Some subordination and superordination results of generalized Srivastava-Attiya operator (Q371928)

In the paper some subordination and superordination-preserving results of the generalized Srivastava-Attyia operator are obtained. The main theorem states that for analytic \(p\)-valent functions \(f\) and \(g\) the subordination condition \[ \frac{\mathcal J_{p,s-1,b}(f)(z)}{\mathcal J_{p,s,b}(f)(z)}\left(\frac{\mathcal J_{p,s,b}(f)(z)}{z^p}\right)^\mu\prec\frac{\mathcal J_{p,s-1,b}(g)(z)}{\mathcal J_{p,s,b}(g)(z)}\left(\frac{\mathcal J_{p,s,b}(g)(z)}{z^p}\right)^\mu \] implies that \[ \left(\frac{\mathcal J_{p,s,b}(f)(z)}{z^p}\right)^\mu\prec\left(\frac{\mathcal J_{p,s,b}(g)(z)}{z^p}\right)^\mu, \] where \( J_{p,s,b}(f)(z)\) is the Liu operator (\(\mu\), \(b\), \(s\), \(p\) are some parameters), \[ \phi(z)=\frac{\mathcal J_{p,s-1,b}(g)(z)}{\mathcal J_{p,s,b}(g)(z)}\left(\frac{\mathcal J_{p,s,b}(g)(z)}{z^p}\right)^\mu, \] with \[ \mathfrak{Re}\left\{1+\frac{z\phi''(z)}{\phi'(z)}\right\}>-\delta, \] and where \[ \delta=\frac{1+\mu^2|b+1|^2-|1-\mu^2(b+1)^2|}{4\mu(1+\mathfrak{Re}\{b\})}. \]