Properties of entire solutions of some linear PDE's

Abstract

Abstract. In this paper, there are improved sufficient conditions of boundedness of the L-index in a direction for entire solutions of some linear partial differential equations. They are new even for the one-dimensional case and L ≡Also, we found a positive continuous function l such that entire solutions of the homogeneous linear differential equation with arbitrary fast growth have a bounded l -index and estimated its growth.

Composition of entire functions and bounded L-index in direction

“…In previous investigations of bounded L-index in direction ( [1,3,4,7]) there was used another equivalent definition of class Q n b . The presented definition appeared for the first time in [6].…”

Boundedness of the L-index in a direction of entire solutions of second order partial differential equation

“…for b 1 = 0, b 2 = 0. For b 1 = 0, b 2 = 0, one can consider a PDE which matches to (8) up to permutations of the variables z 1 and z 2 .…”

“…Recently, an improved version of Theorem 3 was obtained in [8] with restrictions for the values of r.…”

“…Entire functions of bounded index have been used in the theory of value distribution and differential equations [2,4,8,11,[14][15][16]. A full bibliography can be found in [6].…”

Entire Functions of Bounded L-Index: Its Zeros and Behavior of Partial Logarithmic Derivatives

“…In particular, if an entire solution has bounded index [10], then it immediately yields its growth estimates, a uniform distribution of its zeros, a certain regular behavior of the solution, and so forth. A full bibliography about application in theory of ordinary and partial differential equations is in [3,11,12].…”