Convexity of the Generalized Integral Transform and Duality Techniques

TL;DR

This paper investigates conditions under which a generalized integral transform maps functions from a specific class of normalized analytic functions into a subclass characterized by convexity, using duality techniques and integral conditions.

Contribution

It introduces new sufficient conditions on the kernel function for the integral transform to preserve convexity within a broad class of analytic functions.

Findings

Derived sufficient conditions for the integral transform to map into convex functions.

Provided applications for specific kernel functions.

Extended previous results on integral transforms and convexity.

Abstract

Let $\mathcal{W}_{\beta}^\delta(\alpha,\gamma)$ be the class of normalized analytic functions $f$ defined in the domain $|z|<1$ satisfying \begin{align*} {\rm Re\,} e^{i\phi}\left(\dfrac{}{}(1\!-\!\alpha\!+\!2\gamma)\!\left({f}/{z}\right)^\delta +\left(\alpha\!-\!3\gamma+\gamma\left[\dfrac{}{}\left(1-{1}/{\delta}\right)\left({zf'}/{f}\right)+ {1}/{\delta}\left(1+{zf''}/{f'}\right)\right]\right)\right.\\ \left.\dfrac{}{}\left({f}/{z}\right)^\delta \!\left({zf'}/{f}\right)-\beta\right)>0, \end{align*} with the conditions $\alpha\geq 0$, $\beta<1$, $\gamma\geq 0$, $\delta>0$ and $\phi\in\mathbb{R}$. Moreover, for $0<\delta\leq\frac{1}{(1-\zeta)}$, $0\leq\zeta<1$, the class $\mathcal{C}_\delta(\zeta)$ be the subclass of normalized analytic functions such that \begin{align*} {\rm Re}{\,}\left(1/\delta\left(1+zf''/f'\right)+(1-1/\delta)\left({zf'}/{f}\right)\right)>\zeta,\quad |z|<1. \end{align*} In the present work, the sufficient conditions on $\lambda(t)$ are investigated, so that the generalized integral transform \begin{align*} V_{\lambda}^\delta(f)(z)= \left(\int_0^1 \lambda(t) \left({f(tz)}/{t}\right)^\delta dt\right)^{1/\delta},\quad |z|<1, \end{align*} carries the functions from $\mathcal{W}_{\beta}^\delta(\alpha,\gamma)$ into $\mathcal{C}_\delta(\zeta)$. Several interesting applications are provided for special choices of $\lambda(t)$.