ON THE UNIVALENCE OF AN INTEGRAL OPERATOR

A general integral operator J_\gamma_1,\cdots,\gamma_n,\beta(z) is introduced, which is defined on the class ~\mathscrA of normalized analytic functions in ~\mathcalU=\z\in\mathbbC:|z|1 \. Three sufficient conditions for the univalence of this integral operator in the unit disk \mathcalU are provided by applying the well-known Becker univalence criteria, Schwarz lemma and Caratheodory inequality. That is, the integral operator J_\gamma_1,\cdots,\gamma_n,\beta(z) is univalent in the unit disk \mathcalU when the functions f_j(z)(j=1,2,\cdots,n) and the parameters \gamma_1,\cdots,\gamma_n,\beta satisfy some conditions.