Convexity of the Generalized Integral Transform and Duality Techniques

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^{iϕ}\left(\dfrac{}{}(1\!-\!α\!+\!2γ)\!\left({f}/{z}\right)^δ+\left(α\!-\!3γ+γ\left[\dfrac{}{}\left(1-{1}/δ\right)\left({zf'}/{f}\right)+ {1}/δ\left(1+{zf''}/{f'}\right)\right]\right)\right.\\ \left.\dfrac{}{}\left({f}/{z}\right)^δ\!\left({zf'}/{f}\right)-β\right)>0, \end{align*} with the conditions $\alpha \ge 0$, $\beta <1$, $\gamma \ge 0$, $\delta >0$ and $\varphi \in \mathbb{R}$. Moreover, for $0<\delta \le \frac{1}{(1-\zeta )}$, $0\le \zeta <1$, the class ${\mathcal{C}}_{\delta }(\zeta )$ be the subclass of normalized analytic functions such that \begin{align*} {\rm Re}{\,}\left(1/δ\left(1+zf''/f'\right)+(1-1/δ)\left({zf'}/{f}\right)\right)>ζ,\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_λ^δ(f)(z)= \left(\int_0^1 λ(t) \left({f(tz)}/{t}\right)^δdt\right)^{1/δ},\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 $\lambda (t)$.

Convexity results of the generalized integral operator, 25 pages

References in corpus (3)

• Order of Starlikeness and Convexity of certain integral transforms using duality techniques
• Starlikeness of the generalized integral transform using duality techniques
• Inclusion properties of Generalized Integral Transform using Duality Techniques