Differential subordination and Ruscheweyh derivatives (Q1281325)

Let \(f(z)=z+\sum^\infty_{n=2} a_nz^n\) be analytic in the open unit disc \(E\). Put \(D^nf(z)= (z/(1-z)^{n+1})*f(z)\) where * denotes the usual Hadamard convolution. The author proves the following. If \(h\) is convex with \(h(0)=1\), \(\mu>0\), \(\alpha\neq 0\), \(\text{Re} \alpha\geq 0\) and \((1-\alpha) \left({D^nf(z) \over z} \right)^\mu+ \alpha\left( {D^{n+1}f(z)\over z}\right) \left({D^nf(z) \over z} \right)^{\mu-1}\) is subordinate to \(h(z)\) then \(\left( {D^nf(z) \over z} \right)^\mu\) is also subordinate to \(h(z)\). Similar other results involving subordination are also obtained.