Univalence criteria for general integral operator

Let $mathcal{A}$ be the class of all analytic functions which are analytic in the open unit disc $mathcal{U=}left{ z:leftvert zrightvert <1right}$ and [ G_{b}=left{ fin mathcal{A}:leftvert frac{% 1+zf^{prime prime }(z)/f^{prime }(z)}{zf^{prime }(z)/f(z)}-1rightvert <b, zin mathcal{U}right} . ] In this paper, we derive sufficient conditions for the integral operator [ I_{gamma }^{alpha _{i} }(f_{1},...,f_{n})(z)=left{ gamma intlimits_{0}^{z}t^{gamma -1}left( f_{1}^{prime }(t)right) ^{alpha _{1}}left( frac{f_{1}(t)}{t}right) ^{1-alpha _{1}}...left( f_{n}^{prime }(t)right) ^{alpha _{n}}left( frac{f_{n}(t)}{t}right) ^{1-alpha _{n}}dtright} ^{frac{1}{gamma }} ] to be analytic and univalent in the open unit disc$~mathcal{U}$, when $f_{i}in G_{b_{i}}$ for all $i=1,ldots ,n.

On General Integral Operator of Analytic Functions

Let be the integral operator defined by , where each of the functions and is, respectively, analytic functions and functions with positive real part defined in the open unit disk for all . 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