On Certain Generalizations of $\mathcal{S}^*(\psi)$

TL;DR

This paper explores various generalizations of Ma-Minda starlike and convex functions, establishing radius results, distortion theorems, and studying subordination classes with applications to the Bohr phenomenon.

Contribution

It introduces new radius constants, extends distortion theorems, and investigates subordination classes related to Ma-Minda functions with conjectures on the Bohr phenomenon.

Findings

Derived a general radius constant $r_{\psi}$ for majorization in convex Ma-Minda classes.

Established the largest radius for product functions to belong to certain classes.

Proposed conjectures on the Bohr phenomenon for subordination classes.

Abstract

We deal with different kinds of generalizations of $\mathcal{S}^*(\psi)$, the class of Ma-Minda starlike functions, in addition to a majorization result of $\mathcal{C}(\psi),$ the class of Ma-Minda convex functions, which are enlisted as follows: 1. Let $h$ be an analytic function, $f$ be in $\mathcal{C}(\psi)$ and $h$ be majorized by $f$ in the unit disk $\mathbb{D},$ then for a given $\psi,$ we derive a general equation, which yields the radius constant $r_{\psi}$ such that $|h'(z)|\leq |f'(z)|$ in $|z|\leq r_{\psi}$. Consequently, obtain results associating $\mathcal{S}^*(\psi)$ and others. 2. We find the largest radius $r_0$ so that the product function $g(z)h(z)/z$ belongs to a desired class for $|z|<r_0$ whenever $g\in \mathcal{S}^*(\psi_1)$ and $h\in \mathcal{S}^*(\psi_2).$ Also we obtain a condition for the functions to be in $\mathcal{S}^*(\psi)$ 3. We obtain the modified distortion theorem for $\mathcal{S}^*(\psi)$ with a general perspective. 4. For a fixed $f\in \mathcal{S}^*(\psi),$ the class of subordinants $S_{f}(\psi):= \{g : g\prec f \} $ is introduced and studied for the Bohr-phenomenon and a couple of conjectures are also proposed.