Radius of Univalence of Certain Combination of Univalent and Analytic Functions

Let A denote the family of all analytic functions f in the unit disk D with the normalization f (0) = 0 = f ′(0)− 1. Define S = { f ∈ A : f is univalent in D}, U = { f ∈A : ∣∣ f ′(z)(z/ f (z))2−1∣∣< 1 for z ∈ D}, and P(1/2) = { f ∈A : Re( f (z)/z) > 1/2 for z ∈ D}. In this paper, we determine the radius of univalency of F(z) = z f (z)/g(z) whenever f ∈S or U , and g ∈S or P(1/2). Based on our investigations, we conjecture that F is univalent in the disk |z| < 1/3 whenever f ∈S and g ∈P(1/2). We also conjecture that F is univalent in the disk |z|< √ 5−2 whenever both f and g are in S . 2010 Mathematics Subject Classification: 30C45