A property of certain analytic functions involving Ruscheweyh derivatives (Q917734)

A useful theorem involving the Ruscheweyh derivatives is proved. One corollary is: If \(f(z)=z+a_ 2z^ 2+..\). is analytic for \(| z| <1\) and satisfies \[ Re\{zf''(z)/f'(z)\}>2\alpha -1 \] there, where \(0\leq \alpha <1\), then for any \(\beta\) with \[ 0<\beta \leq (1-\alpha)/4,\quad Re(f'(z))^{\beta}>1/(4\beta (1-\alpha)+1). \]