On a linear combination of some expressions in the theory of the univalent functions

Abstract

Let H (α) denote the class of regular functions f(z) normalized so that f (0)=0 and f′ (0)=1 and satisfying in the unit disc E the condition 0$$]]> for fixed α. It is known that H (0) is a particular class NW of close-to-convex univalent functions. The authors show the following results: Theorem 1. Let f(z) ∈ H (α). Then f(z) ∈NW if α≤0 and z ∈ E . Theorem 2 . Let f(z) ∈NW. Then f(z) ∈ H (α) in | z |= r < r α where i) , α≥0 and ii) , α<0. All results are sharp. Theorem 3 . If f(z)=z+a 2 z 2 + a 3 z 3 +... is in H (α) and if μ is an arbitrary complex number, then .