Closed convex hull of the family of multivalently close-to-convex functions of order β

Abstract

In this paper the closed convex hulls of the compact familiesC _β(p), of multivalently close to convex functions of order β andV ^k₀ (p), of multivalent functions of bounded boundary rotation, have been determined, respectively for β≥1 andk≥2(p+1)/p. Extreme points of these convex hulls are partially characterised. For a fixed pointz ₀∈D={z:|z|<1}, a new familyC _β(p, z₀) is defined through Montel normalisation and its closed convex hull is also foud. Sharp coefficient estimates for functions subordinate to or majorised by some function inC _β(p) orC' _β(p) are discussed for β>0. It is shown that iff is subordinate to some function inC _β(p) then each Taylor coefficient off is dominated by the corresponding coefficient of the function$\smallint _0^z pt^{p - 1} (1 + t)^\beta /(1 - t)^{2p + \beta } dt$.