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

Kim, J. A.; Shon, K. H.

Int. J. Math. Math. Sci. 2003, No. 17, 1083-1091 (2003).

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)].

Reviewer: Eligiusz Złotkiewicz (Lublin)

MSC:

30C45

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

Citations:

Zbl 0964.30007