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

For $\alpha \ge 0$, $\beta <1$ and $\gamma \ge 0$, 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.