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DOI QR Code

ANALYTIC AND GEOMETRIC PROPERTIES OF OPEN DOOR FUNCTIONS

  • Li, Ming (Graduate School of Information Sciences Tohoku University) ;
  • Sugawa, Toshiyuki (Graduate School of Information Sciences Tohoku University)
  • Received : 2015.11.30
  • Published : 2017.01.01

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.

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