On the difference of inverse coefficients of convex functions

Young Jae Sim,Derek K Thomas

doi:10.1007/s41478-021-00374-x

Abstract

Let f be analytic in the unit disk {mathbb {D}}={zin {mathbb {C}}:|z|<1 }, and {mathcal {S}} be the subclass of normalised univalent functions given by f(z)=z+sum _{n=2}^{infty }a_n z^n for zin {mathbb {D}}. Let F be the inverse function of f defined in some set |omega |le r_{0}(f), and be given by F(omega )=omega +sum _{n=2}^{infty }A_n omega ^n. We prove the sharp inequalities -1/3 le |A_4|-|A_3| le 1/4 for the class {mathcal {K}}subset {mathcal {S}} of convex functions, thus providing an analogue to the known sharp inequalities -1/3 le |a_4|-|a_3| le 1/4, and giving another example of an invariance property amongst coefficient functionals of convex functions.

Highlights

Introduction and definitionsLet A denote the class of analytic functions f in the unit disk D 1⁄4 fz 2 C : jzj\1g normalized by f ð0Þ 1⁄4 0 1⁄4 f 0ð0Þ À 1
Let f be analytic in the unit disk D 1⁄4 fz 2 C : jzj\P1g, and S be the subclass of normalised univalent functions given by f ðzÞ 1⁄4 z þ
We prove the sharp inequalities À1=3 jA4j À jA3j 1=4 for the class K & S of convex functions, providing an analogue to the known sharp inequalities À1=3 ja4j À ja3j 1=4, and giving another example of an invariance property amongst coefficient functionals of convex functions

Summary

Introduction and definitions

Let A denote the class of analytic functions f in the unit disk D 1⁄4 fz 2 C : jzj\1g normalized by f ð0Þ 1⁄4 0 1⁄4 f 0ð0Þ À 1. 2 by Leung [5] in the case of starlike functions SÃ, sharp upper and lower bounds for janþ1j À janj are only known when n 1⁄4 2 for some subclasses of S, such as the classes K of convex and close-to-convex functions [6, 7]. An exception to this was provided by Ming and Sugawa [8], who showed that if f 2 K, À1=3 ja4j À ja3j 1=4; and that both of these inequalities are sharp. It turns out that finding sharp bounds for jjanþ1j À janjj in the case of convex functions presents a significantly difficult problem. If there exists a positive constant T such that (i) DQ ! T3=2DP, and (ii) TPðtÞ ! QðtÞ for t 2 I, pffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffi the function GðtÞ 1⁄4 PðtÞ À QðtÞ is convex on I

Preliminary results

Main result

À b1 þ Td1