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

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.