Certain subclasses of Analytic Functions with Fixed Second Coefficient

TL;DR

This paper investigates specific subclasses of normalized analytic functions with fixed second coefficients, establishing radii for various starlikeness properties and relating these to existing results in geometric function theory.

Contribution

It introduces new subclasses of analytic functions with fixed second coefficients and derives radii for multiple starlikeness conditions, expanding understanding of their geometric properties.

Findings

Derived radii for strongly starlikeness, lemniscate starlikeness, and parabolic starlikeness.

Established connections between new radii estimates and existing results.

Enhanced the geometric function theory by characterizing these subclasses.

Abstract

This paper aims to pursue some classes of normalized analytic functions $f$ with fixed second coefficient defined on open unit disk, such that ${(1+z)^2f(z)}/{z}$ and ${(1+z)f(z)}/{z}$ are functions having positive real part. The radius of strongly starlikeness, the radius of lemniscate starlikeness, the radius of parabolic starlikeness and other starlikeness estimates are calculated for such functions. As well relevant connections of computed radii estimates with the existing one are also shown.