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.


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


Zbl 0614.30014
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