Preserving properties of subordination and superordination of analytic functions involving the Wright generalized hypergeometric function (Q2868769)

In the paper some subordination and superordination preserving results are obtained. These results are of the following type. Let \(\theta_p^{l,m}[\alpha_1,A_1,B_1]f(z)\) be the Dziok-Raina-Srivastava operator [\textit{J. Dziok} et al., Proc. Jangjeon Math. Soc. 7, No. 1, 43--55 (2004; Zbl 1060.30017)]. Let \(f\), \(g\) be analytic \(p\)-valent functions and let NEWLINE\[NEWLINE \mathfrak{Re}\left\{1+\frac{z\phi''(z)}{\phi'(z)}\right\}>-\delta, NEWLINE\]NEWLINE where NEWLINE\[NEWLINE \phi(z)=\left(\frac{\theta_p^{l,m}[\alpha_1+1,A_1,B_1]g(z)}{\theta_p^{l,m}[\alpha_1,A_1,B_1]g(z)}\right) \left(\frac{\theta_p^{l,m}[\alpha_1,A_1,B_1]g(z)}{z^p}\right)^\mu\;\;(\mu>0; |z|<1), NEWLINE\]NEWLINE and where \(\alpha_1\), \(A_1\), \(\ldots\), \(\alpha_l\), \(A_l\) and \(\beta_1\), \(B_1\), \(\ldots\), \(\beta_m\), \(B_m\) (\(l,m\) positive integer) are positive real parameters such that \(1+B_1+\ldots+B_m-(A_1+\ldots+A_l)>0\), and \(\delta\) is given by NEWLINE\[NEWLINE \delta=\frac{A_1^2+\mu^2\alpha_1^2-|A_1^2-\mu^2\alpha_1^2|}{4\mu A_1\alpha_1}. NEWLINE\]NEWLINE Then the subordination condition NEWLINE\[NEWLINE \left(\frac{\theta_p^{l,m}[\alpha_1+1,A_1,B_1]f(z)}{\theta_p^{l,m}[\alpha_1,A_1,B_1]f(z)}\right) \left(\frac{\theta_p^{l,m}[\alpha_1,A_1,B_1]f(z)}{z^p}\right)^\mu\prec NEWLINE\]NEWLINE NEWLINE\[NEWLINE \left(\frac{\theta_p^{l,m}[\alpha_1+1,A_1,B_1]g(z)}{\theta_p^{l,m}[\alpha_1,A_1,B_1]g(z)}\right) \left(\frac{\theta_p^{l,m}[\alpha_1,A_1,B_1]g(z)}{z^p}\right)^\mu NEWLINE\]NEWLINE implies that NEWLINE\[NEWLINE \left(\frac{\theta_p^{l,m}[\alpha_1,A_1,B_1]f(z)}{z^p}\right)^\mu\prec \left(\frac{\theta_p^{l,m}[\alpha_1,A_1,B_1]g(z)}{z^p}\right)^\mu NEWLINE\]NEWLINE and the function NEWLINE\[NEWLINE \left(\frac{\theta_p^{l,m}[\alpha_1,A_1,B_1]g(z)}{z^p}\right)^\mu NEWLINE\]NEWLINE is the best dominant.