On the univalence of functions defined by certain integral transforms

The integral transform $\text{F(z) = ∝}{}_0{}^{\mathrm{z}}\text{(f′(t))}{}^{\mathrm{\alpha }}\text{(}\text{g(t)}\text{t}\text{)}{}^{\mathrm{\beta }}\text{dt}$, where α and β are real, of pairs of special analytic functions f(z) = z + ···, g(z) = z + ···, univalent in the open unit disc Δ is studied. The transform and our results extend some recent results due to Shirakova.