Generalized uniformly close-to-convex functions of order γ and type β

Abstract

In this paper, a class of analytic functions f defined on the open unit disc satisfying Re z(D n,α λ f(z))0 D n,α λ g(z) > β z(D n,α λ f(z))0 D n,α λ g(z) − 1 + γ, is studied, where β ≥ 0, −1 ≤ γ < 1, β + γ ≥ 0. and g is a certain analytic function associated with conic domains. Among other results, inclusion relations and the coefficients bound are studied. Various known special cases of these results are pointed out. A subclass of uniformly quasi-convex functions is also studied.