In this paper we introduce new subclasses of the class of closeto-convex functions. We call a regular function yfz) an alpha-close-to-convex function if (f(z)f'(z)/z) ¥= 0 for z in E and if for some nonnegative real number a there exists a starlike function a>(z) = z + ■■ • such that Rer(1_a)cí£) + a(£ñf]>0 for z in E. We have proved that all alpha-close-to-convex functions are close-toconvex and have obtained a few coefficient inequalities for a-close-to-convex functions and an integral formula for constructing these functions. Let ?a be the class of regular and normalised functions j(z) which satisfy Re (f'(z) + 0 for z in E.f(z) e % gives Re/'(z) > 0 for z in E provided Rea > 0. A sharp radius of uni valence of the class of functions fiz) for which zf'(z) e 9tt has also been obtained.

No abstract

Abstract.In this paper we introduce new subclasses of the class of closeto-convex functions. We call a regular function yfz) an alpha-close-to-convex function if (f(z)f'(z)/z) ¥= 0 for z in E and if for some nonnegative real number a there exists a starlike function a>(z) = z + ■■ • such that Rer(1_a)cí£) + a(£ñf]>0 for z in E.We have proved that all alpha-close-to-convex functions are close-toconvex and have obtained a few coefficient inequalities for a-close-to-convex functions and an integral formula for constructing these functions.Let ?a be the class of regular and normalised functions j(z) which satisfy Re (f'(z) + 0 for z in E.f(z) e % gives Re/'(z) > 0 for z in E provided Rea > 0. A sharp radius of uni valence of the class of functions fiz) for which zf'(z) e 9tt has also been obtained.

Let f(z)=z + … be regular in the unit disc |z| < 1 (hereafter called E). In a recent paper Trimble [7] has proved that if f(z) be convex in E, then F(z) = (1 − λ)z + λf(z) is starlike with respect to the origin in E for (2/3) ≦ λ ≦ 1. The purpose of this note is to show that if certain additional restrictions be imposed on f(z), then F(z) becomes starlike for all λ, 0 ≦ λ ≦ 1. Also we consider some related problems.

Let f(z) be regular and univalent in |z| < 1 with a power series expansionThen it is proved in (2) thatwhere z1 and z2 are points in |z| < 1.