New subclasses of the class of close-to-convex functions

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) + <xzf"(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.

Convolution properties of a class of starlike functions

“…Finally, denote by 7? the family of functions f e A which satisfy the condition Re(f(z) + zf'(z)) > 0, z e E. Chichra [1] proved that if f e R, then Re f (z) > 0, z G E, and hence / is univalent in E. R. Singh and S. Singh [8] showed that if / G 7? then / is also starlike in E.…”

Construction of Coefficient Inequality for a New Subclass of Class of Starlike Analytic Functions

“…Let be the class of functions of the form (1.1), which are analytic univalent in . In 1916, Bieber Bach ( [7], [8] The inequality (1.2) plays a very important role in determining estimates of higher coefficients for some sub classes (See Chhichra [1], Babalola [6]). Let us define some subclasses of .…”

Construction of Subclasses of Univalent Harmonic Mappings

“…In [5], Chichra introduced the class W of analytic functions f ∈ A which satisfy Re(f ′ (z) + zf ′′ (z)) > 0 for z ∈ D. He proved that the members of W are univalent in D by showing that W ⊂ R. Later Singh and Singh [36] proved that W ⊂ S * . The class W is compact and is closed under convex combination of its members.…”