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

Abstract

We deal with different kinds of generalizations of ${\mathcal{S}}^{\ast }(\psi )$, 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 $|h^{\prime }(z)|\le |f^{\prime }(z)|$ in $|z|\le r_{\psi }$. Consequently, obtain results associating ${\mathcal{S}}^{\ast }(\psi )$ 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 $g\in {\mathcal{S}}^{\ast }({\psi }_1)$ and $h\in {\mathcal{S}}^{\ast }({\psi }_2).$ Also we obtain a condition for the functions to be in ${\mathcal{S}}^{\ast }(\psi )$ 3. We obtain the modified distortion theorem for ${\mathcal{S}}^{\ast }(\psi )$ with a general perspective. 4. For a fixed $f\in {\mathcal{S}}^{\ast }(\psi ),$ 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.

All the results under the Majorization section have been proved to be sharp in this version

References in corpus (2)

• On the Bohr inequality
• A Cardioid Domain and Starlike Functions

Cited by in corpus (1)

• ${\mathcal{S}}^{\ast }(\varphi )$ and $\mathcal{C}(\varphi )$-radii for some special functions