Properties of certain analytic functions associated with two boundary points

TL;DR

This paper investigates specific properties of analytic functions in the unit disk, focusing on boundary points and conditions that ensure certain bounds on the derivative, with illustrative examples.

Contribution

It introduces new boundary conditions involving two points and the derivative, expanding understanding of analytic functions' behavior near the boundary.

Findings

Derived conditions for |f'(z)-1|<ρ|1-α| in the unit disk

Provided examples illustrating these boundary properties

Connected boundary point behavior with derivative bounds

Abstract

For analytic functions f(z) in the closed unit disk \bar{U}, two boundary points z_1 and z_2 such that \alpha = (f'(z_1)+f'(z_2))/2 in f'(U) are considered. The object of the present paper is to discuss some interesting conditions for f(z) to be |f'(z)-1|<\rho|1-\alpha| in U with some examples.