Radius of Univalence of Certain Combination of Univalent and Analytic Functions

Let ₰ denote the family of all analytic functions f in the unit disk D with the normalization f (0) = 0 = f ´(0)-1. Define ₰={ƒє ₰ : ƒ is univalent in D}, μ ={ƒє ₰ A :│ƒ´(z) (z= ƒ (z))^2-1│ 1 for z є D}, and ₰(1/2) = {ƒє ₰ : Re( f (z)/z) 1/2 for z є D}: In this paper, we determine the radius of univalency of F(z)=z ƒ (z)/g(z) whenever ƒє ₰ or ₰, and g є ₰ or ₰(1/2). Based on our investigations, we conjecture that F is univalent in the disk│z│ 1/3 wheneverƒє ₰ and g є ₰(1/2). We also conjecture that F is univalent in the disk │z│ √ 5-2 whenever both f and g are in ₰.