Inclusion properties of Generalized Integral Transform using Duality Techniques

TL;DR

This paper investigates the inclusion properties of a generalized integral transform on classes of analytic functions, establishing conditions under which the transform preserves certain geometric properties and exploring specific applications.

Contribution

It introduces new conditions for the inclusion of transformed functions within specific analytic function classes using duality techniques.

Findings

Derived parameter conditions for inclusion properties

Established applications for specific kernel functions

Extended the understanding of integral transforms in geometric function theory

Abstract

Let $\mathcal{W}_{\beta}^\delta(\alpha,\gamma)$ be the class of normalized analytic functions $f$ defined in the region $|z|<1$ and 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 $\phi\in\mathbb{R}$. For a non-negative and real-valued integrable function $\lambda(t)$ with $\int_0^1\lambda(t) dt=1$, the generalized non-linear integral transform is defined as \begin{align*} V_{\lambda}^\delta(f)(z)= \left(\int_0^1 \lambda(t) \left({f(tz)}/{t}\right)^\delta dt\right)^{1/\delta}. \end{align*} The main aim of the present work is to find conditions on the related parameters such that $V_\lambda^\delta(f)(z)\in\mathcal{W}_{\beta_1}^{\delta_1}(\alpha_1,\gamma_1)$, whenever $f\in\mathcal{W}_{\beta_2}^{\delta_2}(\alpha_2,\gamma_2)$. Further, several interesting applications for specific choices of $\lambda(t)$ are discussed.