Starlikeness of the generalized integral transform using duality techniques

TL;DR

This paper investigates conditions under which a generalized integral transform maps a specific class of analytic functions into starlike functions, extending known results and exploring applications to classical integral operators.

Contribution

It provides new admissible and sufficient conditions on the kernel function for the transform to preserve starlikeness, including applications to well-known integral operators.

Findings

Derived conditions for the integral transform to map into starlike functions

Extended classical results to a broader class of integral transforms

Applied results to specific well-known integral operators

Abstract

For $\alpha\geq 0$, $\delta>0$, $\beta<1$ and $\gamma\geq 0$, the class $\mathcal{W}_{\beta}^\delta(\alpha,\gamma)$ consist of analytic and normalized functions $f$ along with the condition \begin{align*} {\rm Re\,} e^{i\phi}(\dfrac{}{}(1\!-\!\alpha\!+\!2\gamma)\!({f}/{z})^\delta +(\alpha\!-\!3\gamma\!+\!\gamma[\dfrac{}{}(1-{1}/{\delta})({zf'}/{f})+ {1}/{\delta}(1+{zf''}/{f'})]).\\ .\dfrac{}{}({f}/{z})^\delta \!({zf'}/{f})-\beta)>0, \end{align*} where $\phi\in\mathbb{R}$ and $|z|<1$, is taken into consideration. The class $\mathcal{S}^\ast_s(\zeta)$ be the subclass of the univalent functions, defined by the analytic characterization ${\rm Re}{\,}({zf'}/{f})>\zeta$, for $0\leq \zeta< 1$, $0<\delta\leq\frac{1}{(1-\zeta)}$ and $|z|<1$. The admissible and sufficient conditions on $\lambda(t)$ are examined, so that the generalized and non-linear integral transforms \begin{align*} V_{\lambda}^\delta(f)(z)= (\int_0^1 \lambda(t) ({f(tz)}/{t})^\delta dt)^{1/\delta}, \end{align*} maps the function from $\mathcal{W}_{\beta}^\delta(\alpha,\gamma)$ into $\mathcal{S}^\ast_s(\zeta)$. Moreover, several interesting applications for specific choices of $\lambda(t)$ are discussed, that are related to some well-known integral operators.