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A class of analytic functions defined by fractional derivation. (English) Zbl 0813.30016

Let \(T= T_ p(A,B,p^{-1} \alpha,\beta,\lambda)\) denote the class of \(p\)-valent functions, which have the form \[ f(z)= z^{-p}- \sum^ \infty_{n= 1} a_{p+ n} z^{p+ n},\;z\in U= \{z: | z|< 1\},\;a_{n+ p}\geq 0,\;n\in \mathbb{N}, \] and satisfy the condition \[ \left|{\Omega^{(\lambda,p)}_ z f(z)- 1\over B\Omega^{(\lambda, p)}_ z f(z)- [B+ (A- B)(1- p^{-1} \alpha)]}\right|< \beta\quad\text{for }z\in U, \] where \(0\leq p^{-1} \alpha< 1\), \(0< \beta\leq 1\), \(0\leq \lambda\leq 1\), \(-1\leq A\leq 1\), \(0< B\leq 1\) and \[ \Omega^{(\lambda, p)}_ z= {\Gamma(1+ p- \lambda)\over \Gamma(1+ p)} z^{\lambda- p} D^ \lambda_ z f(z), \] where \(D^ \lambda_ z f\) is the fractional derivative operator of order \(\alpha\) [see f.e. 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 \[ F(z)= {c+ p\over z^ c} \int^ z_ 0 t^{c- 1} f(t) dt,\quad c>- p, \] for the class \(T\) are determined.

MSC:

30C45 Special classes of univalent and multivalent functions of one complex variable (starlike, convex, bounded rotation, etc.)
30C75 Extremal problems for conformal and quasiconformal mappings, other methods

Citations:

Zbl 0614.30014
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