New classes of distortion theorems for certain subclasses of analytic functions involving certain fractional derivatives (Q1128335)

Let \(K_p\) denote the class of functions \[ f(z)=z^p-\sum^\infty_{n=1} a_{p+n}z^{p+n} (a_{p+n}\geq 0,\;p=1,2,\dots), \] analytic and \(p\)-valend in \(U=\{z:|z|<1\}\). For a fixed \(a\) and \(b:-1\leq a<b\leq 1\), \(K^*_p(a,b)\) is a subclass of \(K_p\) consisting of such \(f\) that \[ \left|\left( {zf'(z)\over f(z)}- p\right)\left({bzf'(z) \over f(z)}-a p\right)^{-1} \right |<1,\;z\in U. \] With some necessary restrictions on the parameters \(\lambda,\mu, \eta\) and \(f\in K^*_p(a,b)\), the distortion theorems for fractional derivative \[ J_{0,z}^{\lambda, \mu,\eta} f(z)={d\over dz} \left( {z^{\lambda- \mu}\over \Gamma(1-\lambda)} \int^z_0 (z-t)^{- \lambda}f(t)_2F _1\left( \mu-\lambda, 1-\eta,1- \lambda,1-{t \over z}\right) dt\right) \] are obtained. The results are sharp. Some estimates concerning particular cases of the above operator (the Riemann-Liouville and Erdélyi-Kober operators) are noted.