Summary: Let \(\mathcal {P}_{\gamma}(\alpha,\beta)\), denote the class of all normalized analytic functions \(f\) defined in the unit disc \(E=\{z : |z|<1 \}\) such that \[ \mathrm{Re} \, \left\{ e^{i\eta}\left[ (1-\gamma)\left( \frac{f(z)}{z}\right)^{\alpha}+\gamma \left( \frac{zf'(z)}{f(z)}\right) \left( \frac{f(z)}{z}\right)^{\alpha}-\beta \right] \right\} >0, \] for all \(z\in E\) and \(\eta \in \mathbb {R}\), where \(\beta <1\), \(\alpha \geq 0\) and \(0\leq \gamma \leq 1\). For a real-valued non-negative function \(\lambda \) with the normalization \(\int_{0}^{1}\lambda (t)dt=1\), we consider the integral operators \[ V_{\lambda, \mu}(f)(z)=\left( \int_{0}^{1}\lambda (t)\left( \frac{f(tz)}{t}\right)^{\mu}dt\right) ^{\frac{1}{\mu}}, \quad \mu >0 \] and a class \(\mathcal {S}_{\delta}(\nu)\) of normalized analytic functions \(f\) which satisfy the condition \[ \mathrm{Re} \, \left[ \left( 1+\frac{zf''(z)}{f'(z)}\right) +\left( \nu -1\right) \left( \frac{zf'(z)}{f(z)}\right) \right] >\delta, \] for \(\delta <\nu \leq 1+\delta \) and \(0\leq \delta <1\). The aim of this paper is to find the sharp value of \(\beta \) so that the operator \(V_{\lambda, \alpha}(f)\) carries \(\mathcal {P}_{\gamma}(\alpha,\beta)\) into \(\mathcal {S}_{\delta}(\alpha)\). Some interesting applications for different choices of \(\lambda \) are discussed.