Linear combinations of logarithmic derivatives of entire functions with applications to differential equations

Joseph B. Miles and John Rossi

Vol. 174 (1996), No. 1, 195–214

DOI: 10.2140/pjm.1996.174.195

Abstract

Let F₁,F₂,…,F_L be entire functions of finite order and let c₁,c₂,…,c_L be complex numbers whose convex hull does not contain 0. A lower bound in terms of the counting functions of the zeros of the F_j′s is obtained for

valid for r in a set of positive logarithmic density and 𝜃 in a set U_r ⊂ [0,2π] of fixed positive measure. This bound is used to extend a result of Bank and Langley concerning the exponent of convergence of the zero sequences of solutions of certain linear differential equations with entire coefficients.

Mathematical Subject Classification 2000

Primary: 30D35

Secondary: 34A20

Milestones

Received: 4 August 1993

Revised: 23 November 1993

Published: 1 May 1996