Notes on certain class of analytic functions defined by Ruscheweyh derivatives (Q1365326)

The authors introduce the class \(R_n(\alpha)\) consisting of functions \(f(z)= z+ a_2z^2+\cdots\) analytic in the open unit disc for which \[ \text{Re}\Biggl\{{D^{n+1}f(z)\over D^nf(z)}\Biggr\}> {n+\alpha\over n+1}\quad (|z|<1), \] where \(D^\alpha f(z)= {z\over (1-z)^{\alpha+1}}* f(z)\), \(\alpha>-1\). (Here \(*\) denotes the usual Hadamard convolution.) They study various properties of functions in this class. For example if \(f\in R_2(\alpha)\) (\(0\leq \alpha<{1\over 2}\)) it is proved that \(f(z)\) is convex of a certain order \(\beta\) depending on \(\alpha\).