Convex sum of univalent functions

Abstract

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.

A survey of properties of the convex combination of univalent functions

“…A computation shows dV = 2r 2 cos 2 (0/2) + (fc/2) r(l -r 2 )cos(0/2) -r 2 (l + r 2 ) ( ' d0 " 1 -2r 2 cos0 + r 4 The numerator of ( 13) is a quadratic in cos 0/2. We shall see that the root…”

“…If J\\ is a linearly invariant family whose elements satsify \argf'(z)\ ^ k arcsin |z|, \z\ g 2rJ(l + r 0 2 ), then the claims of the theorem hold and are still sharp. we need only find the positive root of 1 + r 2 -6r 4 which is 1/V2.…”

Univalence, strong starlikeness and integral transforms

“…Related problems were considered in [2,12], by imposing an additional condition on f . In Theorem 2.3, we impose conditions on f ∈ A n := {f ∈ A : [2,12] and obtain the starlikeness Univalence and integral transforms…”