On general integral operator of analytic functions (Q2318907)

Summary: Let \(E_\beta\) be the integral operator defined by \(E_\beta \left(z\right) = \left[\beta \int_0^z t^{\beta - 1} \left(f_1^\prime \left(t\right)\right)^{\alpha_1} \left(f_1 \left(t\right) / t\right)^{\gamma_1} P_1^{\zeta_1 \left(t\right)} \cdots \left(f_n^\prime \left(t\right)\right)^{\alpha_n} \left(f_n \left(t\right) / t\right)^{\gamma_n} P_n^{\zeta_n \left(t\right)} d t\right]^{1 / \beta}\), where each of the functions \(f_i\) and \(P_i\) is, respectively, analytic functions and functions with positive real part defined in the open unit disk for all \(i = 1, \ldots, n\). The object of this paper is to obtain several univalence conditions for this integral operator. Our main results contain some interesting corollaries as special cases.