Multivalent functions defined by Ruscheweyh derivatives
Let \(K_{n,p}\) be the classes of functions \(f(z)=z^ p+\sum^{\infty}_{k=1}a_{p+k}z^{p+k}\) analytic in the open unit disc U and satisfying \(Re\{D^{n+p}f(z)/D^{n+p-1}f(z)\}>1/2,\) \(z\in U\), \(n>-p\) where \(D^{n+p}f(z)=(z^ p/(1-z)^{n+p+1})*f(z)\) and (*) is the Hadamard convolution. The authors determine the set of real numbers \(\alpha\) and \(\beta\) such that whenever \[ Re\{[(D^{n+p+1}f(z))/(D^{n+p}f(z))- 1/2]^{\alpha}[(D^{n+p}f(z))/(D^{n+p-1}f(z))-1/2]^{\beta}>0, \] \(z\in U\) then \(f\in K_{n,p}\). Further, let \(H_{n,p}(z)=(D^{n+p- 1}f(z))/(D^{n+p-1}g(z))\). The authors determine the set of real numbers \(\alpha\) and \(\beta\) such that whenever \(f,g\in K_{n,p}\) then \(Re\{H^{\alpha}_{n,p}(z)H^{\beta}_{n+1,p}(z)\}>0\) hold for \(z\in U\). They also study certain integral operators for the classes \(K_{n,p}\).
scientific article; zbMATH DE number 3904918
scientific article