数学专业英语论文含中文版.docx

Introduction and main results
In this paper, we shall assume that the reader is familiar with the fundamental results a. If Some properties of solutions of periodic second order linear differential equations and are two linearly independent solutions of (), then
Or
We remark that the conclusion of Theorem 1 remains valid if we assume
is not equal to a positive integer or infinity, andarbitrary and still assume
,In the case whenis transcendental with its lower order not equal to an integer or infinity andis arbitrary, we need only to consider in,.
Corollary 1 Let,where,andare
entire functions with transcendental and no more than 1/2, and arbitrary.
If f is a non-trivial solution of () with,then and are linearly dependent.
Ifandare any two linearly independent solutions of (), then.
Theorem 2 Letbe a transcendental entire function and its lower order be no more than 1/2. Let,whereand p is an odd positive integer, then for each non-trivial solution f to (). In fact, the stronger conclusion () holds.
We remark that the above conclusion remains valid if
We note that Theorem 2 generalizes Theorem D whenis a positive integer or infinity but . Combining Theorem D with Theorem 2, we have
Corollary 2 Letbe a transcendental entire function. Let where and p is an odd positive integer. Suppose that either (i) or (ii) below holds:
(i) is not a positive integer or infinity;
(ii) ;
thenfor each non-trivial solution f to (). In fact, the stronger conclusion () holds.
Lemmas for the proofs of Theorems
Lemma 1 ([7]) Suppose thatand thatare entire functions of period,and that f is a non-trivial solution of
Suppose further that f satisfies; that is non-constant and rational in，and that if，thenare constants. Then there exists an integer q with such that and are linearly dependent. The same conclusion holds ifis transcendental in，and f satisfies，and if ，then asthrough a setof infinite measure, we havefor.
Lemma 2 ([10]) Letbe a period