Starlikeness of the generalized integral transform using duality techniques

For $\alpha \ge 0$, $\delta >0$, $\beta <1$ and $\gamma \ge 0$, 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 $\varphi \in \mathbb{R}$ and $\mathrm{\mid }z\mathrm{\mid }<1$, is taken into consideration. The class ${\mathcal{S}}_s^{\ast }(\zeta )$ 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 ${\mathcal{S}}_s^{\ast }(\zeta )$. Moreover, several interesting applications for specific choices of $\lambda (t)$ are discussed, that are related to some well-known integral operators.