Certain results on subordinations associated with a differintegral operator (Q2879605)

Let \(\mathcal{A}(p)\) denote the class of functions of the form NEWLINE\[NEWLINEf(z)=\sum_{n=p}^\infty a_nz^n, \quad a_p=1,NEWLINE\]NEWLINE which are analytic and \(p\)-valent in the open unit disc \(\mathbb{U}=\left\{z\in \mathbb{C}:|z|<1\right\}\).NEWLINENEWLINEFor \(f\in\mathcal{A}(p)\) consider the Dziok differintegral operator \(\Omega_\beta^\alpha\) defined, in terms of the gamma function, by NEWLINE\[NEWLINE\Omega_\beta^\alpha f(z)=\sum_{n=p}^\infty\frac{\Gamma(n+\beta)}{\Gamma(n+\beta+\alpha)}a_nz^n.NEWLINE\]NEWLINE Let \(\sigma\in[0,1]\) and \(h(z)=1+a_1z+a_2z^2+\ldots\). Consider the function class NEWLINE\[NEWLINE\mathcal{S}(\sigma,\Omega_\beta^\alpha;h)=\left\{f\in\mathcal{A}(p):\sigma\left[1+\frac{z(\Omega_\beta^\alpha f(z))''}{(\Omega_\beta^\alpha f(z))'}\right]+(1-\sigma)\frac{z(\Omega_\beta^\alpha f(z))'}{\Omega_\beta^\alpha f(z)}\prec ph(z), z\in \mathbb{U}\right\}.NEWLINE\]NEWLINE In this paper, the authors investigate certain properties on subordinations for the class \(\mathcal{S}(\sigma,\Omega_\beta^\alpha;h)\). Some interesting consequences and applications of the main results are also considered.