Univalence and starlikeness of nonlinear integral transform of certain class of analytic functions

Let U(λ, µ) denote the class of all normalized analytic functions f in the unit disk |z| < 1 satisfying the condition $$ \frac{{f(z)}} {z} \ne 0and\left| {f'(z)\left( {\frac{z} {{f(z)}}} \right)^{\mu + 1} - 1} \right| < \lambda ,\left| z \right| < 1. $$ For f ∈ U(λ, µ) with µ ≤ 1 and 0 ≠ µ1 ≤ 1, and for a positive real-valued integrable function φ defined on [0, 1] satisfying the normalized condition ∫ 0 1 φ(t)dt = 1, we consider the transform G φ f (z) defined by $$ G_\phi f(z) = z\left[ {\int_0^1 {\phi (t)\left( {\frac{{zt}} {{f(tz)}}} \right)^\mu dt} } \right]^{ - 1/\mu _1 } ,z \in \Delta . $$ In this paper, we find conditions on the range of parameters λ and µ so that the transform G φ f is univalent or star-like. In addition, for a given univalent function of certain form, we provide a method of obtaining functions in the class U(λ, µ).