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A class of analytic functions defined by fractional derivation. (English) Zbl 0813.30016
Let $T= T\sb p(A,B,p\sp{-1} \alpha,\beta,\lambda)$ denote the class of $p$-valent functions, which have the form $$f(z)= z\sp{-p}- \sum\sp \infty\sb{n= 1} a\sb{p+ n} z\sp{p+ n},\ z\in U= \{z: \vert z\vert< 1\},\ a\sb{n+ p}\ge 0,\ n\in \bbfN,$$ and satisfy the condition $$\left\vert{\Omega\sp{(\lambda,p)}\sb z f(z)- 1\over B\Omega\sp{(\lambda, p)}\sb z f(z)- [B+ (A- B)(1- p\sp{-1} \alpha)]}\right\vert< \beta\quad\text{for }z\in U,$$ where $0\le p\sp{-1} \alpha< 1$, $0< \beta\le 1$, $0\le \lambda\le 1$, $-1\le A\le 1$, $0< B\le 1$ and $$\Omega\sp{(\lambda, p)}\sb z= {\Gamma(1+ p- \lambda)\over \Gamma(1+ p)} z\sp{\lambda- p} D\sp \lambda\sb z f(z),$$ where $D\sp \lambda\sb z f$ is the fractional derivative operator of order $\alpha$ [see f.e. {\it S. Owa}, Fractional calculus, Proc. Workshop, Ross Priory, Univ. Strathclyde/Engl. 1984, Res. Notes Math. 138, 164-175 (1985; Zbl 0614.30014)]. In this paper some results concerning the radii of $p$-valently close-to- convexity, starlikeness and convexity for the class $T$ are obtained. Also some classes preserving integral operator of the form $$F(z)= {c+ p\over z\sp c} \int\sp z\sb 0 t\sp{c- 1} f(t) dt,\quad c>- p,$$ for the class $T$ are determined.

30C45Special classes of univalent and multivalent functions
30C75Extremal problems for conformal and quasiconformal mappings, other methods
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