Note on the Ruscheweyh derivatives (Q1916867)

Let \(A(p)\) denote the class of functions \(f(z)=z^p+\sum^\infty_{k=1}a_{p+k}z^{p+k}\) analytic in the unit disc \(E=\{z:|z|<1\}\). For \(\alpha>-p\) the operator \(D^{\alpha+p-1}\) is defined by \(D^{\alpha+p-1}f(z)={z^p\over (1-z)^{\alpha+p}}* f(z)\), where \(f\in A(p)\) and \(*\) denotes convolution. The author defines a set \(H(\alpha)\) of complex-valued functions \(h(r,s,t)\); \(h(r,s,t): C^3\to C\) and making use of a well-known lemma due to Miller and Mocanu proves a theorem involving differential inequalities for \(f\in A(p)\).