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Complex oscillation of a second-order linear differential equation with entire coefficients of [ p , q ] φ order

Advances in Difference Equations20142014:200

https://doi.org/10.1186/1687-1847-2014-200

  • Received: 22 February 2014
  • Accepted: 22 May 2014
  • Published:

Abstract

In this paper, the authors investigate the interaction between the growth, zeros of solutions with the coefficients of second-order linear differential equations in terms of [ p , q ] φ order and obtain some results in general form.

MSC:30D35, 34A20.

Keywords

  • linear differential equations
  • [ p , q ] φ order
  • [ p , q ] φ exponent of convergence of zero sequence

1 Introduction and notations

In this paper, we shall assume that readers are familiar with the standard notations of Nevanlinna value distribution theory (see [13]). The theory of complex linear equations has been developed since 1960s. Many authors have investigated the second-order linear differential equation
f + A ( z ) f = 0 ,
(1.1)

where A ( z ) is an entire function or a meromorphic function of finite order or finite iterated order, and have obtained many results about the interaction between the solutions and the coefficient of (1.1) (see [47]). What about the case when A ( z ) is an entire function of [ p , q ] -order or more general growth? In the following, we will introduce some notations about [ p , q ] -order, where p and q are two positive integers and satisfy p q 1 throughout this paper (see [811]). Firstly, for r [ 0 , + ) , we define exp 1 r = e r and exp i + 1 r = exp ( exp i r ) , i N , and for all sufficiently large r, we define log 1 r = log r and log i + 1 r = log ( log i r ) , i N . Especially, we have exp 0 r = r = log 0 r and exp 1 r = log 1 r . Secondly, we denote the linear measure and the logarithmic measure of a set E ( 1 , + ) by m E = E d t and m l E = E d t t .

Definition 1.1 ([10])

If f ( z ) is a meromorphic function, the [ p , q ] -order of f ( z ) is defined by
σ [ p , q ] ( f ) = lim ¯ r log p T ( r , f ) log q r .
(1.2)
Especially, if f ( z ) is an entire function, then the [ p , q ] -order of f ( z ) is defined by (see [8, 9, 11, 12])
σ [ p , q ] ( f ) = lim ¯ r log p T ( r , f ) log q r = lim ¯ r log p + 1 M ( r , f ) log q r .
(1.3)

Remark 1.1 We use σ [ 1 , 1 ] ( f ) = σ ( f ) and σ [ p , 1 ] ( f ) = σ p ( f ) to denote the order and the iterated order of a function f ( z ) .

Definition 1.2 ([10, 13])

The growth index (or the finiteness degree) of the iterated order of a meromorphic function f ( z ) is defined by
i ( f ) = { 0 if  f  is rational , min { n N : σ n ( f ) < } if  f  is transcendental and  σ n ( f ) <  for some  n N , if with  σ n ( f ) =  for all  n N .

Remark 1.2 By Definition 1.2, we can similarly give the definition of the growth index of the iterated exponent of convergence of the zero-sequence of a meromorphic function f ( z ) by i λ ( f , 0 ) .

Definition 1.3 ([10, 11])

The [ p , q ] exponent of convergence of the (distinct) zero-sequence of a meromorphic function f ( z ) is respectively defined by
λ [ p , q ] ( f ) = lim ¯ r log p n ( r , 1 f ) log q r = lim ¯ r log p N ( r , 1 f ) log q r ,
(1.4)
λ ¯ [ p , q ] ( f ) = lim ¯ r log p n ¯ ( r , 1 f ) log q r = lim ¯ r log p N ¯ ( r , 1 f ) log q r .
(1.5)

Definition 1.4 ([10])

The [ p , q ] exponent of convergence of the (distinct) pole-sequence of a meromorphic function f ( z ) is respectively defined by
λ [ p , q ] ( 1 f ) = lim ¯ r log p n ( r , f ) log q r ,
(1.6)
λ ¯ [ p , q ] ( 1 f ) = lim ¯ r log p n ¯ ( r , f ) log q r .
(1.7)

Remark 1.3 We use λ [ 1 , 1 ] ( f ) = λ ( f ) , λ [ p , 1 ] ( f ) = λ p ( f ) and λ [ 1 , 1 ] ( 1 f ) = λ ( 1 f ) , λ [ p , 1 ] ( 1 f ) = λ p ( 1 f ) to denote the (iterated) exponent of convergence of the zero-sequence and pole-sequence of a meromorphic function f ( z ) .

Recently, some authors have investigated the exponent of convergence of the zero-sequence and pole-sequence of the solutions of second-order linear differential equations (see [1315]) and have obtained the following results.

Theorem A ([5])

Let A be a transcendental meromorphic function of order σ ( A ) , where 0 < σ ( A ) , and assume that λ ¯ ( A ) < σ ( A ) . Then, if f 0 is a meromorphic solution of (1.1), we have
σ ( A ) max { λ ¯ ( f ) , λ ¯ ( 1 f ) } .

Theorem B ([13])

Let A ( z ) be an entire function with i ( A ) = p N + . Let f 1 , f 2 be two linearly independent solutions of (1.1) and denote F = f 1 f 2 . Then i λ ( F , 0 ) p + 1 and
λ p + 1 ( F , 0 ) = σ p + 1 ( F ) = max { λ p + 1 ( f 1 , 0 ) , λ p + 1 ( f 2 , 0 ) } σ p ( A ) .

If i λ ( F , 0 ) p , then i λ ( f , 0 ) = p + 1 holds for all solutions of type f = c 1 f 1 + c 2 f 2 , where c 1 c 2 0 .

Theorem C ([13])

Let A ( z ) be an entire function with 0 < i ( A ) = p < , let f be any non-trivial solution of (1.1), and assume λ ¯ p ( A , 0 ) < σ p ( A ) 0 . Then λ p + 1 ( f , 0 ) σ p ( A ) λ p ( f , 0 ) .

Theorem D ([13])

Let A ( z ) be an entire function with i ( A ) = p and σ p ( A ) = σ < . Let f 1 and f 2 be two linearly independent solutions of (1.1) such that max { λ p ( f 1 , 0 ) , λ p ( f 2 , 0 ) } < σ . Let Π ( z ) 0 be any entire function for which either i ( Π ) < p or i ( Π ) = p and σ p ( Π ) < σ . Then any two linearly independent solutions g 1 and g 2 of the differential equation y + ( A ( z ) + Π ( z ) ) y = 0 satisfy max { λ p ( g 1 ) , λ p ( g 2 ) } σ .

Theorem E ([14])

Let A be a meromorphic function with i ( A ) = p N + , and assume that λ ¯ p ( A ) < σ p ( A ) . Then, if f is a nonzero meromorphic solution of (1.1), we have
σ p ( A ) max { λ ¯ p ( f ) , λ ¯ p ( 1 f ) } .
In the special case where either δ ( , f ) > 0 or the poles of f are of uniformly bounded multiplicities, we can conclude that
max { λ p + 1 ( f ) , λ p + 1 ( 1 f ) } σ p ( f ) { λ ¯ p ( f ) , λ ¯ p ( 1 f ) } .
In [16], Chyzhykov and his co-authors introduced the definition of φ-order of f ( z ) , where f ( z ) is a meromorphic function in the unit disc and used it to investigate the interaction between the analytic coefficients and solutions of
f ( k ) + A k 1 ( z ) f ( k 1 ) + + A 0 ( z ) f = 0

in the unit disc, where the definition of φ-order of f ( z ) is given as follows.

Definition 1.5 ([16])

Let φ : [ 0 , 1 ) ( 0 , + ) be a non-decreasing unbounded function, the φ-order of a meromorphic function f ( z ) in the unit disc is defined by
σ ( f , φ ) = lim ¯ r 1 log + T ( r , f ) log φ ( r ) .
(1.8)

On the basis of Definition 1.5, it is natural for us to give the [ p , q ] φ order of a meromorphic function f ( z ) in the complex plane.

Definition 1.6 Let φ : [ 0 , + ) ( 0 , + ) be a non-decreasing unbounded function, the [ p , q ] φ order and [ p , q ] φ lower order of a meromorphic function f ( z ) are respectively defined by
σ [ p , q ] ( f , φ ) = lim ¯ r log p T ( r , f ) log q φ ( r ) ,
(1.9)
μ [ p , q ] ( f , φ ) = lim ̲ r log p T ( r , f ) log q φ ( r ) .
(1.10)

Similar to Definition 1.6, we can also define the [ p , q ] φ exponent of convergence of the (distinct) zero-sequence of a meromorphic function f ( z ) .

Definition 1.7 The [ p , q ] φ exponent of convergence of the (distinct) zero-sequence of a meromorphic function f ( z ) is respectively defined by
λ [ p , q ] ( f , φ ) = lim ¯ r log p n ( r , 1 f ) log q φ ( r ) ,
(1.11)
λ ¯ [ p , q ] ( f , φ ) = lim ¯ r log p n ¯ ( r , 1 f ) log q φ ( r ) .
(1.12)
Proposition 1.1 If f 1 ( z ) , f 2 ( z ) are meromorphic functions satisfying σ [ p , q ] ( f 1 , φ ) = a , σ [ p , q ] ( f 2 , φ ) = b , then
  1. (i)

    σ [ p , q ] ( f 1 + f 2 , φ ) max { a , b } , σ [ p , q ] ( f 1 f 2 , φ ) max { a , b } ;

     
  2. (ii)

    If a b , σ [ p , q ] ( f 1 + f 2 , φ ) = max { a , b } , σ [ p , q ] ( f 1 f 2 , φ ) = max { a , b } .

     

In this paper, we add two conditions on φ ( r ) as follows: φ ( r ) : [ 0 , + ) ( 0 , + ) is a non-decreasing unbounded function and satisfies (i) lim r log p + 1 r log q φ ( r ) = 0 , (ii) lim r log q φ ( α r ) log q φ ( r ) = 1 for some α > 1 . Throughout this paper, we assume that φ ( r ) always satisfies the above two conditions without special instruction.

Proposition 1.2 Let φ ( r ) satisfy the above two conditions (i)-(ii).
  1. (i)
    If f ( z ) is an entire function, then
    σ [ p , q ] ( f , φ ) = lim ¯ r log p T ( r , f ) log q φ ( r ) = lim ¯ r log p + 1 M ( r , f ) log q φ ( r ) , μ [ p , q ] ( f , φ ) = lim ̲ r log p T ( r , f ) log q φ ( r ) = lim ̲ r log p + 1 M ( r , f ) log q φ ( r ) .
     
  2. (ii)
    If f ( z ) is a meromorphic function, then
    λ [ p , q ] ( f , φ ) = lim ¯ r log p n ( r , 1 f ) log q φ ( r ) = lim ¯ r log p N ( r , 1 f ) log q φ ( r ) , λ ¯ [ p , q ] ( f , φ ) = lim ¯ r log p n ¯ ( r , 1 f ) log q φ ( r ) = lim ¯ r log p N ¯ ( r , 1 f ) log q φ ( r ) .
     
Proof (i) By the inequality T ( r , f ) log + M ( r , f ) R + r R r T ( R , f ) ( 0 < r < R ), set R = α r ( α > 1 ), we have
T ( r , f ) log + M ( r , f ) α + 1 α 1 T ( α r , f ) .
(1.13)
By (1.13) and lim r log q φ ( α r ) log q φ ( r ) = 1 , it is easy to see that conclusion (i) holds.
  1. (ii)
    Without loss of generality, assume that f ( 0 ) 0 , then N ( r , 1 f ) = 0 r n ( t , 1 f ) t d t . Since
    N ( r , 1 f ) N ( r 0 , 1 f ) = r 0 r n ( t , 1 f ) t d t n ( r , 1 f ) log r r 0 ( 0 < r 0 < r ) ,
    (1.14)
     
then by (1.14) and lim r log p + 1 r log q φ ( r ) = 0 , we have
lim ¯ r log p N ( r , 1 f ) log q φ ( r ) max { lim ¯ r log p n ( r , 1 f ) log q φ ( r ) , lim ¯ r log p + 1 r log q φ ( r ) } = lim ¯ r log p n ( r , 1 f ) log q φ ( r ) .
(1.15)
On the other hand, since α > 1 , we have
N ( α r , 1 f ) = 0 α r n ( t , 1 f ) t d t r α r n ( t , 1 f ) t d t n ( r , 1 f ) log α .
(1.16)
By (1.16) and lim r log q φ ( α r ) log q φ ( r ) = 1 , we have
lim ¯ r log p N ( r , 1 f ) log q φ ( r ) lim ¯ r log p n ( r , 1 f ) log q φ ( r ) .
(1.17)

By (1.15) and (1.17), it is easy to see that λ [ p , q ] ( f , φ ) = lim ¯ r log p n ( r , 1 f ) log q φ ( r ) = lim ¯ r log p N ( r , 1 f ) log q φ ( r ) . By the same proof above, we can obtain the conclusion λ ¯ [ p , q ] ( f , φ ) = lim ¯ r log p n ¯ ( r , 1 f ) log q φ ( r ) = lim ¯ r log p N ¯ ( r , 1 f ) log q φ ( r ) . □

Remark 1.4 If φ ( r ) = r , Definitions 1.1 and 1.3 are special cases of Definitions 1.6 and 1.7.

2 Main results

In this paper, our aim is to make use of the concept of [ p , q ] φ order of entire functions to investigate the growth, zeros of the solutions of equation (1.1).

Theorem 2.1 Let A ( z ) be an entire function satisfying σ [ p , q ] ( A , φ ) > 0 . Then σ [ p + 1 , q ] ( f , φ ) = σ [ p , q ] ( A , φ ) holds for all non-trivial solutions of (1.1).

Theorem 2.2 Let A ( z ) be an entire function satisfying σ [ p , q ] ( A , φ ) > 0 , let f 1 , f 2 be two linearly independent solutions of (1.1) and denote F = f 1 f 2 . Then max { λ [ p + 1 , q ] ( f 1 , φ ) , λ [ p + 1 , q ] ( f 2 , φ ) } = λ [ p + 1 , q ] ( F , φ ) = σ [ p + 1 , q ] ( F , φ ) σ [ p , q ] ( A , φ ) . If σ [ p + 1 , q ] ( F , φ ) < σ [ p , q ] ( A , φ ) , then λ [ p + 1 , q ] ( f , φ ) = σ [ p , q ] ( A , φ ) holds for all solutions of type f = c 1 f 1 + c 2 f 2 , where c 1 c 2 0 .

Theorem 2.3 Let A ( z ) be an entire function satisfying λ ¯ [ p , q ] ( A , φ ) < σ [ p , q ] ( A , φ ) . Then λ [ p + 1 , q ] ( f , φ ) σ [ p , q ] ( A , φ ) λ [ p , q ] ( f , φ ) holds for all non-trivial solutions of (1.1).

Theorem 2.4 Let A ( z ) be an entire function satisfying σ [ p , q ] ( A , φ ) = σ 1 > 0 , let f 1 and f 2 be two linearly independent solutions of (1.1) such that max { λ [ p , q ] ( f 1 , φ ) , λ [ p , q ] ( f 2 , φ ) } < σ 1 . Let Π ( z ) 0 be any entire function satisfying σ [ p , q ] ( Π , φ ) < σ 1 . Then any two linearly independent solutions g 1 and g 2 of the differential equation f + ( A ( z ) + Π ( z ) ) f = 0 satisfy max { λ [ p , q ] ( g 1 , φ ) , λ [ p , q ] ( g 2 , φ ) } σ 1 .

3 Some lemmas

Lemma 3.1 ([1719])

Let f ( z ) be a transcendental entire function, and let z be a point with | z | = r at which | f ( z ) | = M ( r , f ) . Then, for all | z | outside a set E 1 of r of finite logarithmic measure, we have
f ( j ) ( z ) f ( z ) = ( v f ( r ) z ) j ( 1 + o ( 1 ) ) ( j N ) ,
(3.1)

where v f ( r ) is the central index of f ( z ) .

Lemma 3.2 ([7, 19, 20])

Let g : [ 0 , + ) R and h : [ 0 , + ) R be monotone non-decreasing functions such that g ( r ) h ( r ) outside of an exceptional set E 2 of finite linear measure or finite logarithmic measure. Then, for any d > 1 , there exists r 0 > 0 such that g ( r ) h ( d r ) for all r > r 0 .

Lemma 3.3 ([18, 21])

Let f ( z ) = n = 0 a n z n be an entire function, μ ( r ) be the maximum term, i.e., μ ( r ) = max { | a n | r n ; n = 0 , 1 , } , and let v f ( r ) be the central index of f.
  1. (i)
    If | a 0 | 0 , then
    log μ ( r ) = log | a 0 | + 0 r v f ( t ) t d t .
    (3.2)
     
  2. (ii)
    For r < R , we have
    M ( r , f ) < μ ( r ) { v f ( R ) + R R r } .
    (3.3)
     
Lemma 3.4 Let f ( z ) be an entire function satisfying σ [ p , q ] ( f , φ ) = σ 2 and μ [ p , q ] ( f , φ ) = μ 1 , and let v f ( r ) be the central index of f, then
lim ¯ r log p v f ( r ) log q φ ( r ) = σ 2 , lim ̲ r log p v f ( r ) log q φ ( r ) = μ 1 .
Proof Let f ( z ) = n = 0 a n z n . Without loss of generality, we can assume that | a 0 | 0 . From (3.2), for any 1 < α 1 < α , we have
log μ ( α 1 r ) = log | a 0 | + 0 α 1 r v f ( t ) t d t log | a 0 | + r α 1 r v f ( t ) t d t log | a 0 | + v f ( r ) log α 1 .
By the Cauchy inequality, it is easy to see μ ( α 1 r ) M ( α 1 r , f ) , hence
v f ( r ) log α 1 log M ( α 1 r , f ) + c 3 ,
(3.4)
where c 3 > 0 is a constant. By Proposition 1.2, (3.4) and lim r log q φ ( α 1 r ) log q φ ( r ) = 1 ( 1 < α 1 < α ), we have
lim ¯ r log p v f ( r ) log q φ ( r ) lim ¯ r log p + 1 M ( α 1 r , f ) log q φ ( α 1 r ) lim ¯ r log q φ ( α 1 r ) log q φ ( r ) = σ [ p , q ] ( f , φ ) ,
(3.5)
lim ̲ r log p v f ( r ) log q φ ( r ) lim ̲ r log p + 1 M ( α 1 r , f ) log q φ ( α 1 r ) lim r log q φ ( α 1 r ) log q φ ( r ) = μ [ p , q ] ( f , φ ) .
(3.6)
On the other hand, set R = α 1 r , by (3.3), we have
M ( r , f ) < μ ( r ) ( v f ( α 1 r ) + α 1 α 1 1 ) = | a v f ( α 1 r ) | r v f ( α 1 r ) ( v f ( α 1 r ) + α 1 α 1 1 ) .
(3.7)
Since { | a n | } n = 1 is a bounded sequence, by (3.7), we have
log p + 1 M ( r , f ) log p v f ( α 1 r ) [ 1 + log p + 1 v f ( α 1 r ) log p v f ( α 1 r ) ] + log p + 1 r + c 4 ,
(3.8)
where c 4 > 0 is a constant. By Proposition 1.2, (3.8), lim r log q φ ( α 1 r ) log q φ ( r ) = 1 ( 1 < α 1 < α ) and lim r log p + 1 r log q φ ( r ) = 0 , we have
σ [ p , q ] ( f , φ ) = lim ¯ r log p + 1 M ( r , f ) log q φ ( r ) lim ¯ r log p v f ( α 1 r ) log q φ ( α 1 r ) = lim ¯ r log p v f ( r ) log q φ ( r ) ,
(3.9)
μ [ p , q ] ( f , φ ) = lim ̲ r log p + 1 M ( r , f ) log q φ ( r ) lim ̲ r log p v f ( α 1 r ) log q φ ( α 1 r ) = lim ̲ r log p v f ( r ) log q φ ( r ) .
(3.10)

By (3.5), (3.6), (3.9) and (3.10), we obtain the conclusion of Lemma 3.4. □

Lemma 3.5 Let f 1 ( z ) and f 2 ( z ) be entire functions of [ p , q ] φ order and denote F = f 1 f 2 . Then
λ [ p , q ] ( F , φ ) = max { λ [ p , q ] ( f 1 , φ ) , λ [ p , q ] ( f 2 , φ ) } .
Proof Let n ( r , F ) , n ( r , f 1 ) and n ( r , f 2 ) be unintegrated counting functions for the number of zeros of F ( z ) , f 1 ( z ) and f 2 ( z ) . For any r > 0 , it is easy to see
n ( r , F ) max { n ( r , f 1 ) , n ( r , f 2 ) } .
(3.11)
By Definition 1.7 and (3.11), we have
λ [ p , q ] ( F , φ ) max { λ [ p , q ] ( f 1 , φ ) , λ [ p , q ] ( f 2 , φ ) } .
(3.12)
On the other hand, since the zeros of F ( z ) must be the zeros of f 1 ( z ) or the zeros of f 2 ( z ) , for any r > 0 , we have
n ( r , F ) n ( r , f 1 ) + n ( r , f 2 ) 2 max { n ( r , f 1 ) , n ( r , f 2 ) } .
(3.13)
By Definition 1.7 and (3.13), we have
λ [ p , q ] ( F , φ ) max { λ [ p , q ] ( f 1 , φ ) , λ [ p , q ] ( f 2 , φ ) } .
(3.14)

Therefore, by (3.12) and (3.14), we have λ [ p , q ] ( F , φ ) = { λ [ p , q ] ( f 1 , φ ) , λ [ p , q ] ( f 2 , φ ) } . □

Lemma 3.6 Let f ( z ) be a transcendental meromorphic function satisfying σ [ p , q ] ( f , φ ) = σ 3 , where φ ( r ) only satisfies log p + 1 r log q φ ( r ) = 0 , and let k be any positive integer. Then, for any ε > 0 , there exists a set E 3 having finite linear measure such that for all r E 3 , we have
m ( r , f ( k ) f ) = O { exp p 1 { ( σ 3 + ε ) log q φ ( r ) } } .
Proof Set k = 1 , since σ [ p , q ] ( f , φ ) = σ 3 < , for sufficiently large r and for any given ε > 0 , we have
T ( r , f ) < exp p { ( σ 3 + ε ) log q φ ( r ) } .
(3.15)
By the lemma of logarithmic derivative, we have
m ( r , f f ) = O { log T ( r , f ) + log r } ( r E 3 ) ,
(3.16)

where E 3 [ 0 , + ) is a set of finite linear measure, not necessarily the same at each occurrence. By (3.15), (3.16) and log p + 1 r log q φ ( r ) = 0 , we have m ( r , f f ) = O { exp p 1 { ( σ + ε ) log q φ ( r ) } } ( r E 3 ).

We assume that m ( r , f ( k ) f ) = O { exp p 1 { ( σ 3 + ε ) log q φ ( r ) } } ( r E 3 ) holds for any positive integer k. By N ( r , f ( k ) ) ( k + 1 ) N ( r , f ) , for all r E 3 , we have
T ( r , f ( k ) ) = m ( r , f ( k ) ) + N ( r , f ( k ) ) m ( r , f ( k ) f ) + m ( r , f ) + ( k + 1 ) N ( r , f ) ( k + 1 ) T ( r , f ) + O { exp p 1 { ( σ 3 + ε ) log q φ ( r ) } } .
(3.17)
By (3.16) and (3.17), for r E 3 , we have
m ( r , f ( k + 1 ) f ) m ( r , f ( k + 1 ) f ( k ) ) + m ( r , f ( k ) f ) = O { exp p 1 { ( σ 3 + ε ) log q φ ( r ) } } .

 □

Lemma 3.7 ([19])

Let f ( z ) be an entire function of [ p , q ] -order, and f ( z ) can be represented by the form
f ( z ) = U ( z ) e V ( z ) ,
where U ( z ) and V ( z ) are entire functions such that
λ [ p , q ] ( f ) = λ [ p , q ] ( U ) = σ [ p , q ] ( U ) , σ [ p , q ] ( f ) = max { σ [ p , q ] ( U ) , σ [ p , q ] ( e V ) } .

If f ( z ) is an entire function of [ p , q ] φ order, we have a similar result as follows.

Lemma 3.8 Let f ( z ) be an entire function of [ p , q ] φ order, and f ( z ) can be represented by the form
f ( z ) = U ( z ) e V ( z ) ,
where U ( z ) and V ( z ) are entire functions of [ p , q ] φ order such that
λ [ p , q ] ( f , φ ) = λ [ p , q ] ( U , φ ) = σ [ p , q ] ( U , φ ) , σ [ p , q ] ( f , φ ) = max { σ [ p , q ] ( U , φ ) , σ [ p , q ] ( e V , φ ) } .

4 Proofs of Theorems 2.1-2.4

Proof of Theorem 2.1 Set σ [ p , q ] ( A , φ ) = σ 4 > 0 . First, we prove that every solution of (1.1) satisfies σ [ p + 1 , q ] ( f , φ ) σ 4 . If f ( z ) is a polynomial solution of (1.1), it is easy to know that σ [ p + 1 , q ] ( f , φ ) = 0 σ 4 holds. If f ( z ) is a transcendental solution of (1.1), by (1.1) and Lemma 3.1, there exists a set E 1 ( 1 , + ) having finite logarithmic measure such that for all z satisfying | z | = r [ 0 , 1 ] E 1 and | f ( z ) | = M ( r , f ) , we have
( v f ( r ) r ) 2 ( 1 + o ( 1 ) ) exp p + 1 { ( σ 4 + ε 2 ) log q φ ( r ) } .
And hence, we have
v f ( r ) r exp p + 1 { ( σ 4 + ε ) log q φ ( r ) } ( r E 1 ) .
(4.1)
By (4.1) and Lemma 3.2, there exists some α 1 ( 1 < α 1 < α ) such that for all r r 0 , we have
v f ( r ) α 1 r exp p + 1 { ( σ 4 + ε ) log q φ ( α 1 r ) } .
(4.2)
By Lemma 3.4, (4.2) and the two conditions on φ ( r ) , we have
σ [ p + 1 , q ] ( f , φ ) = lim ¯ r log p + 1 v f ( r ) log q φ ( r ) σ 4 .
(4.3)
On the other hand, by (1.1), we have
m ( r , A ) = m ( r , f f ) = O { log r T ( r , f ) } .
(4.4)

By (4.4), we have σ [ p , q ] ( A , φ ) σ [ p + 1 , q ] ( f , φ ) . Therefore, we have that σ [ p + 1 , q ] ( f , φ ) = σ [ p , q ] ( A , φ ) holds for all non-trivial solutions of (1.1). □

Proof of Theorem 2.2 Set σ [ p , q ] ( A , φ ) = σ 5 > 0 , by Theorem 2.1, we have σ [ p + 1 , q ] ( f 1 , φ ) = σ [ p + 1 , q ] ( f 2 , φ ) = σ [ p , q ] ( A , φ ) = σ 5 . Hence, we have
λ [ p + 1 , q ] ( F , φ ) σ [ p + 1 , q ] ( F , φ ) max { σ [ p + 1 , q ] ( f 1 , φ ) , σ [ p + 1 , q ] ( f 2 , φ ) } = σ [ p , q ] ( A , φ ) .
(4.5)
By Lemma 3.5 and (4.5), we have
max { λ [ p + 1 , q ] ( f 1 , φ ) , λ [ p + 1 , q ] ( f 2 , φ ) } = λ [ p + 1 , q ] ( F , φ ) σ [ p + 1 , q ] ( F , φ ) σ [ p , q ] ( A , φ ) .
(4.6)
It remains to show that λ [ p + 1 , q ] ( F , φ ) = σ [ p + 1 , q ] ( F , φ ) . By (1.1), we have (see [[13], pp.76-77]) that all zeros of F ( z ) are simple and that
F 2 = C 2 ( ( F F ) 2 2 ( F F ) 4 A ) 1 ,
(4.7)
where C 0 is a constant. Hence,
2 T ( r , F ) = T ( r , ( F F ) 2 2 ( F F ) 4 A ) + O ( 1 ) O ( N ¯ ( r , 1 F ) + m ( r , F F ) + m ( r , F F ) + m ( r , A ) ) .
(4.8)
By Lemma 3.6, for all r E 3 , we have m ( r , A ) = m ( r , f f ) = O { exp p { ( σ 5 + ε ) log q φ ( r ) } } , m ( r , F F ) = O { exp p { ( σ 5 + ε ) log q φ ( r ) } } and m ( r , F F ) = O { exp p { ( σ 5 + ε ) log q φ ( r ) } } . By (4.8), for all r E 3 , we have
T ( r , F ) = O { N ¯ ( r , 1 F ) + exp p { ( σ 5 + ε ) log q φ ( r ) } } .
(4.9)
Let us assume λ [ p + 1 , q ] ( F , φ ) < β < σ [ p + 1 , q ] ( F , φ ) . Since all zeros of F ( z ) are simple, we have
N ¯ ( r , 1 F ) = N ( r , 1 F ) = O { exp p + 1 { β log q φ ( r ) } } .
(4.10)
By (4.9) and (4.10), for all r E 3 , we have
T ( r , F ) = O { exp p + 1 { β log q φ ( r ) } } .

By Definition 1.6 and Lemma 3.2, we have σ [ p + 1 , q ] ( F , φ ) β < σ [ p + 1 , q ] ( F , φ ) , this is a contradiction. Therefore, the first assertion is proved.

If σ [ p + 1 , q ] ( F , φ ) < σ [ p , q ] ( A , φ ) , let us assume that λ [ p + 1 , q ] ( f , φ ) < σ [ p , q ] ( A , φ ) holds for any solution of type f = c 1 f 1 + c 2 f 2 ( c 1 c 2 0 ). We denote F = f 1 f 2 and F 1 = f f 1 , then we have λ [ p + 1 , q ] ( F , φ ) < σ [ p , q ] ( A , φ ) and λ [ p + 1 , q ] ( F 1 , φ ) < σ [ p , q ] ( A , φ ) . Since (4.9) holds for F ( z ) and F 1 ( z ) and F 1 = f f 1 = ( c 1 f 1 + c 2 f 2 ) f 1 = c 1 f 1 2 + c 2 F , we have
T ( r , f 1 ) = O ( T ( r , F 1 ) + T ( r , F ) ) = O { N ¯ ( r , 1 F 1 ) + N ¯ ( r , 1 F ) + exp p { ( σ 5 + ε ) log q φ ( r ) } } .
(4.11)
By λ [ p + 1 , q ] ( F , φ ) < σ [ p , q ] ( A , φ ) , λ [ p + 1 , q ] ( F 1 , φ ) < σ [ p , q ] ( A , φ ) and (4.10), for some β < σ [ p , q ] ( A , φ ) , we have
T ( r , f 1 ) = O { exp p + 1 { β log q φ ( r ) } } .
(4.12)

By Definition 1.6 and (4.12), we have σ [ p + 1 , q ] ( f 1 , φ ) β < σ [ p , q ] ( A , φ ) , this is a contradiction with Theorem 2.1. Therefore, we have that λ [ p + 1 , q ] ( f , φ ) = σ [ p , q ] ( A , φ ) holds for all solutions of type f = c 1 f 1 + c 2 f 2 , where c 1 c 2 0 . □

Proof of Theorem 2.3 By Theorem 2.1 and λ [ p + 1 , q ] ( f , φ ) σ [ p + 1 , q ] ( f , φ ) , it is easy to know that λ [ p + 1 , q ] ( f , φ ) σ [ p , q ] ( A , φ ) holds. It remains to show that σ [ p , q ] ( A , φ ) λ [ p , q ] ( f , φ ) . Let us assume σ [ p , q ] ( A , φ ) > λ [ p , q ] ( f , φ ) . By (1.1) and a similar proof of Theorem 5.6 in [[13], p.82], we have
T ( r , f f ) = O { N ¯ ( r , 1 f ) + N ¯ ( r , 1 A ) } ( r E 3 ) .
(4.13)
By (4.13), the assumption σ [ p , q ] ( A , φ ) > λ [ p , q ] ( f , φ ) and λ ¯ [ p , q ] ( A , φ ) σ [ p , q ] ( A , φ ) , for some β < σ [ p , q ] ( A , φ ) , we have
T ( r , f f ) = O { exp p { β log q φ ( r ) } } .
(4.14)
By Definition 1.6 and (4.14), we have σ [ p , q ] ( f f , φ ) = σ [ p , q ] ( f f , φ ) β < σ [ p , q ] ( A , φ ) . By
A ( z ) = ( f f ) + ( f f ) 2 ,

we have σ [ p , q ] ( A , φ ) σ [ p , q ] ( f f , φ ) < σ [ p , q ] ( A , φ ) , this is a contradiction. Therefore, we have that λ [ p + 1 , q ] ( f , φ ) σ [ p , q ] ( A , φ ) λ [ p , q ] ( f , φ ) holds for all non-trivial solutions of (1.1). □

Proof of Theorem 2.4 As a similar proof of Theorem 3.1 in [6], we denote F = f 1 f 2 and F 2 = g 1 g 2 . Let us assume
λ [ p , q ] ( F 2 , φ ) = max { λ [ p , q ] ( g 1 , φ ) , λ [ p , q ] ( g 2 , φ ) } < σ 1 .
By Theorem 2.1, we have σ [ p + 1 , q ] ( F , φ ) max { σ [ p + 1 , q ] ( f 1 , φ ) , σ [ p + 1 , q ] ( f 2 , φ ) } = σ 1 , and hence, by Lemma 3.6, for any integer k 1 and for any ε > 0 , we have
m ( r , F ( k ) F ) = O { exp p { ( σ 1 + ε ) log q φ ( r ) } } ( r E 3 ) .
Furthermore, by Theorem 2.1, we have λ [ p , q ] ( F , φ ) = max { λ [ p , q ] ( f 1 , φ ) , λ [ p , q ] ( f 2 , φ ) } < σ 1 , and hence we have N ¯ ( r , 1 F ) = O { exp p { β log q φ ( r ) } } for some β < σ 1 . And the [ p , q ] φ order of the function A ( z ) implies that
T ( r , A ) = O { exp p { ( σ 1 + ε ) log q φ ( r ) } } ( r ) .
By (4.9), we obtain
T ( r , F ) = O { N ¯ ( r , 1 F ) + exp p { ( σ 1 + ε ) log q φ ( r ) } } = O { exp p { ( β log q φ ( r ) } } .
(4.15)
By Definition 1.6 and (4.15), we have σ [ p , q ] ( F , φ ) σ 1 . On the other hand, by
4 A = ( F F ) 2 2 F F 1 F 2 ,
(4.16)
we have σ [ p , q ] ( A , φ ) = σ 1 σ [ p , q ] ( F , φ ) , hence σ [ p , q ] ( F , φ ) = σ 1 . The same reasoning is valid for the function F 2 , we have
4 ( A + Π ) = ( F 2 F 2 ) 2 2 F 2 F 2 1 F 2 2 ,
(4.17)
and σ [ p , q ] ( F 2 , φ ) = σ 1 . Since λ [ p , q ] ( F , φ ) < σ 1 and λ [ p , q ] ( F 2 , φ ) < σ 1 , by Lemma 3.8, we may write
F = Q e P , F 2 = R e S ,
(4.18)
where P, Q, R, S are entire functions satisfying σ [ p , q ] ( Q , φ ) = λ [ p , q ] ( F , φ ) < σ 1 , σ [ p , q ] ( R , φ ) = λ [ p , q ] ( F 2 , φ ) < σ 1 and σ [ p , q ] ( e P , φ ) = σ [ p , q ] ( e S , φ ) = σ 1 . Substituting (4.18) into (4.16) and (4.17), we have
4 A = 1 Q 2 e 2 P + G 1 ( z ) ,
(4.19)
4 ( A + π ) = 1 R 2 e 2 S + G 2 ( z ) ,
(4.20)
where G 1 ( z ) and G 2 ( z ) are meromorphic functions satisfying σ [ p , q ] ( G j , φ ) < σ 1 ( j = 1 , 2 ). Equation (4.19) subtracting (4.20), we have
1 R 2 e 2 S 1 Q 2 e 2 P = G 3 ( z ) ,
(4.21)
where G 3 ( z ) is a meromorphic function satisfying σ [ p , q ] ( G 3 , φ ) < σ 1 . From (4.21), we have
e 2 S + H 1 e 2 P = H 2 ,
(4.22)
where H 1 ( z ) and H 2 ( z ) are meromorphic functions satisfying σ [ p , q ] ( H j , φ ) < σ 1 ( j = 1 , 2 ), and H 1 = R 2 Q 2 . Deriving (4.22), we have
2 S e 2 S + ( H 1 2 P H 1 ) e 2 P = H 3 ,
(4.23)
where H 3 ( z ) is a meromorphic function satisfying σ [ p , q ] ( H 3 , φ ) < σ 1 . Eliminating e 2 S by (4.22) and (4.23), we have
( H 1 2 ( P S ) H 1 ) e 2 P = H 4 ,
(4.24)
where H 4 ( z ) is a meromorphic function satisfying σ [ p , q ] ( H 4 , φ ) < σ 1 . Since σ [ p , q ] ( e P , φ ) = σ 1 , therefore by (4.24), we have H 1 2 ( P S ) H 1 0 , thus we have H 1 = c e 2 ( P S ) , c 0 . Hence
F 2 F 2 2 = Q 2 R 2 e 2 ( P S ) = 1 c .
(4.25)
From (4.16), (4.17) and (4.25), we have
4 ( A + Π + 1 c A ) = ( F 2 F 2 ) 2 2 F 2 F 2 + 1 c ( F F ) 2 2 c F F .
By Lemma 3.6, we obtain
T ( r , ( 1 + 1 c ) A + Π ) = m ( r , ( 1 + 1 c ) A + Π ) = O { exp p 1 { ( σ 1 + ε ) log q φ ( r ) } } ( r ) .
This implies
σ [ p , q ] ( ( 1 + 1 c ) A + Π , φ ) = 0 .
Hence, by Proposition 1.1, we have c = 1 . Since F 2 = F 2 2 , we have
F F = F 2 F 2 , F F = F 2 F 2 .

From (4.13) and (4.17), we have Π 0 , this is a contradiction. Therefore, we obtain the conclusion of Theorem 2.4. □

Declarations

Acknowledgements

The authors thank the referee for his/her valuable suggestions to improve the present article. This work was supported by the NSFC (11301233), the Natural Science Foundation of Jiang-Xi Province in China (20132BAB211001, 20132BAB211002), and the Foundation of Education Department of Jiangxi (GJJ14272, GJJ14644) of China.

Authors’ Affiliations

(1)
College of Science, Jiujiang University, JiuJiang, 332005, China
(2)
College of Mathematics and Information Science, Jiangxi Normal University, Nanchang, 330022, China
(3)
Department of Informatics and Engineering, Jingdezhen Ceramic Institute, Jingdezhen, 333403, China

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