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Certain properties of the class of univalent functions defined by Ruscheweyh derivative. (English) Zbl 1093.30012

Let $A$ denote the class of analytic functions in $U=\left\{z:|z|<1\right\}$ and $T\subset A$ denote the class of functions of the form

$f\left(z\right)=z-\sum _{n=2}^{\infty }{a}_{n}{z}^{n},\phantom{\rule{4pt}{0ex}}{a}_{n}\ge 0,\phantom{\rule{4pt}{0ex}}n=1,2,\cdots ·$

Let us put

${D}^{\lambda }f\left(z\right):=f\left(z\right)*\frac{z}{{\left(1-z\right)}^{1+\lambda }}=z-\sum _{n=2}^{\infty }{a}_{n}{B}_{n}\left(\lambda \right){z}^{n},$

where

${B}_{n}\left(\lambda \right)=\frac{{\Gamma }\left(n+\lambda \right)}{\left(n-1\right)!{\Gamma }\left(1+\lambda \right)}$

and let

$D\left(\alpha ,\beta ,\lambda \right):\left\{f\in T:\text{Re}\frac{z{\left({D}^{\lambda }f\left(z\right)\right)}^{\text{'}}}{{D}^{\lambda }f\left(z\right)}>\alpha \left|\frac{z{\left({D}^{\lambda }f\left(z\right)\right)}^{\text{'}}}{{D}^{\lambda }f\left(z\right)}-1\right|+\beta z\in U\right\},$

where $\alpha \ge 0$, $0\le \beta <1$, $\lambda >-1$. In this paper some special properties of the classes $D\left(\alpha ,\beta ,\lambda \right)$ are investigated. It is shown that $D\left(\alpha ,\beta ,\lambda \right)$ are the convex sets. It is proved that if $f\in D\left(\alpha ,\beta ,\lambda \right)$ then for $\delta >0$ the function

${G}_{\delta }\left(z\right)=\left(1-\delta \right)f\left(z\right)+\delta {\int }_{0}^{z}\frac{f\left(t\right)}{t}\phantom{\rule{0.166667em}{0ex}}dt$

also belongs to $D\left(\alpha ,\beta ,\lambda \right)$. Some estimations for the functional integral of order $\delta$, $\delta <0$, defined by

${D}_{z}^{\delta }f\left(z\right):=\frac{1}{{\Gamma }\left(-\delta \right)}{\int }_{0}^{z}\frac{f\left(t\right)}{{\left(z-t\right)}^{1+\delta }}\phantom{\rule{0.166667em}{0ex}}dt$

where $f\in D\left(\alpha ,\beta ,\lambda \right)$ are given.

MSC:
 30C45 Special classes of univalent and multivalent functions