Double integral operators concerning starlike of order \(\beta \) (Q606215)

Summary: Double integral operators which were considered by \textit{S. S. Miller} and \textit{P. T. Mocanu} [Integral Transforms Spec. Funct. 19, No. 8, 591--597 (2008; Zbl 1156.30014)] are discussed. In order to show the analytic function \(f(z)\) is starlike of order \(\beta \) in the open unit disk \(\mathbb U\), the theory of differential subordinations for analytic functions is applied. The object of the present paper is to discuss some interesting conditions for \(f(z)\) to be starlike of order \(\beta \) in \(\mathbb U\) concerned with second-order differential inequalities and double integral operators.