Abstract
S. B. Bank has shown that there is no uniform growth estimate for meromorphic solutions of algebraic differential equations with meromorphic coefficients in the unit disk. We give conditions under which such solutions must have a finite order of growth.
Highlights
Consider the algebraic differential equation aα (z)y α0 (y )α1 · · · y (k) αk = 0, α∈I (1.1)where I is a finite set of distinct tuples (α0, α1, . . . , αk) for which each αi is a nonnegative integer, and the aαare meromorphic functions in D = {z | |z| < 1}
In [1], Bank investigated (1.1) where I consists of 2-tuples and the aαare arbitrary analytic functions of finite order in the unit disk. He observed that such equations could possess analytic solutions of infinite order in the unit disk, but obtained a uniform growth estimate for all such solutions
Heittokangas [3] showed for certain sets I that each meromorphic solution of (1.1) has finite order when the aαare polynomial functions
Summary
In [1], Bank investigated (1.1) where I consists of 2-tuples and the aαare arbitrary analytic functions of finite order in the unit disk He observed that such equations could possess analytic solutions of infinite order in the unit disk, but obtained a uniform growth estimate for all such solutions. Heittokangas [3] showed for certain sets I that each meromorphic solution of (1.1) has finite order when the aαare polynomial functions. He and Wulan [5] studied the equation (y )n = bα (z)y α0 (y )α1 · · · y (k) αk , α∈I (1.2). Our first theorem is similar in character to the result of Wulan and Heittokangas, while our other theorems take into account the nature of the zeros or poles of the aαcoefficient functions
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