SOME ESTIMATES FOR HOLOMORPHIC FUNCTIONS AT THE BOUNDARY OF THE UNIT DISC

In this paper, for holomorphic function f(z) = z + c2z + c3z + · · · belong to the class of N (λ), it has been estimated from below the modulus of the angular derivative of the function zf ′(z) f(z) on the boundary point of the unit disc.

The equality in (1.3) occurs for the function The following boundary version of the Schwarz lemma was proved in 1938 by Unkelbach in [21] and then rediscovered and partially improved by Osserman in [17].

The equality in (1.4) holds if and only if f is of the form
for some constant a ∈ (−1, 0].
with equality only if f is of the form where θ is a real number.
Vladimir N. Dubinin has continued this line and has made a refinement on the boundary Schwar lemma under the assumption that f (z) = c p z p + c p+1 z p+1 + · · · , with a zero set {z k } (see [5]).
S. G. Krantz and D. M. Burns [3] and D. Chelst [4] studied the uniqueness part of the Schwarz lemma. According to M. Mateljević's studies, some other types of results which are related to the subject can be found in ( [13,14] and [12]). In addition, [11] was posed on ResearchGate where is discussed concerning results in more general aspects.
Also, M. Jeong [10] showed some inequalities at a boundary point for different form of holomorphic functions and found the condition for equality and in [9] a holomorphic self map defined on the closed unit disc with fixed points only on the boundary of the unit disc.

Main Results
In this section, for holomorphic function f (z) = z + c 2 z 2 + c 3 z 3 + · · · belong to the class of N (λ), it has been estimated from below the modulus of the angular derivative of the function zf (z) f (z) on the boundary point of the unit disc.
Then we have the inequality The equality in (2.1) occurs for the function is a holomorphic function in the unit disc E and φ(0) = 0. From the Jack's lemma and since f (z) ∈ N (λ), we obtain |φ(z)| < 1 for |z| < 1. Also, we have |φ(b)| = 1 for b ∈ ∂E.
From (1.5), we obtain So, we take the inequality (2.1). Now, we shall show that the inequality (2.1) is sharp. Let Then, we have The inequality (2.2) is sharp with equality for the function where λ = β β+α . Proof. Let φ(z) be as in the proof of Theorem 2.1. Using the inequality (1.4) for the function φ(z), we obtain So, we obtain the inequality (2.2).
To show that the inequality (2.2) is sharp, take the holomorphic function Then and |h (i)| = λ. Since |c 2 | = 2λ is satisfied with equality. That is; Then we have the inequality

The inequality (2.3) is sharp with equality for the function
Proof. Let φ(z) be as in the proof of Theorem 2.1. By the maximum principle for each z ∈ E, we have |φ(z)| ≤ |z|. So, is a holomorphic function in E and |ψ(z)| < 1 for |z| < 1. For any real number µ = 1 λ that is not a non-negative integer Thus, we take Moreover, it can be seen that The function is a holomorphic in the unit disc E, |Φ(z)| < 1 for |z| < 1, Φ(0) = 0 and |Φ(b)| = 1 for b ∈ ∂E. From (1.4), we obtain Therefore, we obtain So, we obtain the inequality (2.3).
To show that the inequality (2.3) is sharp, take the holomorphic function Then Since |c 2 | = 2λ, (2.3) is satisfied with equality.
λ − 1 has no zeros different from z = 0 in Theorem 2.3, the inequality (2.3) can be further strengthened. This is given by the following theorem.

Moreover, the result is sharp and the extremal function is
where λ = β β+α . Proof. Let c 2 > 0 . Using the inequality (1.5) for the function Φ(z), we obtain Replacing arg 2 ϕ(b) by zero, then we have Thus, we obtain the inequality (2.6) with an obvious equality case.