Order of Starlikeness and Convexity of certain integral transforms using duality techniques

Abstract

For $\alpha\geq 0$, $\beta<1$ and $\gamma\geq 0$, the class $\mathcal{W}_{\beta}(\alpha,\gamma)$ satisfies the condition \begin{align*} {\rm Re\,} \left( e^{i\phi}\left((1-\alpha+2\gamma)f/z+(\alpha-2\gamma)f'+ \gamma zf''-\beta\right)\frac{}{}\right)>0, \quad \phi\in {\mathbb{R}},{\,}z\in {\mathbb{D}}; \end{align*} is taken into consideration. The Pascu class of $\xi$-convex functions of order $\sigma$ $(M(\sigma,{\,}\xi))$, having analytic characterization \begin{align*} {\rm Re\,}\frac{\xi z(zf'(z))'+(1-\xi)zf'(z)}{\xi zf'(z)+(1-\xi)f(z)}>\sigma,\quad 0\leq \sigma< 1,\quad z\in {\mathbb{D}}, \end{align*} unifies starlike and convex functions class of order $\sigma$.The admissible and sufficient conditions on $\lambda(t)$ are investigated so that the integral transforms \begin{align*} V_{\lambda}(f)(z)= \int_0^1 \lambda(t) \frac{f(tz)}{t} dt, \end{align*} maps the function from $\mathcal{W}_{\beta}(\alpha,\gamma)$ into $M(\sigma,{\,}\xi)$. Further several interesting applications, for specific choice of $\lambda(t)$ are discussed which are related to the classical integral transform.