An application of the fractional calculus. IV (Q2266139)

[For part III see the author in J. Korean Math. Soc. 20, 133-140 (1983; Zbl 0558.30017).] The class A(\(\alpha)\) of analytic functions \(f(z)=z+\sum^{\infty}_{n=2}a_ nz^ n\) which satisfy \(| (f(z)/z)-1| <\alpha\) \((z\in U=\{z: | z| <1\})\), for some \(\alpha\) \((0<\alpha \leq 1)\) was introduced by \textit{K. S. Padmanabhan} [J. Indian Math. Soc., New. Ser. 29, 71-80 (1965; Zbl 0145.087)], and \textit{S. Chandra} and \textit{P. Singh} [Indian J. Pure Appl. Math. 4, 745-748 (1973; Zbl 0289.30014)]. Let \(D_ z^{\lambda}f(z)\) and \(D_ z^{-\lambda}f(z)\) denote the fractional derivative and the fractional integral of order \(\lambda\) of f(z), respectively. The author derives some distortion inequalities for fractional calculus (that is, fractional derivative and fractional integral) of functions belonging to A(\(\alpha)\). Theorem 1. Let the function f(z) belong to A(\(\alpha)\). Then \[ (| z|^{1- \lambda}/\Gamma (2-\lambda))(1-\alpha | z|)\leq | D_ z^{\lambda}f(z)| \leq (| z|^{1-\lambda}/\Gamma (2- \lambda))(1+\alpha | z|) \] for \(0<\alpha \leq 1\), \(0<\lambda <1\) and \(z\in U\). The result is best possible.