Mapping properties for convolutions involving hypergeometric functions.(English)Zbl 1019.30008

Let $$\mu$$ be a non-negative number, let $$_2F_1(a,b,c;z)$$ denote the hypergeometric function, $$G(z)= z(_2F_1(a,b, c;z))$$ and let $$f_\mu(z)= (1-\mu)z G(z) +G'(z)$$, $$I_\lambda(z) \int^z_0t^{-1} f_\lambda(t) dt$$. An objective of this note is to give some sufficient conditions under which operators $$I_\lambda (z)$$, $$I_\lambda* f(z)$$ are functions univalent and starlike or convex in the unit disk. Justifications are based on a result of N. Shukla and P. Shukla [Soochow J. Math. 25, 29-36 (1999; Zbl 0964.30007)].

MSC:

 30C45 Special classes of univalent and multivalent functions of one complex variable (starlike, convex, bounded rotation, etc.)

Zbl 0964.30007
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