Some properties of convolution operators in the class \({\mathcal P}_ \alpha(\beta)\) (Q1340566)

Making use of several families of convolution operators, we introduce and study a certain general class \({\mathcal P}_ \alpha(\beta)\) \((0\leq \alpha< 1;\;\beta\geq 0)\) of analytic functions in the open unit disk \({\mathbf U}\). We also investigate the relationships between the class \({\mathcal P}_ \alpha(\beta)\) and the Hardy space \({\mathcal H}^ \infty\) (of bounded analytic functions in \({\mathbf U}\)). Finally, we consider some interesting applications of the results presented here to a class of generalized hypergeometric functions.