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 ~$\mathscr{A}$ of normalized analytic functions in ~$\mathcal{U}=\{z\in\mathbb{C}:|z|1 \}$. Three sufficient conditions for the univalence of this integral operator in the unit disk $\mathcal{U}$ 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 $\mathcal{U}$ when the functions $f_{j}(z)(j=1,2,\cdots,n)$ and the parameters $\gamma_{1},\cdots,\gamma_{n},\beta$ satisfy some conditions.