Sufficiency for Gaussian hypergeometric functions to be uniformly convex
The authors consider subclasses of the class \(A\) of functions holomorphic in the unit disk \(D\) and normalized by \(f(0)=f'(0)-1=0\), among them the class of uniformly convex functions, \(UCV\), and the classes \[ R_\eta (\beta)= \{f\in A\mid \min\text{Re}(e^{i\eta} \bigl(f'(z)-\beta)\bigr)> 0\quad \text{for}\quad z\in D\}. \] For the classical hypergeometric function \(F(a,b,c;z)\) they find conditions which imply that \(zF(a,b;c;z)\) belongs to one of the above classes. Further they derive conditions under which \(f\in R_\eta(\beta)\) implies \(zF(a,b,c;z)* f(z)\in UCV\), where * denotes the Hadamard products.
scientific article; zbMATH DE number 1417975