Analytic and geometric properties of open door functions

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Abstract

In this paper, we study analytic and geometric properties of the solution q(z) to the differential equation q(z) + zq’(z)/q(z) = h(z) with the initial condition q(0) = 1 for a given analytic function h(z) on the unit disk |z| < 1 in the complex plane with h(0) = 1. In particular, we investigate the possible largest constant c > 0 such that the condition |Im[zf’'(z)/f’(z)]| < c on |z| < 1 implies starlikeness of an analytic function f(z) on |z| < 1 with f(0) = f’(0)-1 = 0.

Original languageEnglish
Pages (from-to)267-280
Number of pages14
JournalJournal of the Korean Mathematical Society
Volume54
Issue number1
DOIs
Publication statusPublished - 2017 Jan 1

Keywords

  • Convex function
  • Dilogarithm function
  • Open door lemma
  • Strongly star-like function
  • Subordination

ASJC Scopus subject areas

  • Mathematics(all)

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