Double Integral Operators Concerning Starlike of Order β

Abstract

Double integral operators which were considered by S. S. Miller and P. T. Mocanu (Integral Transform. Spec. Funct. 19(2008), 591–597) are discussed. In order to show the analytic function $f\left(z\right)$ is starlike of order $\beta$ in the open unit disk $𝕌$, the theory of differential subordinations for analytic functions is applied. The object of the present paper is to discuss some interesting conditions for $f\left(z\right)$ to be starlike of order $\beta$ in $𝕌$ concerned with second-order differential inequalities and double integral operators.