On a Linear Combination of Some Expressions in the Theory of the Univalent Functions By

Let H (a) denote the class of regular functions f(z) normalized so that f(0) = 0 andf ' (0) = 1 and satisfying in the unit disc E the condition Re {(1-- ~)f ' (z) + ~ (1 + zf " (z)/f " (z))}:> 0 for fixed a. I t is known that H(0) is a part icular class NW of close-to-convex univalent functions. The authors show the following results: Theorem 1. Let f(z)eH(~). Then f ( z)eNW if ~< 0 and zeE. Theorem 2. Let f ( z)eNW. Then.f(z)eH(cr in l z]=r<ra where i) ra= ( l+12/~)- l /2, ~/>0, and ii) ra=] / ~ , a<0. All results are sharp. Theorem 3. I f f (z) = z + a2z ~ + asz 3 +... is in H(a) and if # is an arbitrary complex number, then [1 + ~] la3--~a~[<(2/3)max[1,]l-+- 2~--3/2/~(1 + ~)[]