Integral transforms of functions to be in the Pascu class using duality techniques

TL;DR

This paper establishes conditions under which integral transforms map functions from a specific analytic class into Pascu's class of onvex functions, unifying starlike and convex functions, with applications to known operators.

Contribution

It provides new criteria for integral transforms to map functions into Pascu's class, extending and unifying previous results on starlike and convex functions.

Findings

Derived conditions on mbda(t) for the transforms to preserve Pascu class membership.

Identified specific integral operators that map functions from W_{eta}(lpha,gamma) into M(ta).

Extended results to a generalized operator related to V_mbda(f).

Abstract

Let $W_{\beta}(\alpha,\gamma)$, $\beta<1$, denote the class of all normalized analytic functions $f$ in the unit disc ${\mathbb{D}}=\{z\in {\mathbb{C}}: |z|<1\}$ such that \begin{align*} {\rm Re\,} \left(e^{i\phi}\left((1-\alpha+2\gamma)\frac{f}{z}+(\alpha-2\gamma)f'+\gamma zf"-\beta\right)\frac{}{}\right)>0, \quad z\in {\mathbb{D}}, \end{align*} for some $\phi\in {\mathbb{R}}$ with $\alpha\geq 0$, $\gamma\geq 0$ and $\beta< 1$. Let $M(\xi)$, $0\leq \xi\leq 1$, denote the Pascu class of $\xi$-convex functions given by the analytic condition \begin{align*} {\rm Re\,}\frac{\xi z(zf'(z))'+(1-\xi)zf'(z)}{\xi zf'(z)+(1-\xi)f(z)}>0 \end{align*} which unifies the class of starlike and convex functions. The aim of this paper is to find conditions on $\lambda(t)$ so that the integral transforms of the form \begin{align*} V_{\lambda}(f)(z)= \int_0^1 \lambda(t) \frac{f(tz)}{t} dt. \end{align*} carry functions from $W_{\beta}(\alpha,\gamma)$ into $M(\xi)$. As applications, for specific values of $\lambda(t)$, it is found that several known integral operators carry functions from $W_{\beta}(\alpha,\gamma)$ into $M(\xi)$. Results for a more generalized operator related to $V_\lambda(f)(z)$ are also given.