ON THE HYPER-ORDER OF SOLUTIONS OF SECOND ORDER LINEAR DIFFERENTIAL EQUATIONS WITH MEROMORPHIC COEFFICIENTS

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230

ON THE HYPER-ORDER OF SOLUTIONS OF SECOND ORDER LINEAR DIFFERENTIAL EQUATIONS WITH MEROMORPHIC COEFFICIENTS

Jianren Long

School of Mathematics and Computer Science, Guizhou Normal University, Guiyang, 550001, P.R.China ABSTRACT

In this paper, we investigate the growth of solutions of the linear differential equation

f   A f   Bf = 0

, where

) (z

A

and

B ( z )(  0)

are meromorphic functions. More specifically, we estimate the lower bounded of hyper- order of solutions of the equation with respect to the conditions of

A (z )

^and

B ( z )(  0)

if solutions

f ( 0

^{) of}

the equation is of infinite order.

keywords: Deficient value, Complex differential equations, Meromorphic function, Hyper-order.

2010 Mathematical Subject Classification: 34M10; 30D35.

1. INTRODUCTION AND MAIN RESULTS

We shall assume that the reader is familiar with the fundamental results and the standard notations of Nevanlinna theory of meromorphic functions (see [10], [14] or [17]). In addition, for a meromorphic function

f (z )

^{in the}

complex plane



, we shall use the notation

 ( f )

and

 ( f )

to denote its order and lower order respectively.

In order to estimate the rate of growth of meromorphic function of infinite order more precisely, we recall the following definition.

Definition A.([19]) Let

f (z )

be a meromorphic function in

C

, then we define the hyper-order



₂

( f )

^of

f (z )

log .

) , log ( lim log

= )

2

(

r f r f

r

T









Consider the second order linear differential equation

0,

= ) ( )

( z f B z f A

f    

(1.1)

where

A (z )

and

B (z )

are meromorphic functions. It is well known that if

A (z )

is entire function and

B (z )

is transcendental entire function, and

f

₁,

f

₂ are two linearly independent solutions of equation (1.1), then at least one of

f

₁^,

f

₂ must have infinite order. On the other hand, there are some equations of the form (1.1) that possess a solution

f ( 0)

of finite order; for example,

f ( z ) = e

^z satisfies

f   e

^^z

f   ( e

^^z

 1) f = 0

. Thus a natural question is: what conditions on

A (z )

and

B (z )

can guarantee that every solution

f (  0)

of the equation (1.1) has infinite order? There are many work done on this subject. From the works of Gundersen (see [4]), Hellerstein, Miles and Rossi (see [8]), we know that if

A (z )

and

B (z )

are entire functions with

 ( A ) <  ( B )

; or if

A (z )

is a polynomial, and

B (z )

is transcendental; or if

2 ) 1 (

<

)

( B  A 



, then every solution

f ( 0)

^{of the}

equation (1.1) has infinite order. More results can be found in [6], [9], [11] and [16]. Then for a great deal of solutions with infinite order, more precise estimates for their rate of growth is a very important aspect. There are many authors investigate the hyper-order



₂

( f )

of solutions of (1.1) (see, [2-3] and [12-13]). Ki-Ho Kwon investigate the problem and obtained the following result in [12].

*The work is supported by the united technology Foundation of Science and Technology Department of Guizhou Province and Guizhou Normal University(Grant No.LKS[2012]12), and National Natural Science Foundation of China (Grant No.11171080).

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231

Theorem B.([12]) Let

A (z )

and

B (z )

be entire functions such that

 ( A ) <  ( B )

^or

2 < 1 ) (

<

)

( B  A



.

Then every solution

f  0

of (1.1) satisfies



₂

( f )  max {  ( A ),  ( B )}

.

It seems that there are few work done on the equation (1.1), where

A (z )

^and

B (z )

are meromorphic functions, but the growth of meromorphic solution of the equation (1.1) is a very important aspect (see [1-2]). It would be interesting to get some relations between the equation (1.1) and some deep results in value distribution theory of meromorphic functions. Thus, we shall introduce the deficient value into the studies of the equation (1.1). It is well- know that deficient value plays fundamental role in the theory of value distribution of meromorphic function and many important work done on this aspect (for example [18]). There are some results on the value distribution theory of the solutions of the equation (1.1) having connections with deficient values (see [6] or [11]). Furthermore, we mention that the author consider the equation (1.1) and obtain the result following.

Theorem C.([15]) Let

A (z )

be an meromorphic function having a finite deficient value, and let

B (z )

be a transcendental meromorphic function with

2 < 1 )

 (B

^and

 (  , B ) = 1

. Then every solution

f ( 0)

^of

equation

(1.1)

is of infinite order.

We will consider the equation (1.1) again, and obtain some precise estimates of hyper-order of solutions, when the solutions

f ( 0

) of the equation (1.1) is of infinite order. Here we shall prove some results in which the deficient value is involved.

Theorem 1.1. Let

A (z )

B (z )

2 < 1 )

 (B

^and

 (  , B ) = 1

f ( 0)

^of

equation

(1.1)

satisfies



₂

( f )   ( B )

.

A (z )

B (z )

2 < 1 )

 (B

^and

 (  , B ) = 1

f ( 0)

^of

equation

(1.1)

satisfies



₂

( f )   ( B )

.

Using the same method of the proof of Theorem 1.1, we can easily obtain the following result.

A (z )

B ( z )(  0)

be an meromorphic function. Suppose that there exist two constants

 > 0

^and

 > 0

, for any given

 > 0

, two finite set of real numbers

{ 

_k

}

and

{ 

_k

}

that satisfy



₁

< 

₁

< 

₂

< 

₂

<  < 

_m

< 

_m

< 

_m_₁

( 

_m_₁

= 

₁

 2  )

and

,

<

) (

₁

1

=







_k _k

m

k







^(1.2)

such that

}

|

| (1)) {(1

exp

| ) (

| B z   o  z

^ (1.3)

as

z  

ⁱⁿ



_k

 arg z  

_k

( k = 1,2,  , m )

f (  0)

of equation

(1.1)

satisfies





₂

( f ) 

.

The paper is organized as the following. In Section 2, we shall state and prove some lemmas related with our Theorems. In Section 3, we shall prove Theorem 1.1-1.3.

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232 2. LEMMAS

For the proofs of our theorems, we need the following lemmas.

Lemma 2.1. (5) Let

( f ,  )

denote a pair that consists of a transcendental meromorphic function

f (z )

and a finite set.

)}

, ( , ), , ( ), , {(

= k

₁

j

₁

k

₂

j

₂

 k

_q

j

_q



of distinct pairs of integers that satisfies

k

_i

> j

_i

 0

for

i = 1,  , q

. Let

 > 1

and

 > 0

be given real constants. Then the following three statements hold.

(i)There exists a set

E

₁

 [0,2  )

that has linear measure zero, and there exists a constant

c > 0

that depend only on



^and



, such that if



₀

 [0,2  )  E

₁, then there is a constant

R

₀

= R

₀

( 

₀

) > 0

such that for all

z

satisfying

arg z = 

₀ and

| z |= r  R

₀, and for all

( k , j )  

, we have

. )) , ( log log

) , ( ( ) | (

)

| (

) (

j k j

k

f r T r r

f r c T z f

z

f

_

 

_



(2.1) (ii) There exists a set

E

₂

 (1,  )

that has finite logarithmic measure, and there exists a constant

c > 0

that depend only on



^and



, such that for all

z

satisfying

| ^z |= ^r ^ ( ^E

₂

 [0,1])

and for all

( k , j )  

^{, the}

inequality (2.1) holds.

(iii) There exiets a set

E

₃

 [0,  )

that has finite linear measure, and there exists a constant

c > 0

that depend only on



and



, such that for all

z

satisfying

| z |= r  E

₃ and for all

( k , j )  

, we have

. )) , ( log ) , ( ( ) | (

)

| (

) (

j k j

k

f r T r f r T z c

f z

f

_

 

^



(2.2)

To state the following lemma, we define the upper and lower logarithmic measure of

E  [1, )

respectively by

log , ]) [1, (

= lim

log r

r E E m

dens

^l

r







and

log , ]) [1, lim (

=

log r

r E E m

dens

^l

r







where

t r dt

E

m

l

( [1, ]) = 

E [1,r]

 

^.

Lemma 2.2. (7) Suppose that

g (z )

is a transcendental and meromorphic in the complex plane, of lower order

1 <

< 



, and define

L ( r , g ) = min {| g ( z ) |:| z |= r }

and

)}

, ( 1) ) , ( cos

(

>

) , ( log : 1

>

{

= r L r g g T r g

E      

^,



 

= sin

^{. Then}

E

has upper logarithmic density at least 1 -





.

Remark 1. In Lemma 2.2, if the order of

g (z )

is less than

2

1

, then, for every

  (  ( g ),1)

, there exists a set

[1, )



E

such that



 ( ) 1

log g

E

dens  

^{, where}

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233

)}

, ( 1) ) , ( cos

(

>

) , ( log : 1

>

{

= r L r g g T r g

E      

^,



 

= sin

.

Lemma 2.3. (15) Let

g (z )

2 < 1 ) (

<

0  g

^and

 (  , g ) = 1

^,

and let

A (z )

be a meromorphic function with

 ( A ) < 

^{. If}

A (z )

has a finite deficient value

a

^with

deficiency

 =  ( a , A )

, then for any given constant

 > 0

, there exists a sequence

( R

_n

)

with

R

_n

< R

_n_₁ and



n



R

(

n  

), such that the following two inequalities

), [0,2

}, {

exp

|>

) (

| g R

_n

e

ⁱ^

R

_n^⁽^g⁾^^

  

0 >

)}

, 4 (

| ) (

| log : ) [0,2 {

=:

)

( F mes A R e a T R A d

mes

_n

  

_n ⁱ^

   

_n



hold for all sufficiently large

n

^{, where}

d

is a constant depending only on

 ( A ),  ( g )

^and



^.

Remark 2. If

g

is a transcendental meromorphic function with

 (g ) = 0

^and

 (  , g ) = 1

, according to Lemma 2.2, we only need to give an appropriate modification and Lemma 2.3 still holds.

Remark 3. If

g

is a meromorphic function with

2 < 1 )

 (g

^and

 (  , g ) = 1

, according to Remark 1, we only need to give an appropriate modification for method of proof of Lemma 2.3 and Lemma 2.3 still holds.

3. PROOF OF THEOREMS In this section,we prove our Theorems.

Proof of Theorem 1.1. From the equation (1.1), we have the following inequalities

. ) | (

)

|| ( ) (

| ) | (

)

| (

| ) (

|

' '

z f

z z f

z A f

z z f

B 

^



(3.1)

By using Lemma 2.1 that there exists a set

E

₁

 [0,2  )

that has linear measure zero, if



₀

 [0,2  )  E

₁, then there is a constant

R

₀

= R

₀

( 

₀

) > 0

such that for all

z

satisfying

arg z = 

₀ and

| z |= r  R

₀, we have

3 )

(

) , (2 ) |

( )

| ( T r f

z f

z

f

^k



(3.2)

holds for

k = 1

^{and 2.}

In the following, we treat two cases.

Case 1.

2 < 1 ) (

<

0  B

. Suppose that

A (z )

a

with deficiency

 =  ( a , A )

^{. By}

using Lemma 2.3, we have that for any given constant

 > 0

, there exists a sequence

( r

_n

)

with

r

_n

< r

_n_₁ and



n



r

(

n  

), such that for every

n

, we have

0,

>

)}

, 4 (

| ) (

| log : ) [0,2 {

=:

)

( F mes A r e a T r A d

mes

_n

  

_n ⁱ^

   

_n



^(3.3)

), [0,2

}, {

exp

|>

) (

| B r

_n

e

ⁱ^

r

_n^⁽^B⁾^^

  

(3.4)

where

d

is a constant depending only on

 ( A ),  ( B )

^and



^.

For every

n  n

₀, there exists a constant

R

₁

> 0

, such that

r

_n

> R

₁ (

n  n

₀), we have (3.3) and (3.4) hold. Thus

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234

there must exist real numbers

^

₀

 ([0,2 ^ )  F

_n

)  E

₁^{, and}

^R

²

^> ^max ^{ ^R

⁰

^, ^R

¹

^}

, such that for all

z

satisfying

=

0

arg z 

and

| z |= r  R

₂, we have (3.2), (3.3) and (3.4) hold.

Hence, calculating at the points

z

_n

= r

_n

e

ⁱ^⁰ with

r

_n

 R

₂(

n > n

₀), we get from (3.1), (3.2), (3.3) and (3.4) that

|).

| )) , 4 ( ( exp (1 ) , (2 ) (

exp r

_n_⁽^B⁾^_

 T r

_n

f

³

   T r

_n

A  a

Thus

).

log (

) , ( log

lim log B

r f r T

r ^ ^

 

(3.5)

Therefore, from (3.5), we have



₂

( f )   ( B )

.

Case 2:

 (B ) = 0

. Obviously, we have



₂

( f )   ( B )

. The proof of Theorem is completed.

Proof of Theorem 1.2. It follows from the Remark 3 and same method of proof of Theorem 1.1, we can easily obtain



₂

( f )   ( B )

, so, we omit the detail.

Proof of Theorem 1.3. Suppose that

A (z )

a

with deficiency

 =  ( a , A )

. By using Lemma 2.1 that there exists a set

E

₂

 [0,2  )

that has linear measure zero, if



₀

 [0,2  )  E

₂, then there is a constant

R

₃

= R

₃

( 

₀

) > 0

such that for all

z

satisfying

arg z = 

₀ and

| z |= r  R

₃, we have (3.2) holds for

1 =

k

and 2. From the proof of Lemma 2.3, there still exists a sequence

( r

_n

)

that satisfies (3.3).

Let

0 <  < d 2

. Then for every positive integer

n

, we choose

( [ , ])

1

= k k

m n k

n

F  

 ^  

. So there exists a

integer

k  {1,2,  , m }

such that for every integer

n

satisfies



_n

^ F

_n

 [ 

_k

, 

_k

]

(otherwise we use the subsequence

nj



instead of



_n). Thus form our hypothesis, we have

}, (1)) {(1

exp

| ) (

| B r

_n

e

ⁱ^ⁿ

  o  r

_n^ (3.6)

as

n  

^.

Hence, calculating at the points

z

_n

= r

_n

e

ⁱ^ⁿ with

r

_n

> R

₃

( n  n

₀

)

, we get from (3.1), (3.2), (3.3) and (3.6) that

|),

| )) , 4 ( ( exp (1 ) , (2 } (1)) {(1

exp  o  r

_n^

 T r

_n

f

³

   T r

_n

A  a

(3.7)

holds for

n  n

₀. Thus

log .

) , ( log

lim



log

^ ^

 

r f r T

r (3.8)

Therefore, from (3.8), we have



₂

( f )  

. The proof of Theorem is completed.

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