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

Abstract

In this paper we introduce new subclasses of the class of close-to-convex functions. We call a regular function f ( z ) f(z) an alpha-close-to-convex function if ( f ( z ) f ′ ( z ) / z ) ≠ 0 (f(z)f’(z)/z) \ne 0 for z in E and if for some nonnegative real number α \alpha there exists a starlike function ϕ ( z ) = z + ⋯ \phi (z) = z + \cdots such that \[ Re [ ( 1 − α ) z f ′ ( z ) ϕ ( z ) + α ( z f ′ ( z ) ) ′ ϕ ′ ( z ) ] > 0 \operatorname {Re} \;\left [ {(1 - \alpha )\frac {{zf’(z)}}{{\phi (z)}} + \alpha \frac {{(zf’(z))’}}{{\phi ’(z)}}} \right ] > 0 \] for z in E. We have proved that all alpha-close-to-convex functions are close-to-convex and have obtained a few coefficient inequalities for α \alpha -close-to-convex functions and an integral formula for constructing these functions. Let F α {\mathfrak {F}_\alpha } be the class of regular and normalised functions f ( z ) f(z) which satisfy Re ( f ′ ( z ) + α z f ( z ) ) > 0 \operatorname {Re} \;(f’(z) + \alpha zf(z)) > 0 for z in E. f ( z ) ∈ F α f(z) \in {\mathfrak {F}_\alpha } gives Re ⁡ f ′ ( z ) > 0 \operatorname {Re} f’(z) > 0 for z in E provided Re ⁡ α ⩾ 0 \operatorname {Re} \alpha \geqslant 0 . A sharp radius of univalence of the class of functions f ( z ) f(z) for which z f ′ ( z ) ∈ F α zf’(z) \in {\mathfrak {F}_\alpha } has also been obtained.