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

Let U(lambda, mu) denote the class of all normalized analytic functions f in the unit disk |z| < 1 satisfying the condition f(z)/z not equal 0 and |f'(z)(z/f(z))(mu+1) - 1| < lambda, |z| < 1. For f is an element of U(lambda, mu) with mu <= 1 and 0 not equal mu(1) <= 1, and for a positive real-valued integrable function. defined on [0, 1] satisfying the normalized condition integral(1)(0) phi(t) dt = 1, we consider the transform G(phi)f (z) defined by G(phi)f (z) = z [integral(1)(0) phi(t) (zt/f(tz))(mu) dt](-1/mu 1), z is an element of Delta. In this paper, we find conditions on the range of parameters lambda and mu so that the transform G(phi)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(lambda, mu).