二阶线性微分方程英文翻译

Some Properties of Solutions of Periodic Second Order

Linear Differential Equations

1. Introduction and main results

In this paper, we shall assume that the reader is familiar with the fundamental results and the stardard notations of the Nevanlinna's value distribution theory of meromorphic functions [12, 14,

(f)and (f)to denote respectively the order 16]. In addition, we will use the notation (f)，

of growth, the lower order of growth and the exponent of convergence of the zeros of a meromorphic function f， e(f)（[see 8]），the e-type order of f(z), is defined to be

e(f) limlogT(r,f) r r

Similarly, e(f)，the e-type exponent of convergence of the zeros of meromorphic function f, is defined to be

log N(r,1/f) e(f) lim r r

We say thatf(z)has regular order of growth if a meromorphic functionf(z)satisfies

(f) limlogT(r,f) r logr

We consider the second order linear differential equation

f Af 0

Where A(z) B(e z)is a periodic entire function with period 2 i/ . The complex oscillation theory of (1.1) was first investigated by Bank and Laine [6]. Studies concerning (1.1) have een carried on and various oscillation theorems have been obtained [2{11, 13, 17{19]. WhenA(z)is rational in e，Bank and Laine [6] proved the following theorem

Theorem A LetA(z) B(e z)be a periodic entire function with period 2 i/ and rational in e z z.IfB( )has poles of odd order at both and 0, then for every solutionf(z)( 0)of (1.1), (f)

Bank [5] generalized this result: The above conclusion still holds if we just suppose that both and 0are poles ofB( ), and at least one is of odd order. In addition, the stronger conclusion

log N(r,1/f) o(r) (1.2)

holds. WhenA(z)is transcendental ine, Gao [10] proved the following theorem

Theorem B Let B( ) g(1/ ) z pjb ,whereg(t)is a transcendental entire function jj 1

zwith (g) 1, p is an odd positive integer andbp 0，Let A(z) B(e).Then any

non-trivia solution fof (1.1) must have (f) . In fact, the stronger conclusion (1.2) holds.

An example was given in [10] showing that Theorem B does not hold when (g)is any positive integer. If the order (g) 1 , but is not a positive integer, what can we say? Chiang and Gao [8] obtained the following theorems

zTheorem C Let A(z) B(e),whereB( ) g1(1/ ) g2( ),g1andg2are entire

functionsg2transcendental and (g2)not equal to a positive integer or infinity, andg1arbitrary. (i) (g2) 1. (a) If f is a non-trivial solution of (1.1) with e(f) (g2);

thenf(z)andf(z 2 i)are linearly dependent. (b) Iff1andf2are any two linearly Suppose

independent solutions of (1.1), then

(ii) Suppose e(f) (g2). (g2) 1 (a) If f is a non-trivial solution of (1.1)

with e(f) 1,f(z)andf(z 2 i)are linearly dependent. Iff1andf2are any two linearly independent solutions of (1.1),then e(f1f2) 1.

Theorem D Letg( )be a transcendental entire function and its order be not a positive integer or infinity. LetA(z) B(e z); whereB( ) g(1/ ) jb j 1jand p is an odd positive p

integer. Then (f) or each non-trivial solution f to (1.1). In fact, the stronger conclusion (1.2) holds.

Examples were also given in [8] showing that Theorem D is no longer valid when (g)is infinity.

The main purpose of this paper is to improve above results in the case whenB( )is transcendental. Specially, we find a condition under which Theorem D still holds in the case when (g)is a positive integer or infinity. We will prove the following results in Section 3.

Theorem 1 Let A(z) B(e),whereB( ) g1(1/ ) g2( ),g1andg2are entire functions withg2transcendental and z (g2)not equal to a positive integer or infinity, andg1arbitrary. If Some properties of solutions of periodic second order linear differential equationsf(z) and f(z 2 i)are two linearly independent solutions of (1.1), then

e(f)

Or

e(f) 1 (g2) 1 2

We remark that the conclusion of Theorem 1 remains valid if we assume (g1)

is not equal to a positive integer or infinity, andg2arbitrary and still assumeB( ) g1(1/ ) g2( ),In the case wheng1is transcendental with its lower order not equal to an integer or infinity andg2is arbitrary, we need only to consider B*( ) B(1/ ) g1( ) g2(1/ )in0 , 1/ .

Corollary 1 LetA(z) B(e z),whereB( ) g1(1/ ) g2( ),g1andg2are

entire functions with g2 transcendental and

(a)

(b) (g2)no more than 1/2, and g1 arbitrary. If f is a non-trivial solution of (1.1) with e(f) ,thenf(z) and f(z 2 i)are linearly dependent. Iff1andf2are any two linearly independent solutions of (1.1),

then e(f1f2) .

Theorem 2 Letg( )be a transcendental entire function and its lower order be no more than 1/2.

zLetA(z) B(e),whereB( ) g(1/ ) j 1bj

ppjand p is an odd positive integer, then (f) for each non-trivial solution f to (1.1). In fact, the stronger conclusion (1.2) holds. We remark that the above conclusion remains valid if

B( ) g( ) b j j

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