1. INTRODUCTION

In recent years, boundary version of Schwarz lemma was investigated in D. M. Burns and S. G. Krantz [6], R. Osserman [8], V. N. Dubinin [2], M. Jeong [4, 5], H. P. Boas [1] and other’s studies. On the other hand, in the book [7], Sharp Real-Parts Theorem’s (in particular Carathéodory’s inequalities), which are frequently used in the theory of entire functions and analytic function theory, have been studied.

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 [3, p.329]. It is an elementary consequence of Schwarz lemma that if f extends continuously to some boundary point z0 with |z0| = 1, and if |f(z0)| = 1 and f'(z0) exists, then |f'(z0)| ≥ 1, which is known as the Schwarz lemma on the boundary.

In this paper, we studied “boundary Carathéodory’s inequalities” as analog the Schwarz lemma at the boundary [8].

The Carathéodory’s inequality states that, if the function f is holomorphic on the unit disc D with f(0) = 0 and ℜf ≤ A in D, then the inequality

holds for all z ∈ D, and moreover

Equality is achieved in (1.1) (for some nonzero z ∈ D ) or in (1.2) if and only if f(z) is the function of the form

where θ is a real number [7, pp.3-4].

Robert Osserman 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 z0 is a boundary point to which f extends continuously and |f(z0)| = 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 z0. Also, the equality in (1.4) holds if and only if f(z) = zeiθ, where θ is a real number.

Moreover, if then

It follows that

with equality only if f is of the form f(z) = zpeiθ, θ real [8].

If, in addition, the function f has an angular limit f(z0) at z0 ∈ ∂D, |f(z0)| = 1, then by the Julia-Wolff lemma the angular derivative f'(z0) exists and 1 ≤ |f'(z0)| ≤ ∞ (see [11]).

The inequality (1.5) is a particular case of a result due to Vladimir N. Dubinin in (see [2]), who strengthened the inequality |f'(z0)| ≥ 1 by involving zeros of the function f. Some other types of strengthening inequalities are obtained in (see [9], [10]).

We have following results, which can be offered as the boundary refinement of the Carathéodory’s inequality.

Theorem 1.1. Let f be a holomorphic function in the unit disc D, f(0) = 0 and for |z| < 1. Further assume that, for some z0 ∈ ∂D, f has an angular limit f(z0) at z0, ℜf(z0) = A. Then

Moreover, the equality in (1.7) holds if and if

wehere θ is a real number.

Proof. The function

is holomorphic in the unit disc D, |ϕ(z)| < 1, ϕ(0) = 0 and |ϕ(z0)| = 1 for z0 ∈ ∂D.

That is,

|f(z) − 2A|2 = |f(z)|2 − 2ℜ (f(z)2A) + 4A2 = |f(z)|2 − 4Aℜ (f(z)) + 4A2.

From the hypothesis, since ℜf(z) ≤ A and 4Aℜf(z) ≤ 4A2, we take

|2A − f(z)|2 ≥ |f(z)|2 − 4Aℜf(z) + 4Aℜf(z) = |f(z)|2.

Therefore, we obtain

From (1.4), we obtain

So, we take

If from (1.9) and |ϕ'(z0)| = 1, we obtain

Theorem 1.2. Let f be a holomorphic function in the unit disc D, f(0) = 0 and for |z| < 1. Further assume that, for some z0 ∈ ∂D, f has an angular limit f(z0) at z0, ℜf(z0) = A. Then

The inequality (1.10) is sharp, with equality for the function

where is an arbitrary number on [0, 1] (see (1.2)).

Proof. Using the inequality (1.3) for the function (1.8), we obtain

and

Now, we shall show that the inequality (1.10) is sharp. Choose arbitrary a ∈ [0, 1]

Let

Then

and

Since |f'(0)| = 2Aa, (1.10) is satisfied with equality.

An interesting special case of Theorem1.2 is when f''(0) = 0, in which case inequality (1.10) implies Clearly equality holds for θ real.

Now, if is a holomorphic function in the unit disc D and for |z| < 1, it can be seen that Carathéodory’s inequality can be obtained with standard methods as follows:

and

The following result is a generalization of Theorem1.1.

Theorem 1.3. Let cp ≠ 0, p ≥ 1 be a holomorphic function in the unit disc D and for |z| < 1. Further assume that, for some z0 ∈ ∂D, f has an angular limit f(z0) at z0, ℜf(z0) = A. Then

In addition, the equality in (1.12) holds if and if

wehere θ is a real number.

Proof. Using the inequality (1.6) for the function (1.8), we obtain

Therefore, we take

If from (1.13) and |ϕ' (z0)| = p, we obtain

Theorem 1.4. Under hypotheses of Theorem1.3, we have

The inequality (1.14) is sharp, with equality for the function

where is arbitrary number from [0, 1] (see (1.11)).

Proof. Using the inequality (1.5) for the function (1.8), we obtain

where Since

we may write

Thus, we take

The equality in (1.14) is obtained for function

as show simple calculations.

Consider the following product:

B(z) is called a finite Blaschke product, where . Let the function satisfy the conditions of Carathéodory’s inequality and also have zeros a1, a2, ..., an with order k1, k2, . . . , kn, respectively. Thus, one can see that Carathéodory’s inequality can be strengthened with the standard methods as follows:

and

The inequalities (1.15) and (1.16) show that the inequalities (1.1) and (1.2) will be able to be strengthened, if the zeros of function which are different from origin of f(z) in the (1.12) and (1.14) are taken into account.

Theorem 1.5. Let cp ≠ 0, p ≥ 1 be a holomorphic function in the unit disc D, and for |z| < 1. Assume that for some z0 ∈ ∂D, f has an angular limit f(z0) at z0, ℜf(z0) = A. Let a1, a2, ..., an be zeros of the function f in D that are different from zero. Then we have the inequality

In addition, the equality in (1.17) occurs for the function

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

Proof. Let ϕ(z) be as in the proof of Theorem1.1 and a1, a2, ..., an be zeros of the function f in D that are different from zero.

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

|ϕ(z)| ≤ |B(z)| .

The auxiliary function

is holomorphic in D, and |φ(z)| < 1 for |z| < 1, φ(0) = 0 and |φ(z0)| = 1 for z0 ∈ ∂D.

Moreover, it can be seen that

Besides, with the simple calculations, we take

From (1.5), we obtain

where Since

we may write

Therefore, we have

Now, we shall show that the inequality (1.17) is sharp. Let

Then

and

Since a1, a2, ..., an are positive real numbers, we take

Since (1.17) is satisfied with equality.