Mapping properties for convolutions involving hypergeometric functions (Q1811887)

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 \textit{N. Shukla} and \textit{P. Shukla} [Soochow J. Math. 25, 29-36 (1999; Zbl 0964.30007)].