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

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.