On the initial coefficients for certain class of functions analytic in the unit disc

TL;DR

This paper establishes sharp bounds on the second, third, and fourth coefficients of certain analytic functions in the unit disk, constrained by a specific argument condition involving parameters alpha and gamma.

Contribution

It provides new sharp bounds for initial coefficients of functions satisfying a particular argument inequality, extending understanding of their geometric properties.

Findings

Sharp bounds for the second coefficient.

Sharp bounds for the third coefficient.

Sharp bounds for the fourth coefficient.

Abstract

Let function $f$ be analytic in the unit disk ${\mathbb D}$ and be normalized so that $f(z)=z+a_2z^2+a_3z^3+\cdots$. In this paper we give sharp bounds of the modulus of its second, third and fourth coefficient, if $f$ satisfies \[ \left|\arg \left[\left(\frac{z}{f(z)}\right)^{1+\alpha}f'(z) \right] \right|<\gamma\frac{\pi}{2} \quad\quad (z\in {\mathbb D}),\] for $0<\alpha<1$ and $0<\gamma\leq1$.