzp+cp+1zp+1+..., cp ≠ 0 be a holomorphic function in the unit disc D, and |f(z) − 1| < 1 for |z| < 1, where 0 < a < 2. Assume that, for some b ∈ ∂D, f has an angular limit f(b) at b, f(b) = 2. Then

Moreover, the equality in (1.5) occurs for the function

Proof. Consider the functions

φ(z) and B(z) are holomorphic functions in D, and |φ(z)| < 1, |B(z)| < 1 for |z| < 1. By the maximum principle for each z ∈ D, we have

Therefore,

is holomorphic function in D and |p(z)| ≤ 1 for |z| < 1. In particular, we have

and

Moreover, it can be seen that for b ∈ ∂D

The function

is holomorphic in the unit disc D, |ϕ(z)| < 1, ϕ(0) = 0 and |ϕ(b)| = 1 for b ∈ ∂D. From (1.3), we obtain

and since

we obtain

Since h(0) = f(0) − 1 = a − 1 and h(b) = 1 for b ∈ ∂D,

So, we take the inequality (1.5).

Now, we shall show that the inequality (1.5) is sharp. Let

Then

and

Also,

and

Since |cp| = a(2 − a), (1.5) is satisfied with equality.                               ☐

If f(z)−a have no zeros different from z = 0 in Theorem 2.1, the inequality (1.5) can be further strengthened. This is given by the following Theorem.

Theorem 2.2. Let f(z) = a+cpzp+cp+1zp+1+..., cp > 0 be a holomorphic function in the unit disc D, f(z) − a has no zeros in D except z = 0 and |f(z) − 1| < 1 for |z| < 1, where 0 < a < 2. Assume that, for some b ∈ ∂D, f has an angular limit f(b) at b, f(b) = 2. Then

and

In addition, the equality in (1.7) occurs for the function

and the equality in (1.8) occurs for the function

where 0 < cp < 1 and ln .

Proof. Let cp > 0. Let φ(z) and p(z) be as in the proof of Theorem2.1. Having inequality (1.6) in mind , we denote by ln p(z) the holomorphic branch of the logarithm normed by the condition

The composite function

is holomorphic in the unit disc D, |Φ(z)| < 1 for |z| < 1, Φ(0) = 0 and |Φ(b)| = 1 for b ∈ ∂D. From (1.3), we obtain

and

It can be seen that

and

Therefore, we obtain

Replacing arg2 p(b) by zero, we take

and we obtain (1.7) with an obvious equality case. Similary, function Φ(z) satisfies the assumptions of the Schwarz lemma, we obtain

and

Therefore, we have the inequality (1.8).

Now, we shall show that the inequality (1.8) is sharp. Let

where

Then

Under the simple calculations, we take

Therefore, we obtain

                              ☐

If f(z) − a have zeros different from z = 0, taking into account these zeros, the inequality (1.5) can be strengthened in another way. This is given by the following Theorem.

Theorem 2.3. Let f(z) = a+cpzp+cp+1zp+1+..., cp ≠ 0 be a holomorphic function in the unit disc D, and |f(z) − 1| < 1 for |z| < 1, where 0 < a < 2. Assume that, for some b ∈ ∂D, f has an angular limit f(b) at b, f(b) = 2. Let a1, a2, ..., an be zeros of the function f(z) − a in D that are different from zero. Then we have the inequality

In addition, the equality in (1.9) occurs for the function

where a1, a2, ..., an are positive real numbers.

Proof. Let φ(z) be as in the proof of Theorem2.1 and a1, a2, ..., an be zeros of the function f(z) − a in D that are different from zero.

is a holomorphic function in D and |B0(z)| < 1 for |z| < 1. By the maximum principle for each z ∈ D, we have

The function

is holomorphic in D and |k(z)| ≤ 1 for |z| < 1. In particular, we have

and

Moreover, it can be seen that

Besides, with the simple calculations, we take

The auxiliary function

is holomorphic in the unit disc D, , and for b ∈ ∂D. From (1.3), we obtain

Since

and

we take

Thus, we take the inequality (1.9) with an obvious equality case.                               ☐