Open Access

Complex linear differential equations with certain analytic coefficients of [ p , q ] -order in the unit disc

Advances in Difference Equations20142014:167

https://doi.org/10.1186/1687-1847-2014-167

Received: 20 March 2014

Accepted: 27 May 2014

Published: 12 June 2014

Abstract

In this paper, the authors study some growth properties of analytic functions of [ p , q ] -order in the disc and apply them to investigating the growth and zeros of solutions of complex linear differential equations with analytic coefficients of [ p , q ] -order satisfying certain growth conditions in the unit disc, and they obtain some results which are generalizations and improvements of some previous results.

MSC:30D35, 34M10.

Keywords

linear differential equations unit disc [ p , q ] -order [ p , q ] -exponent of convergence of zero-sequence

1 Notations and results

We assume that readers are familiar with the fundamental results and the standard notations of the Nevanlinna value distribution theory of meromorphic functions in the unit disc (see [14]). Firstly, we introduce some notations. Let us define inductively, for r ( 0 , + ) , exp 1 r = e r and exp i + 1 r = exp ( exp i r ) , i N . For all r sufficiently large in ( 0 , + ) , we define log 1 r = log r and log i + 1 r = log ( log i r ) , i N . We also denote exp 0 r = r = log 0 r and exp 1 r = log 1 r . Moreover, we denote the linear measure of a set E [ 0 , 1 ) by m E = E d t , and the upper and lower density of E [ 0 , 1 ) are defined, respectively, by
dens ¯ Δ E = lim ¯ r 1 m ( E [ r , 1 ) ) 1 r , dens ̲ Δ E = lim ̲ r 1 m ( E [ r , 1 ) ) 1 r .
The complex oscillation theory of linear differential equations
f ( k ) + A k 1 ( z ) f ( k 1 ) + + A 0 ( z ) f = 0
(1.1)
and
f ( k ) + A k 1 ( z ) f ( k 1 ) + + A 0 ( z ) f = F ( z )
(1.2)

in the unit disc has been developed since the 1980s (see [5]). After that, many important results have been obtained (see [611]). After the work of Liu et al. in [12], there has been an increasing interest in studying the interaction between the analytic coefficients of [ p , q ] -order and the solutions of (1.1) and (1.2) (see [1316]). In this paper, the authors continue to focus on studying the growth and zeros of solutions of (1.1), (1.2) with analytic coefficients of [ p , q ] -order which satisfy certain growth conditions in the unit disc.

We use p, q to denote positive integers, and we use Δ = { z : | z | < 1 } to denote the unit disc. In the following, we recall some notations of meromorphic functions and analytic functions in Δ.

Definition 1.1 (see [3, 10])

Let f ( z ) be a meromorphic function in Δ, and set
D ( f ) = lim ¯ r 1 T ( r , f ) log ( 1 r ) .
If f ( z ) is an analytic function in Δ,
D M ( f ) = lim ¯ r 1 log M ( r , f ) log ( 1 r ) .

If D ( f ) = , we say that f is admissible, if D ( f ) < , we say that f is non-admissible. If D M ( f ) = , we say that f is of infinite degree, if D M ( f ) < , we say that f is of finite degree.

Definition 1.2 (see [7, 11])

The iterated p-order of a meromorphic function f ( z ) in Δ is defined by
σ p ( f ) = lim ¯ r 1 log p T ( r , f ) log ( 1 r ) .
For an analytic function f ( z ) in Δ, we also define
σ M , p ( f ) = lim ¯ r 1 log p + 1 M ( r , f ) log ( 1 r ) .

Remark 1.1 If p = 1 , we denote σ 1 ( f ) = σ ( f ) and σ M , 1 ( f ) = σ M ( f ) , and we have σ ( f ) σ M ( f ) σ ( f ) + 1 (see [3]) and σ M , p ( f ) = σ p ( f ) for p > 1 (see [7, 11]).

Definition 1.3 (see [1315])

Let 1 q p or 2 q = p + 1 , and f ( z ) be a meromorphic function in Δ, then the [ p , q ] -order of f ( z ) is defined by
σ [ p , q ] ( f ) = lim ¯ r 1 log p T ( r , f ) log q ( 1 1 r ) .
For an analytic function f ( z ) in Δ, we also define
σ M , [ p , q ] ( f ) = lim ¯ r 1 log p + 1 M ( r , f ) log q ( 1 1 r ) .

Definition 1.4 (see [13])

Let 1 q p or 2 q = p + 1 , we use N ( r , 1 f ) ( N ¯ ( r , 1 f ) ) to denote the integrated counting function for the (distinct) zero-sequence of a meromorphic function f ( z ) in Δ. Then the [ p , q ] -exponents of convergence of (distinct) zero-sequence of f ( z ) about N ( r , 1 f ) ( N ¯ ( r , 1 f ) ) are defined, respectively, by
λ [ p , q ] N ( f ) = lim ¯ r 1 log p N ( r , 1 f ) log q ( 1 1 r ) , λ ¯ [ p , q ] N ¯ ( f ) = lim ¯ r 1 log p N ¯ ( r , 1 f ) log q ( 1 1 r ) .

By the above definitions, the following propositions about the analytic function of [ p , q ] -order in the unit disc can easily be deduced.

Proposition 1.1 Let f ( z ) be an analytic function of [ p , q ] -order in Δ. Then the following five statements hold:
  1. (i)

    If p = q = 1 , then σ ( f ) σ M ( f ) 1 + σ ( f ) .

     
  2. (ii)

    If p = q 2 and σ [ p , q ] ( f ) < 1 , then σ [ p , q ] ( f ) σ M , [ p , q ] ( f ) 1 .

     
  3. (iii)

    If p = q 2 and σ [ p , q ] ( f ) 1 , or p > q 1 , then σ [ p , q ] ( f ) = σ M , [ p , q ] ( f ) .

     
  4. (iv)

    If p 1 and σ [ p , p + 1 ] ( f ) > 1 , then D ( f ) = ; if σ [ p , p + 1 ] ( f ) < 1 , then D ( f ) = 0 .

     
  5. (v)

    If p 1 and σ M , [ p , p + 1 ] ( f ) > 1 , then D M ( f ) = ; if σ M , [ p , p + 1 ] ( f ) < 1 , then D M ( f ) = 0 .

     
Proof (i), (iv), (v) hold obviously, we prove (ii) and (iii).
  1. (ii)
    By the standard inequality T ( r , f ) log + M ( r , f ) 1 + 3 r 1 r T ( 1 + r 2 , f ) ( 0 < r < 1 ) (see [1, 2, 4]), we get
    log p T ( r , f ) log p + 1 + M ( r , f ) max { log p ( 4 1 r ) , log p T ( 1 + r 2 , f ) } .
    (1.3)
     
If p = q 2 and σ [ p , q ] ( f ) < 1 , from (1.3) we have σ [ p , q ] ( f ) σ M , [ p , q ] ( f ) 1 .
  1. (iii)

    If p = q 2 and σ [ p , q ] ( f ) 1 , or p > q 1 , from (1.3), we have σ [ p , q ] ( f ) = σ M , [ p , q ] ( f ) . □

     
Proposition 1.2 Let f ( z ) be a meromorphic function of [ p , q ] -order in Δ. Then the following statements hold:
  1. (i)

    If p > q 1 , then λ ¯ [ p , q ] N ¯ ( f ) = λ ¯ [ p , q ] n ¯ ( f ) .

     
  2. (ii)

    If p = q = 1 , then λ ¯ N ¯ ( f ) λ ¯ n ¯ ( f ) λ ¯ N ¯ ( f ) + 1 .

     
  3. (iii)

    If p = q 2 , then λ ¯ [ p , p ] N ¯ ( f ) λ ¯ [ p , p ] n ¯ ( f ) max { λ ¯ [ p , p ] N ¯ ( f ) , 1 } . Furthermore, we have λ ¯ [ p , p ] N ¯ ( f ) = λ ¯ [ p , p ] n ( f ) if λ ¯ [ p , p ] N ¯ ( f ) 1 , and if λ ¯ [ p , p ] N ¯ ( f ) < 1 then λ ¯ [ p , p ] N ¯ ( f ) λ ¯ [ p , p ] n ¯ ( f ) 1 .

     
Proof Without loss of generality, assuming that f ( 0 ) 0 , by N ¯ ( r , 1 f ) = 0 r n ¯ ( t , 1 f ) t d t , we have
n ¯ ( r , 1 f ) 1 log ( 1 + 1 r 2 r ) r r + 1 r 2 n ¯ ( t , 1 f ) t d t 1 log ( 1 + 1 r 2 r ) N ¯ ( 1 + r 2 , 1 f ) ( 0 < r < 1 ) ,
since log ( 1 + 1 r 2 r ) 1 r 2 r ( r 1 ), we obtain
lim ¯ r 1 log p n ¯ ( r , 1 f ) log q ( 1 1 r ) max { lim ¯ r 1 log p N ¯ ( 1 + r 2 , 1 f ) log q ( 1 1 r ) , lim ¯ r 1 log p ( 2 r 1 r ) log q ( 1 1 r ) } .
By the above inequality, we obtain:
  1. (i)

    if p > q 1 , then λ ¯ [ p , q ] n ¯ ( f ) λ ¯ [ p , q ] N ¯ ( f ) ;

     
  2. (ii)

    if p = q = 1 , then λ ¯ n ¯ ( f ) λ ¯ N ¯ ( f ) + 1 ;

     
  3. (iii)

    if p = q 2 , then λ ¯ [ p , p ] n ¯ ( f ) max { λ ¯ [ p , p ] N ¯ ( f ) , 1 } .

     
On the other hand, by
N ¯ ( r , 1 f ) = r 0 r n ¯ ( t , 1 f ) t d t + N ¯ ( r 0 , 1 f ) n ¯ ( r , 1 f ) log ( r r 0 ) + N ¯ ( r 0 , 1 f ) ( 0 < r 0 < r < 1 ) ,

we can easily get λ ¯ [ p , q ] N ¯ ( f ) λ ¯ [ p , q ] n ¯ ( f ) ( p q 1 ). Therefore, the conclusions of Proposition 1.2 hold. □

In recent years, Belaïdi has investigated the growth of solutions of (1.1), (1.2) with analytic coefficients of [ p , q ] -order in the unit disc and obtained the following results.

Theorem A (see [13])

Let p q 1 be integers and H 1 be a set of complex numbers satisfying dens ¯ Δ { | z | : z H 1 Δ } > 0 , and let A 0 , A 1 , , A k 1 be analytic functions in Δ satisfying max { σ M , [ p , q ] ( A j ) : j = 1 , , k 1 } σ M , [ p , q ] ( A 0 ) = σ 1 . Suppose that there exists a real number α 1 satisfying 0 α 1 < σ 1 such that, for any given ε ( 0 < ε < σ 1 α 1 ), we have
| A 0 ( z ) | exp p + 1 { ( σ 1 ε ) log q ( 1 1 | z | ) }
and
| A j ( z ) | exp p + 1 { α 1 log q ( 1 1 | z | ) } ( j = 1 , 2 , , k 1 ) ,

as | z | 1 for z H 1 . Then every solution f 0 of (1.1) satisfies σ [ p , q ] ( f ) = σ M , [ p , q ] ( f ) = and σ [ p + 1 , q ] ( f ) = σ M , [ p + 1 , q ] ( f ) = σ M , [ p , q ] ( A 0 ) = σ 1 .

Theorem B (see [15])

Let p q 1 be integers and H 2 be a set of complex numbers satisfying dens ¯ Δ { | z | : z H 2 Δ } > 0 , and let A 0 , A 1 , , A k 1 be analytic functions in Δ satisfying max { σ [ p , q ] ( A j ) : j = 1 , , k 1 } σ [ p , q ] ( A 0 ) = σ 2 . Suppose that there exists a real number β 1 satisfying 0 β 1 < σ 2 such that, for any given ε ( 0 < ε < σ 2 β 1 ), we have
T ( r , A 0 ) exp p { ( σ 2 ε ) log q ( 1 1 | z | ) }
and
T ( r , A j ) exp p { β 1 log q ( 1 1 | z | ) } ( j = 1 , 2 , , k 1 ) ,

as | z | 1 for z H 2 . Then every solution f 0 of (1.1) satisfies σ [ p , q ] ( f ) = σ M , [ p , q ] ( f ) = and σ [ p , q ] ( A 0 ) σ [ p + 1 , q ] ( f ) = σ M , [ p + 1 , q ] ( f ) max { σ M , [ p , q ] ( A j ) : j = 0 , 1 , , k 1 } . Furthermore, if p > q , then σ [ p + 1 , q ] ( f ) = σ M , [ p + 1 , q ] ( f ) = σ [ p , q ] ( A 0 ) .

Theorem C (see [13])

Suppose that the assumptions of Theorem  A are satisfied, and let F 0 be an analytic function in Δ of [ p , q ] -order. Then the following two statements hold:
  1. (i)

    If σ [ p + 1 , q ] ( F ) < σ M , [ p , q ] ( A 0 ) , then every solution f of (1.2) satisfies λ ¯ [ p + 1 , q ] ( f ) = λ [ p + 1 , q ] ( f ) = σ [ p + 1 , q ] ( f ) = σ M , [ p , q ] ( A 0 ) with at most one exceptional solution f 0 satisfying σ [ p + 1 , q ] ( f 0 ) < σ M , [ p , q ] ( A 0 ) .

     
  2. (ii)

    If σ [ p + 1 , q ] ( F ) > σ M , [ p , q ] ( A 0 ) , then every solution f of (1.2) satisfies σ [ p + 1 , q ] ( f ) = σ [ p + 1 , q ] ( F ) .

     

From Theorems A-C, we obtain the following results.

Theorem 1.1 Let H 3 be a complex set satisfying dens ¯ Δ { | z | : z H 3 Δ } > 0 . If A j ( z ) ( j = 0 , 1 , , k 1 ) are analytic functions in Δ satisfying max { σ M , [ p , q ] ( A j ) | j = 0 , 1 , , k 1 } σ 3 ( 0 < σ 3 < ), and if there exist two positive constants α 2 , β 2 ( 0 < β 2 < α 2 ) such that, for all z H 3 and | z | 1 , we have
| A 0 ( z ) | exp p { α 2 [ log q 1 ( 1 1 r ) ] σ 3 }
and
| A j ( z ) | exp p { β 2 [ log q 1 ( 1 1 r ) ] σ 3 } ( j = 1 , , k 1 ) .
Then the following statements hold:
  1. (i)

    If p q 1 , then every solution f ( z ) 0 of (1.1) satisfies σ [ p + 1 , q ] ( f ) = σ M , [ p + 1 , q ] ( f ) = σ 3 .

     
  2. (ii)

    If 2 q = p + 1 and σ 3 > 1 , then every solution f ( z ) 0 of (1.1) satisfies σ [ p + 1 , p + 1 ] ( f ) = σ M , [ p + 1 , p + 1 ] ( f ) = σ 3 .

     
Theorem 1.2 Let H 4 be a complex set satisfying dens ¯ Δ { | z | : z H 4 Δ } > 0 . If A j ( z ) ( j = 0 , 1 , , k 1 ) are analytic functions in Δ satisfying max { σ M , [ p , q ] ( A j ) | j = 0 , 1 , , k 1 } σ 4 ( 0 < σ 4 < ), and there exist two positive constants α 3 , β 3 such that, for all z H 4 and | z | 1 , we have
T ( r , A 0 ( z ) ) exp p 1 { α 3 [ log q 1 ( 1 1 r ) ] σ 4 }
and
T ( r , A j ( z ) ) exp p 1 { β 3 [ log q 1 ( 1 1 r ) ] σ 4 } ( j = 1 , , k 1 ) .
Then the following statements hold:
  1. (i)

    If p q 2 and 0 < β 3 < α 3 , then every solution f ( z ) 0 of (1.1) satisfies σ [ p + 1 , q ] ( f ) = σ M , [ p + 1 , q ] ( f ) = σ 4 .

     
  2. (ii)

    If 3 q = p + 1 , 0 < β 3 < α 3 and σ 4 > 1 , then every solution f ( z ) 0 of (1.1) satisfies σ [ p + 1 , p + 1 ] ( f ) = σ M , [ p + 1 , p + 1 ] ( f ) = σ 4 .

     
  3. (iii)

    If p = 1 , q = 2 , 0 < k β 3 < α 3 and σ 4 > 1 , then every solution f ( z ) 0 of (1.1) satisfies σ [ 2 , 2 ] ( f ) = σ M , [ 2 , 2 ] ( f ) = σ 4 .

     
Theorem 1.3 Let F ( z ) 0 , A j ( z ) ( j = 0 , 1 , , k 1 ) be analytic functions in Δ. Suppose that H 3 , A j ( z ) ( j = 0 , 1 , , k 1 ) satisfy the hypotheses in Theorem  1.1, then we have the following statements:
  1. (i)

    Let 1 q p , if σ [ p + 1 , q ] ( F ) > σ 3 , then all solutions of (1.2) satisfy σ [ p + 1 , q ] ( f ) = σ [ p + 1 , q ] ( F ) ; if σ [ p + 1 , q ] ( F ) σ 3 , then all solutions of (1.2) satisfy λ ¯ [ p + 1 , q ] N ¯ ( f ) = λ [ p + 1 , q ] N ( f ) = σ [ p + 1 , q ] ( f ) = σ 3 with at most one exceptional solution f 0 satisfying σ [ p + 1 , q ] ( f 0 ) < σ 3 .

     
  2. (ii)

    Let 2 q = p + 1 , σ 3 > 1 , if σ [ p + 1 , p + 1 ] ( F ) > σ 3 , then all solutions of (1.2) satisfy σ [ p + 1 , p + 1 ] ( f ) = σ [ p + 1 , p + 1 ] ( F ) ; if σ [ p + 1 , p + 1 ] ( F ) σ 3 , then all solutions of (1.2) satisfy λ ¯ [ p + 1 , p + 1 ] N ¯ ( f ) = λ [ p + 1 , p + 1 ] N ( f ) = σ [ p + 1 , p + 1 ] ( f ) = σ 3 , with at most one exceptional solution f 0 satisfying σ [ p + 1 , p + 1 ] ( f 0 ) < σ 3 .

     
Corollary 1.4 Let F ( z ) 0 , A j ( z ) ( j = 0 , 1 , , k 1 ) be analytic functions in Δ. Suppose that H 4 , A j ( z ) ( j = 0 , 1 , , k 1 ) satisfy the hypotheses in Theorem  1.2, then we have the following statements:
  1. (i)

    Let 2 q p , 0 < β 3 < α 3 , if σ [ p + 1 , q ] ( F ) > σ 4 , then all solutions of (1.2) satisfy σ [ p + 1 , q ] ( f ) = σ [ p + 1 , q ] ( F ) ; if σ [ p + 1 , q ] ( F ) σ 4 , then all solutions of (1.2) satisfy λ ¯ [ p + 1 , q ] N ¯ ( f ) = λ [ p + 1 , q ] N ( f ) = σ [ p + 1 , q ] ( f ) = σ 4 with at most one exceptional solution f 0 satisfying σ [ p + 1 , q ] ( f 0 ) < σ 4 .

     
  2. (ii)

    Let 3 q = p + 1 , 0 < β 3 < α 3 , σ 4 > 1 , if σ [ p + 1 , p + 1 ] ( F ) > σ 4 , then all solutions of (1.2) satisfy σ [ p + 1 , p + 1 ] ( f ) = σ [ p + 1 , p + 1 ] ( F ) ; if σ [ p + 1 , p + 1 ] ( F ) σ 4 , then all solutions of (1.2) satisfy λ ¯ [ p + 1 , p + 1 ] N ¯ ( f ) = λ [ p + 1 , p + 1 ] N ( f ) = σ [ p + 1 , p + 1 ] ( f ) = σ 4 with at most one exceptional solution f 0 satisfying σ [ p + 1 , p + 1 ] ( f 0 ) < σ 4 .

     
  3. (iii)

    Let p = 1 , q = 2 , 0 < k β 3 < α 3 , σ 4 > 1 , if σ [ 2 , 2 ] ( F ) > σ 4 , then all solutions of (1.2) satisfy σ [ 2 , 2 ] ( f ) = σ [ 2 , 2 ] ( F ) ; if σ [ 2 , 2 ] ( F ) σ 4 , then all solutions of (1.2) satisfy λ ¯ [ 2 , 2 ] N ¯ ( f ) = λ [ 2 , 2 ] N ( f ) = σ [ 2 , 2 ] ( f ) = σ 4 with at most one exceptional solution f 0 satisfying σ [ 2 , 2 ] ( f 0 ) < σ 4 .

     

Remark 1.2 If a set E [ 0 , 1 ) satisfies dens ¯ E > 0 , then E d t 1 t = + .

2 Preliminary lemmas

Lemma 2.1 (see [10])

Let f ( z ) be a meromorphic function in Δ, and let k 1 be an integer. Then
m ( r , f ( k ) f ) = S ( r , f ) ,

where S ( r , f ) = O { log + T ( r , f ) + log ( 1 1 r ) } , possibly outside a set E 1 [ 0 , 1 ) with E 1 d t 1 t < .

Remark 2.1 Throughout this paper, we use E 1 [ 0 , 1 ) to denote a set satisfying E 1 d t 1 t < , not always the same at each occurrence.

Lemma 2.2 (see [9])

Let k and j be integers satisfying k > j 0 , and let ε > 0 and d ( 0 , 1 ) . If f is a meromorphic function in Δ such that f ( j ) does not vanish identically, then
| f ( k ) ( z ) f ( j ) ( z ) | ( ( 1 1 | z | ) 2 + ε max { log ( 1 1 | z | ) , T ( s ( | z | ) , f ) } ) k j ( | z | E 1 ) ,

where s ( | z | ) = 1 d ( 1 | z | ) .

Lemma 2.3 (see [14])

Let 1 q p be integers. If A 0 ( z ) , , A k 1 ( z ) are analytic functions of [ p , q ] -order in the unit disc. Then every solution f of (1.1) satisfies σ [ p + 1 , q ] ( f ) = σ M , [ p + 1 , q ] ( f ) max { σ M , [ p , q ] ( A j ) | j = 0 , 1 , , k 1 } .

By a similar proof to Lemma 2.3, we have the following lemma.

Lemma 2.4 If A 0 ( z ) , , A k 1 ( z ) are analytic functions of [ p , p + 1 ] -order in the unit disc with max { σ M , [ p , p + 1 ] ( A j ) | j = 0 , 1 , , k 1 } < . Then every solution f of (1.1) satisfies σ [ p + 1 , p + 1 ] ( f ) σ M , [ p + 1 , p + 1 ] ( f ) max { σ M , [ p , p + 1 ] ( A j ) | j = 0 , 1 , , k 1 } .

Lemma 2.5 Let 1 q p or 2 q = p + 1 and f ( z ) be an analytic function in Δ satisfying 0 σ [ p , q ] ( f ) = σ 5 (or 0 σ M , [ p , q ] ( f ) = σ 5 ), then there exists a set E 2 [ 0 , 1 ) satisfying E 2 d t 1 t = + such that, for all r E 2 , we have
lim r 1 log p T ( r , f ) log q ( 1 1 r ) = σ 5 ( lim r 1 log p + 1 M ( r , f ) log q ( 1 1 r ) = σ 5 ) .
Proof If 1 q p , by Definition 1.3, there exists a sequence { r n } n = 1 1 satisfying 1 d ( 1 r n ) < r n + 1 ( 0 < d < 1 ) and
lim n log p T ( r n , f ) log q ( 1 1 r n ) = σ [ p , q ] ( f ) = σ 5 .
Therefore there exists an n 1 ( N ) such that, for n n 1 and for any r E 2 = n = n 1 [ r n , 1 d ( 1 r n ) ] , we have
log p T ( r , f ) log q ( 1 1 r ) log p T ( r n , f ) log q [ 1 1 [ 1 d ( 1 r n ) ] ] = log p T ( r n , f ) log q [ 1 d ( 1 r n ) ] .
Hence
lim ̲ r 1 log p T ( r , f ) log q ( 1 1 r ) σ 5 ( r E 2 ) .
Since σ [ p , q ] ( f ) = σ 5 , for any r E 2 , we have
lim r 1 log p T ( r , f ) log q ( 1 1 r ) = σ 5 ,
where
m l E 2 = n = n 1 r n 1 d ( 1 r n ) d t 1 t = n = n 1 log 1 d = + .

We also can prove lim r 1 log p + 1 M ( r , f ) log q ( 1 1 r ) = σ 5 ( r E 2 ) by the above proof.

By the above proof, this lemma also holds for the case 2 q = p + 1 . □

Lemma 2.6 Let A j ( z ) ( j = 0 , 1 , , k 1 ), F ( z ) 0 be analytic functions in Δ. Then the following statements hold:
  1. (i)

    If p q 1 and f ( z ) is a solution of (1.2) satisfying max { σ [ p , q ] ( A j ) , σ [ p , q ] ( F ) | j = 0 , 1 , , k 1 } < σ [ p , q ] ( f ) , then λ ¯ [ p , q ] N ¯ ( f ) = λ [ p , q ] N ( f ) = σ [ p , q ] ( f ) .

     
  2. (ii)

    If f ( z ) is a solution of (1.2) satisfying max { σ [ p , p + 1 ] ( A j ) , σ [ p , p + 1 ] ( F ) , 1 | j = 0 , 1 , , k 1 } < σ [ p , p + 1 ] ( f ) , then λ ¯ [ p , p + 1 ] N ¯ ( f ) = λ [ p , p + 1 ] N ( f ) = σ [ p , p + 1 ] ( f ) .

     
Proof (i) Suppose that f ( z ) 0 is a solution of (1.2). By (1.2), we get
1 f = 1 F ( f ( k ) f + A k 1 f ( k 1 ) f + + A 0 ) ,
(2.1)
and it is easy to see that if f has a zero at z 0 of order α ( α > k ), and A 0 , , A k 1 are analytic at z 0 , then F must have a zero at z 0 of order α k , hence
n ( r , 1 f ) k n ¯ ( r , 1 f ) + n ( r , 1 F )
and
N ( r , 1 f ) k N ¯ ( r , 1 f ) + N ( r , 1 F ) .
(2.2)
By Lemma 2.1 and (2.1), we have
m ( r , 1 f ) m ( r , 1 F ) + j = 0 k 1 m ( r , A j ) + O { log + T ( r , f ) + log ( 1 1 r ) } ( r E 1 ) .
(2.3)
By (2.2)-(2.3), we get
T ( r , f ) = T ( r , 1 f ) + O ( 1 ) k N ¯ ( r , 1 f ) + T ( r , F ) + j = 0 k 1 T ( r , A j ) + O { log + T ( r , f ) + log ( 1 1 r ) } ( r E 1 ) .
(2.4)
Since max { σ [ p , q ] ( F ) , σ [ p , q ] ( A j ) | j = 0 , 1 , , k 1 } < σ [ p , q ] ( f ) , by Lemma 2.5 and Definition 1.3, there exists a set E 3 with E 3 d t 1 t = + such that
max { T ( r , F ) T ( r , f ) , T ( r , A j ) T ( r , f ) } 0 ( r 1 , r E 3 , j = 0 , , k 1 ) .
(2.5)
By (2.4)-(2.5), for all | z | = r E 3 E 1 , we have
( 1 o ( 1 ) ) T ( r , f ) k N ¯ ( r , 1 f ) + O { log + T ( r , f ) + log ( 1 1 r ) } ,
then we get σ [ p , q ] ( f ) λ ¯ [ p , q ] N ¯ ( f ) . Therefore λ ¯ [ p , q ] N ¯ ( f ) = λ [ p , q ] N ( f ) = σ [ p , q ] ( f ) .
  1. (ii)

    By a similar proof to case (i), we can easily obtain the conclusion of case (ii). □

     

Lemma 2.7 (see [17])

Let g : ( 0 , 1 ) R and h : ( 0 , 1 ) R be monotone increasing functions such that g ( r ) h ( r ) holds outside of an exceptional set E 1 [ 0 , 1 ) with E 1 d t 1 t < . Then there exists a constant d ( 0 , 1 ) such that if s ( r ) = 1 d ( 1 r ) , then g ( r ) h ( s ( r ) ) for all r [ 0 , 1 ) .

Lemma 2.8 (see [18])

Suppose that f ( z ) is meromorphic in Δ with f ( 0 ) = 0 . Then
m ( r , f ) [ 1 + φ ( r R ) ] T ( R , f ) + N ( R , f ) ,
(2.6)

where 0 < r < R < 1 , φ ( t ) = 1 π log 1 + t 1 t .

Lemma 2.9 Let f ( z ) be an analytic function of [ p , q ] -order in Δ. Then the following statements hold:
  1. (i)

    If p q 1 , then σ [ p , q ] ( f ) = σ [ p , q ] ( f ) .

     
  2. (ii)

    If 3 q = p + 1 , then σ [ p , p + 1 ] ( f ) max { σ [ p , p + 1 ] ( f ) , 1 } and σ [ p , p + 1 ] ( f ) max { σ [ p , p + 1 ] ( f ) , 1 } .

     
  3. (iii)

    If p = 1 , q = 2 , then σ [ 1 , 2 ] ( f ) max { σ [ 1 , 2 ] ( f ) , 1 } and σ [ 1 , 2 ] ( f ) 1 + σ [ 1 , 2 ] ( f ) .

     
Proof By Lemma 2.1, we have
T ( r , f ) 2 T ( r , f ) + m ( r , f f ) 3 T ( r , f ) + O { log 1 1 r } ( 0 < r < 1 , r E 1 ) .
(2.7)
By (2.7) and Lemma 2.7, it is easy to see σ [ p , q ] ( f ) σ [ p , q ] ( f ) ( p q 1 ) and σ [ p , p + 1 ] ( f ) max { σ [ p , p + 1 ] ( f ) , 1 } . On the other hand, set R = 1 + r 2 , 0 < r < 1 , by Lemma 2.8, we have
T ( r , f ) < ( 3 + log 4 ( 1 r ) ) T ( 1 + r 2 , f ) .
(2.8)

By (2.8), we have, if p q 1 , then σ [ p , q ] ( f ) σ [ p , q ] ( f ) and if 3 q = p + 1 , then σ [ p , p + 1 ] ( f ) max { σ [ p , p + 1 ] ( f ) , 1 } ; and we can easily obtain the conclusion (iii) by (2.7) and (2.8). Therefore Lemma 2.9 holds. □

3 Proofs of Theorems 1.1-1.3

Proof of Theorem 1.1 (i) Let H 5 = { | z | : z H 3 Δ } , since dens ¯ Δ { | z | : z H 3 Δ } > 0 , then by Remark 1.2, H 5 is a set of r with H 5 d t 1 t = + . For any | z | = r H 5 and r 1 , we have
| A 0 ( z ) | exp p { α 2 [ log q 1 ( 1 1 r ) ] σ 3 } , | A j ( z ) | exp p { β 2 [ log q 1 ( 1 1 r ) ] σ 3 } ( j = 1 , , k 1 ) .
(3.1)
If f 0 , from (1.1), we get
| A 0 | | f ( k ) f | + | A k 1 | | f ( k 1 ) f | + + | A 1 | | f f | .
(3.2)
By Lemma 2.2, for | z | = r E 1 , we get
| f ( j ) ( z ) f ( z ) | ( 1 1 r ) M T ( s ( r ) , f ) j ( j = 1 , , k ) ,
(3.3)
where M denotes a positive constant, not always the same at each occurrence. By (3.1)-(3.3), for all z satisfying | z | = r H 5 E 1 and r 1 , we have
exp p { α 2 [ log q 1 ( 1 1 r ) ] σ 3 } k exp p { β 2 [ log q 1 ( 1 1 r ) ] σ 3 } ( 1 1 r ) M T ( s ( r ) , f ) k .
(3.4)
If p q 1 , by (3.4), then σ 3 σ [ p + 1 , q ] ( f ) . On the other hand, by Lemma 2.3, we have σ [ p + 1 , q ] ( f ) max { σ M , [ p , q ] ( A j ) | j = 0 , 1 , , k 1 } σ 3 . Therefore every solution f ( z ) 0 of (1.1) satisfies σ [ p + 1 , q ] ( f ) = σ M , [ p + 1 , q ] ( f ) = σ 3 .
  1. (ii)

    If 2 q = p + 1 and σ 3 > 1 , by a similar proof to case (i), we obtain the conclusion. □

     
Proof of Theorem 1.2 (i) Let H 6 = { | z | : z H 4 Δ } , since dens ¯ Δ { | z | : z H 4 Δ } > 0 , then by Remark 1.2, H 6 is a set of r with H 6 d t 1 t = + . For any | z | = r H 6 and r 1 , we have
T ( r , A 0 ) exp p 1 { α 3 [ log q 1 ( 1 1 r ) ] σ 4 } , T ( r , A j ) exp p 1 { β 3 [ log q 1 ( 1 1 r ) ] σ 4 } ( j = 1 , , k 1 ) .
(3.5)
If f 0 , from (1.1), we get
A 0 ( z ) = f ( k ) ( z ) f ( z ) + + A j ( z ) f ( j ) ( z ) f ( z ) + + A 1 ( z ) f ( z ) f ( z ) ,
then
T ( r , A 0 ) i = 1 k 1 T ( r , A j ) + j = 1 k m ( r , f ( k ) f ) + O ( 1 ) .
(3.6)
By Lemma 2.1 and (3.6), there exists a set E 1 [ 0 , 1 ) with E 1 d t 1 t < such that, for all z satisfying | z | = r E 1 , we have
T ( r , A 0 ) i = 1 k 1 T ( r , A j ) + O { log + T ( r , f ) + log ( 1 1 r ) } .
(3.7)
By (3.5), (3.7), for | z | = r H 6 E 1 and r 1 , we have
exp p 1 { α 3 [ log q 1 ( 1 1 r ) ] σ 4 } ( k 1 ) exp p 1 { β 3 [ log q 1 ( 1 1 r ) ] σ 4 } + O { log + T ( r , f ) + log ( 1 1 r ) } .

If p q 2 and 0 < β 3 < α 3 , then every solution f ( z ) 0 of (1.1) satisfies σ 4 σ [ p + 1 , q ] ( f ) = σ M , [ p + 1 , q ] ( f ) . On the other hand, by Lemma 2.3, all solutions of (1.1) satisfy σ [ p + 1 , q ] ( f ) = σ M , [ p + 1 , q ] ( f ) max { σ M , [ p , q ] ( A j ) | j = 0 , 1 , , k 1 } σ 4 . Therefore every solution f ( z ) 0 of (1.1) satisfies σ [ p + 1 , q ] ( f ) = σ 4 .

(ii)-(iii) By a similar proof to case (i), we obtain the conclusions of (ii)-(iii). □

Proof of Theorem 1.3 (i) For 1 q p , assume that f is a solution of (1.2), by the elementary theory of differential equations, thus all the solutions of (1.2) have the form
f = f + C 1 f 1 + C 2 f 2 + + C k f k ,
where C 1 , , C k are complex constants, f 1 , , f k is a solution base of (1.1), f is a solution of (1.2) and has the form
f = D 1 f 1 + D 2 f 2 + + D k f k ,
(3.8)
where D 1 , , D k are certain analytic functions in Δ satisfying
D j = F G j ( f 1 , , f k ) W ( f 1 , , f k ) 1 ( j = 1 , , k ) ,
(3.9)

where G j ( f 1 , , f k ) are differential polynomials in f 1 , , f k and their derivative with constant coefficients, and W ( f 1 , , f k ) is the Wronskian of f 1 , , f k .

If σ [ p + 1 , q ] ( F ) > σ 3 , by Lemma 2.3, Lemma 2.9, and (3.8)-(3.9), we find that all solutions of (1.2) satisfy
σ [ p + 1 , q ] ( f ) max { σ [ p + 1 , q ] ( f j ) , σ [ p + 1 , q ] ( F ) | j = 1 , , k } = max { σ 3 , σ [ p + 1 , q ] ( F ) } σ [ p + 1 , q ] ( F ) .
On the other hand, by a simple order comparison from (1.2), we see that all solutions of (1.2) satisfy σ [ p + 1 , q ] ( f ) σ [ p + 1 , q ] ( F ) . Therefore all solutions of (1.2) satisfy
σ [ p + 1 , q ] ( f ) = σ [ p + 1 , q ] ( F ) .
If σ [ p + 1 , q ] ( F ) σ 3 , by the above proof in (3.8)-(3.9), we can find that all solutions of (1.2) satisfy σ [ p + 1 , q ] ( f ) σ 3 . We affirm that (1.2) can only possess at most one exceptional solution f 0 satisfying σ [ p + 1 , q ] ( f 0 ) < σ 3 . In fact, if f is another solution satisfying σ [ p + 1 , q ] ( f ) < σ 3 , then σ [ p + 1 , q ] ( f 0 f ) < σ 3 . But f 0 f is a solution of (1.1) and satisfies σ [ p + 1 , q ] ( f 0 f ) = σ 3 by Theorem 1.1(i), this is a contradiction. Then σ [ p + 1 , q ] ( f ) = σ 3 holds for all solutions of (1.2) with at most one exceptional solution f 0 satisfying σ [ p + 1 , q ] ( f 0 ) < σ 3 . By Lemma 2.6(i), we get
λ ¯ [ p + 1 , q ] N ¯ ( f ) = λ [ p + 1 , q ] N ( f ) = σ [ p + 1 , q ] ( f )
holds for all solutions satisfying σ [ p + 1 , q ] ( f ) = σ 3 with at most one exceptional solution f 0 satisfying σ [ p + 1 , q ] ( f 0 ) < σ 3 .
  1. (ii)

    For 2 q = p + 1 , σ 3 > 1 , by a similar proof to case (i), we draw the conclusions of case (ii). □

     

Declarations

Acknowledgements

This project is supported by the Natural Science Foundation of Jiangxi Province in China (20132BAB211002) and by the Science and Technology Plan Program of Education Bureau of Jiangxi Province (GJJ14271, GJJ14272). Zu-Xing Xuan is supported in part by Beijing Natural Science Foundation (No. 1132013) and The Project of Construction of Innovative Teams and Teacher Career Development for Universities and Colleges under Beijing Municipality (CIT and TCD20130513).

Authors’ Affiliations

(1)
College of Mathematics and Information Science, Jiangxi Normal University
(2)
Beijing Key Laboratory of Information Service Engineering, Department of General Education, Beijing Union University

References

  1. Hayman W: Meromorphic Functions. Clarendon, Oxford; 1964.MATHGoogle Scholar
  2. Laine I: Nevanlinna Theory and Complex Differential Equations. de Gruyter, Berlin; 1993.View ArticleMATHGoogle Scholar
  3. Tsuji M: Potential Theory in Modern Function Theory. Chelsea, New York; 1975. (reprint of the 1959 edition)MATHGoogle Scholar
  4. Yang L: Value Distribution Theory and Its New Research. Science Press, Beijing; 1982. (in Chinese)Google Scholar
  5. Pommerenke C: On the mean growth of the solutions of complex linear differential equations in the disc. Complex Var. Elliptic Equ. 1982, 1(1):23–38. 10.1080/17476938208814004MathSciNetView ArticleMATHGoogle Scholar
  6. Benbourenane D, Sons LR: On global solutions of linear differential equations in the unit disc. Complex Var. Elliptic Equ. 2004, 49: 913–925.MathSciNetMATHGoogle Scholar
  7. Cao TB, Yi HX: The growth of solutions of complex differential equations in the unit disc. J. Math. Anal. Appl. 2006, 319: 278–294. 10.1016/j.jmaa.2005.09.050MathSciNetView ArticleMATHGoogle Scholar
  8. Chen ZX, Shon KH: The growth of solutions of differential equations with coefficients of small growth in the unit disc. J. Math. Anal. Appl. 2004, 297: 285–304. 10.1016/j.jmaa.2004.05.007MathSciNetView ArticleMATHGoogle Scholar
  9. Chyzhykov I, Gundersen G, Heittokangas J: Linear differential equations and logarithmic derivate of estimates. Proc. Lond. Math. Soc. 2003, 86: 735–754. 10.1112/S0024611502013965MathSciNetView ArticleMATHGoogle Scholar
  10. Heittokangas J: On complex differential equations in the unit disc. Ann. Acad. Sci. Fenn., Math. Diss. 2000, 122: 1–54.MathSciNetMATHGoogle Scholar
  11. Heittokangas J, Korhonen R, Rättyä J: Fast growing solutions of linear differential equations in the unit disc. Results Math. 2006, 49: 265–278. 10.1007/s00025-006-0223-3MathSciNetView ArticleMATHGoogle Scholar
  12. Liu J, Tu J, Shi LZ:Linear differential equations with coefficients of ( p , q ) -order in the complex plane. J. Math. Anal. Appl. 2010, 372: 55–67. 10.1016/j.jmaa.2010.05.014MathSciNetView ArticleMATHGoogle Scholar
  13. Belaïdi B:Growth and oscillation theory of [ p , q ] -order analytic solutions of linear equations in the unit disc. J. Math. Anal. 2012, 3: 6–7.MathSciNetMATHGoogle Scholar
  14. Belaïdi B:Growth of solutions to linear equations with analytic coefficients of [ p , q ] -order in the unit disc. Electron. J. Differ. Equ. 2011., 2011: Article ID 156Google Scholar
  15. Belaïdi B:On the [ p , q ] -order of analytic solutions of linear equations in the unit disc. Novi Sad J. Math. 2012, 42: 117–129.MathSciNetMATHGoogle Scholar
  16. Li LM, Cao TB:Solutions for linear differential equations with meromorphic coefficients of ( p , q ) -order in the plane. Electron. J. Differ. Equ. 2012., 2012: Article ID 195Google Scholar
  17. Bank S: A general theorem concerning the growth of solutions of first-order algebraic differential equations. Compos. Math. 1972, 25: 61–70.MathSciNetMATHGoogle Scholar
  18. Shea DF, Sons LR: Value distribution theory for meromorphic functions of slow growth in the disk. Houst. J. Math. 1986, 12(2):249–266.MathSciNetMATHGoogle Scholar

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© Tu and Xuan; licensee Springer 2014

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