On value distribution and uniqueness of meromorphic function with finite logarithmic order concerning its derivative and q-shift difference

Abstract

In this paper, we study the value distribution of a meromorphic function [InlineEquation not available: see fulltext.] concerning its derivative [InlineEquation not available: see fulltext.] and q-shift difference [InlineEquation not available: see fulltext.], where [InlineEquation not available: see fulltext.] is of finite logarithmic order. We also investigate the uniqueness of differential-q-shift-difference polynomials with more general forms of entire functions of order zero.

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R E S E A R C H Open Access

On value distribution and uniqueness of meromorphic function with nite logarithmic order concerning its derivative and q-shift difference

Xiu-Min Zheng1* and Hong-Yan Xu2

*Correspondence: mailto:[email protected]

Web End [email protected]

1Institute of Mathematics and Information Science, Jiangxi Normal University, Nanchang, 330022, China Full list of author information is available at the end of the article

Abstract

In this paper, we study the value distribution of a meromorphic function f(z) concerning its derivative f (z) and q-shift dierence f(qz + c), where f(z) is of nite logarithmic order. We also investigate the uniqueness of dierential-q-shift-dierence polynomials with more general forms of entire functions of order zero.

Keywords: dierential-q-shift-dierence; meromorphic function; logarithmic order

1 Introduction and main results

The fundamental theorems and the standard notations of the Nevanlinna value distribution theory of meromorphic functions will be used (see e.g. Hayman [], Yang [] and Yi and Yang []). In addition, for a meromorphic function f (z), we use S(r, f ) to denote any quantity satisfying S(r, f ) = o(T(r, f )) for all r outside a possible exceptional set E of nite logarithmic measure limr

(,r]Edtt < and also use S(r, f ) to denote any quantity satisfying S(r, f ) = o(T(r, f )) for all r on a set F of logarithmic density , where the logarithmic density of a set F is dened by

lim

r

(,r]Ft dt.

The order of a meromorphic function f (z) is dened by

(f ) = lim sup

r

log T(r, f ) log r .

The logarithmic order of a meromorphic function f (z) is dened by (see [])

log(f ) = lim sup

r

log r

log T(r, f ) log log r .

If log(f ) < , then f (z) is said to be of nite logarithmic order. It is clear that if a meromorphic function f (z) has nite logarithmic order, then f (z) has order zero.

2014 Zheng and Xu; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0

Web End =http://creativecommons.org/licenses/by/2.0 ), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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Remark . [, ] If f (z) is a meromorphic function of nite positive logarithmic order log(f ), then T(r, f ) has proximate logarithmic order log(f ). The logarithmic-type function of T(r, f ) is dened as U(r, f ) = (log r)log(f). We have T(r, f ) U(r, f ) for suciently large r.

The logarithmic exponent of convergence of a-points of f (z) is equal to the logarithmic order of n(r,

f a ), which is dened as

log(a) = lim sup

r

order of n(r,

Moreover, we assume in the whole paper that m, n, k, tm, tn are positive integers, q C\{}, c C, and a(z) is a non-zero small function with respect to f (z), that is, a(z) is a non-zero meromorphic function of growth S(r, f ).

Many mathematicians were interested in the value distribution of dierent expressions of meromorphic functions and obtained lots of important theorems (see e.g. [, ]). Especially, Hayman [] discussed Picards values of meromorphic functions and their derivatives, and he obtained the following famous theorem in .

Theorem . [] Let f (z) be a transcendental entire function. Then(a) for n , f (z)f (z)n assumes all nite values except possibly zero innitely often;

(b) for n and a = , f (z) af (z)n assumes all nite values innitely often.

Further, for a transcendental meromorphic function f (z), Chen and Fang [] obtained the following result.

Theorem . [, Theorem ] Let f (z) be a transcendental meromorphic function. If n is a positive integer, f (z)f (z)n has innitely many zeros.

Recently, with the establishments of dierence analogies of the Nevanlinna theory (see e.g. []), many mathematicians focused on studying dierence analogies of Theorems . and .. The main purpose of these results (see e.g. [, ]) is to get the sharp estimation of the value of n to make dierence polynomials f (z + c)f (z)n a and

f (z + c) af (z)n b admit innitely many zeros.

Meantime, q-dierence analogies of the Nevanlinna theory and their applications on the value distribution of q-dierence polynomials and q-shift-dierence equations are also studied (see e.g. []). Especially, for a transcendental meromorphic (resp. entire) function f (z) of order zero, Zhang and Korhonen [] studied the value distribution of qdierence polynomials of f (z) and found that if n (resp. n ), then f (qz)f (z)n assumes every non-zero value a C innitely often (see [, Theorem .]).

Further, Xu and Zhang [] investigated the zeros of q-shift dierence polynomials of meromorphic functions of nite logarithmic order and obtained the following result in .

Theorem . [, Theorem .] If f (z) is a transcendental meromorphic function of nite logarithmic order log(f ), with the logarithmic exponent of convergence of poles less than

log n(r,

log log r .

We see by [] that for a meromorphic function f (z) of nite positive logarithmic order log(f ), the logarithmic order of N(r,

f a ) is log(a) + , where log(a) is the logarithmic

f a )

f a ).

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log(f ) and q, c are non-zero complex constants, then for n , f (z)nf (qz + c) assumes every value b C innitely often.

One main aim of this paper is to investigate the zeros of dierential-q-shift-dierence polynomials about f (z), f (z), and f (qz+c), where f (z) is of nite positive logarithmic order.

Theorem . Let f (z) be a transcendental meromorphic function of nite logarithmic order log(f ), with the logarithmic exponent of convergence of poles less than log(f ) . Set

F(z) = f (qz + c)nf (z).

If n , then F(z) a(z) has innitely many zeros.

We also deal with the value distribution of a dierential-q-shift-dierence polynomial with another form about f (z), f (z) and f (qz + c), and we obtain the following result.

Theorem . Let f (z) be a transcendental meromorphic function of nite logarithmic order log(f ), with the logarithmic exponent of convergence of poles less than log(f ) . Set

F(z) = f (qz + c)n + f (z) + f (z).

If n , then F(z) a has innitely many zeros, where a C.

Some more general dierential-q-shift-dierence polynomials are investigated in the following.

Theorem . Let f (z) be a transcendental meromorphic function of nite logarithmic order log(f ), with the logarithmic exponent of convergence of poles less than log(f ) . Set

F(z) = f (z)mf (qz + c)nf (z).

If m, n satisfy m n + or n m + , then F(z) a(z) has innitely many zeros.

Let

Pn(z) = anzn + anzn + + az + a

be a non-zero polynomial, where a, a, . . . , an ( = ) are complex constants and tn is the number of the distinct zeros of Pn(z). Then we also obtain the following results.

Theorem . Let f (z) be a transcendental meromorphic function of nite logarithmic order log(f ), with the logarithmic exponent of convergence of poles less than log(f ) . Set

F(z) = f (z)mPn

If m n + k + , then F(z) a(z) has innitely many zeros.

f (qz + c)

[parenrightbig]

k

j=f (j)(z).

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Theorem . Let f (z) be a transcendental meromorphic function of nite logarithmic order log(f ), with the logarithmic exponent of convergence of poles less than log(f ) . Set

F(z) = Pm

If m n + k + , then F(z) a(z) has innitely many zeros.

Next, we investigate the uniqueness of dierential-q-shift-dierence polynomials of entire functions of order zero and obtain the following results.

Theorem . Let f (z) and g(z) be transcendental entire functions of order zero and n . If f (qz + c)nf (z) and g(qz + c)ng (z) share a non-zero polynomial p(z) CM, then f (qz + c)nf (z) = g(qz + c)ng (z).

Theorem . Let f (z) and g(z) be transcendental entire functions of order zero and m n+tn +. If f (z)mPn(f (qz+c))f (z) and g(z)mPn(g(qz+c))g (z) share a non-zero polynomial p(z) CM, then f (z)mPn(f (qz + c))f (z) = g(z)mPn(g(qz + c))g (z).

Theorem . Let f (z) and g(z) be transcendental entire functions of order zero and n m + tm + . If Pm(f (z))f (qz + c)nf (z) and Pm(g(z))g(qz + c)ng (z) share a non-zero polynomial p(z) CM, then Pm(f (z))f (qz + c)nf (z) = Pm(g(z))g(qz + c)ng (z).

2 Some lemmas

To prove the above theorems, we need some lemmas as follows.

Lemma . [] Let f (z) be a non-constant meromorphic function and P(f ) = a + af +

+ anf n, where a, a, . . . , an are complex constants and an = , then

T

r, P(f )

[parenrightbig]= nT(r, f ) + S(r, f ).

Lemma . [] Let f (z) be a transcendental meromorphic function of nite logarithmic order and q, be two non-zero complex constants. Then we have

T

r, f (qz + )

[parenrightbig]= T(r, f ) + S(r, f ),

N

r, f (qz + )

[parenrightbig]= N(r, f ) + S(r, f ), N

r, f[parenrightbigg]+ S(r, f ).

Lemma . [, Theorem .] Let f (z) be a non-constant zero-order meromorphic function and q C \ {}. Then

m

r, f (qz +) f (z)

[parenrightbigg]

r, f (l)

f (z)

f (qz + c)n

k

j=f (j)(z).

r, f (qz + )

= S(r, f ).

Lemma . [, p.] Let f (z) be a non-constant meromorphic function in the complex plane and l be a positive integer. Then

T

r, f (l)

[parenrightbig]

T(r, f ) + lN(r, f ) + S(r, f ), N

[parenrightbigg]

= N

[parenrightbig]

= N(r, f ) + lN(r, f ).

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Lemma . [] If f (z) is a transcendental meromorphic function of nite logarithmic order log(f ), then for any two distinct small functions a(z) and b(z) with respect to f (z), we have

T(r, f ) N

r, f a

[parenrightbigg]

+ N

r, f b[parenrightbigg]+ o

U(r, f )

,

where U(r, f ) = (log r)log(f) is a logarithmic-type function of T(r, f ). Furthermore, if T(r, f ) has a nite lower logarithmic order

= lim inf

r

log T(r, f ) log log r ,

with log(f ) < , then

T(r, f ) N

r, f a

[parenrightbigg]

+ N

r, f b[parenrightbigg]+ o

T(r, f )

.

Remark . From the proof of Lemma . (see [, Theorem .]), we can easily see that complex values a and b can be changed into a(z) and b(z), where a(z) and b(z) are two distinct small functions with respect to f (z).

Lemma . Let f (z) be a transcendental meromorphic function of order zero, F(z) = f (qz+ c)nf (z). Then we have

(n )T(r, f ) + S(r, f ) (n )T(r, f ) N(r, f ) + S(r, f )

T(r, F) (n + )T(r, f ) + S(r, f ). ()

Proof If f (z) is a meromorphic function of order zero, from Lemmas . and ., we have

T(r, F) nT

r, f (qz + c)

[parenrightbig]+ T

(n + )T(r, f ) + S(r, f ).

On the other hand, from Lemmas . and . again, we have

(n + )T(r, f ) = T

r, f (qz + c)n+

[parenrightbig]

+ S(r, f )

r, f

[parenrightbig]

T(r, F) + T

r, f (qz + c) f (z)

[parenrightbigg]

+ S(r, f )

T(r, F) + T(r, f ) + N(r, f ) + S(r, f )

T(r, F) + T(r, f ) + S(r, f ).

Thus, we get ().

Lemma . Let f (z) be a transcendental meromorphic function of zero order, F(z) =

f (z)mf (qz + c)nf (z). Then we have

T(r, F) (m + n + )T(r, f ) + S(r, f ) ()

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and

T(r, F) |m n|T(r, f ) N(r, f ) + S(r, f )

T(r, f ) + S(r, f ). ()

Proof If f (z) is a meromorphic function of order zero, from Lemmas . and ., we have

T(r, F) mT(r, f ) + nT

r, f (qz + c)

[parenrightbig]+ T

|m n|

(m + n + )T(r, f ) + S(r, f ),

that is, we have (). On the other hand, from Lemmas . and ., we have

(n + m + )T(r, f ) = T r, f n+m+

[parenrightbig]

= T

r, f

[parenrightbig]

r, f (z)n+F(z)f (qz + c)nf (z)

[parenrightbigg]

r, f (z)nf (qz + c)n[parenrightbigg]

T(r, F) + (n + )T(r, f ) + N(r, f ) + S(r, f )

T(r, F) + (n + )T(r, f ) + S(r, f ),

that is, we have (), where we assume m n without loss of generality.

Lemma . Let f (z) be a transcendental meromorphic function of order zero, F(z) =

f (z)mPn(f (qz + c))

kj= f (j)(z). Then we have

T(r, F) + T

r, f f

[parenrightbigg]

+ T

(m n k)T(r, f ) T(r, F) +

k(k + )

N(r, f ) + S(r, f ),

T(r, F)

m + n + k(k + )

T(r, f ) + S(r, f ).

Proof Since f (z) is a transcendental meromorphic function of order zero, by Lemmas ., ., and ., we can easily get the second inequality. On the other hand, it follows by Lemmas ., ., and . that

(m + k)T(r, f ) = T

r, f m+k

[parenrightbig]

T

r, f (z)kF(z) Pn(f (qz + c))

kj= f (j)(z)

[parenrightbigg]

T(r, F) + T

r, Pn

f (qz + c)

[parenrightbig][parenrightbig]+ T

r,k

j=f (j) f

[parenrightBigg]

N(r, f ) + S(r, f ).

Thus, this completes the proof of Lemma ..

Similar to Lemma ., we have the following lemma.

Lemma . Let f (z) be a transcendental meromorphic function of zero order, F(z) =

Pm(f (z))f (qz + c)n

kj= f (j)(z). Then we have

T(r, F) + (n + k)T(r, f ) +

k(k + )

(n m k)T(r, f ) T(r, F) +

k(k + )

N(r, f ) + S(r, f ),

T(r, F)

m + n + k(k + )

T(r, f ) + S(r, f ).

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3 Proofs of Theorems 1.4-1.83.1 Proof of Theorem 1.4

It follows by Lemma . that T(r, F) = O(T(r, f )) holds for all r on a set of logarithmic density . Since f (z) is transcendental and n , F(z) is transcendental by Lemma .

again. Since the logarithmic exponent of convergence of poles of f (z) less than log(f ) ,

we have

lim sup

r

log N(r, f ) log log r <

log(f ).

Assume that F(z) a(z) has only nitely many zeros. Thus, by Lemmas ., .-., we have

(n )T(r, f ) T(r, F) + S(r, f )

N(r, F) + N

r, F a

[parenrightbigg]

+ o

U(r, f )

[parenrightbig]+ S(r, f )

= N

r, f (qz + c)nf (z)

[parenrightbig]

+ N

r, F a

[parenrightbigg]

+ o

U(r, f )

[parenrightbig]+ S(r, f )

(n + )N(r, f ) + N

r, F a

[parenrightbigg]

+ o

U(r, f )

[parenrightbig]+ S(r, f ).

Thus, it follows that

(n )T(r, f ) (n + )N(r, f ) + o

U(r, f )

[parenrightbig]+ S(r, f ).

Since n , the above inequality implies

lim sup

r

log T(r, f ) log log r

lim sup

r

log N(r, f ) log log r <

log(f ),

which contradicts the fact that T(r, f ) has nite logarithmic order log(f ). Thus, F(z)a(z) has innitely many zeros, that is, f (qz + c)nf (z) a(z) has innitely many zeros.

This completes the proof of Theorem ..

3.2 Proof of Theorem 1.5

Since f (z) is a transcendental meromorphic function of nite logarithmic order, we rst claim that f (z)+f (z)a . In fact, if f (z)+f (z)a , that is,

f (z)f (z)a . By solving the above equation, we have f (z) = Aez + a, where A is a non-zero complex constant. Thus, we have (f ) = , which contradicts the fact that f (z) is of order zero. Thus, set

F(z) = a f (z) f (z)f (qz + c)n .

It follows by Lemmas . and . that

nT(r, f ) = T

r, f (qz + c)n

[parenrightbig]

+ S(r, f )

T

r, F(z)a f (z) f (z)

[parenrightbigg]

+ S(r, f )

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T(r, F) + T(r, f ) + T

+ S(r, f )

T(r, F) + T(r, f ) + N(r, f ) + S(r, f ),

that is,

(n )T(r, f ) T(r, F) + N(r, f ) + S(r, f ). ()

On the other hand, we can easily get

T(r, F) T

r, f (qz + c)n

r, f

[parenrightbig]

[parenrightbig]

+ T(r, f ) + T

r, f

[parenrightbig]+ O() (n + )T(r, f ) + S(r, f ). ()

And it follows from () and n that

T(r, F) = O T(r, f )

[parenrightbig]

holds for all r on a set of logarithmic density . By Lemma ., we have

N r,

F

[parenrightbigg]

= N

r, f (qz + c)nf (qz + c)n + f (z) + f (z) a [parenrightbigg]

N

r, f (qz + c)n + f (z) + f (z) a

[parenrightbigg]

+ nN(r, f ) + S(r, f ).

Assume that f (qz + c)n + f (z) + f (z) a has nitely many zeros, then

N r,

F

[parenrightbigg]

nN(r, f ) + S(r, f ). ()

Since the logarithmic exponent of convergence of poles of f (z) is less than log(f ) , we

have

lim sup

r

log N(r, f ) log log r <

log(f ).

Then, by Lemmas ., ., and (), we have

T(r, F) N

r, F

[parenrightbigg]

+ N

r,

F

[parenrightbigg]+ o

U(r, f )

[parenrightbig]

N

r, a f f

[parenrightbigg]

+ nN(r, f ) + S(r, f ) + o

U(r, f )

[parenrightbig]

T(r, f ) + (n + )N(r, f ) + S(r, f ) + o

U(r, f )

.

It follows by the above inequality and () that

(n )T(r, f ) (n + )N(r, f ) + S(r, f ) + o

U(r, f )

.

Since n , the above inequality implies

lim sup

r

log T(r, f ) log log r

lim sup

r

log N(r, f ) log log r <

log(f ),

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which contradicts that T(r, f ) has nite logarithmic order log(f ). Thus, f (qz + c)n + f (z) + f (z) a has innitely many zeros.

This completes the proof of Theorem ..

3.3 Proofs of Theorems 1.6, 1.7, and 1.8

Similar to the argument as in Theorem ., by applying Lemmas ., ., and . instead, we can easily prove Theorems ., ., and . respectively.

4 Proofs of Theorems 1.9-1.11

Here, we only give the proof of Theorem . because the methods of the proofs of Theorems ., ., and . are very similar.

4.1 Proof of Theorem 1.10

Denote

F(z) = f (z)mPn

f (qz + c)

f (z), G(z) = g(z)mPn

g(qz + c)

g (z).

Since f (z) is a transcendental entire function of order zero, by Lemmas ., ., and ., we have

T(r, F) T

r, f m

[parenrightbig]

+ T(r, Pn

f (qz + c)

[parenrightbig]+ T

r, f

[parenrightbig]+ O()

(n + m + )T(r, f ) + S(r, f ), ()

and

(m + )T(r, f ) = T

r, f m+

[parenrightbig]

= T

r, f (z)F(z) Pn(f (qz + c))f (z)

[parenrightbigg]

T(r, F) + T

r, f f

[parenrightbigg]

+ T

r,

Pn(f (qz + c))

[parenrightbigg]

+ S(r, f )

T(r, F) + (n + )T(r, f ) + S(r, f ). ()

Then it follows from () and () that

(m n)T(r, f ) + S(r, f ) T(r, F) (n + m + )T(r, f ) + S(r, f ). ()

We have by () that S(r, F) = S(r, f ). Similarly, we have S(r, G) = S(r, g) and

(m n)T(r, g) + S(r, g) T(r, G) (n + m + )T(r, g) + S(r, g). ()

Since f (z) and g(z) are entire functions of order zero and share p(z) CM, we have

F(z) p(z)

G(z) p(z) =

, ()

where is a non-zero constant. If = , then we have F(z) = G(z), that is, f (z)mPn(f (qz +

c))f (z) = g(z)mPn(g(qz + c))g (z).

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If = , then we have

F(z) G(z) = p(z)( ). ()

Since Pn(z) has tn distinct zeros, by using the second main theorem and Lemma ., we have

T(r, F) N(r, F) + N

r, F

[parenrightbigg]

+ N

r,

F p(z)( )

[parenrightbigg]

+ S(r, F)

N

r,

Pn(f (qz + c))

[parenrightbigg]+ N

r, f

[parenrightbigg]

+ N

r, f

[parenrightbigg]

+ N

r,

G[parenrightbigg]+ S(r, f )

tn

j=

N

r, f (qz + c) j

[parenrightbigg]

+ N

r, f

[parenrightbigg]

+ N

r, f

[parenrightbigg]

+ S(r, f )

+

tn

j=

N

r, g(qz + c) j

[parenrightbigg]

+ N

r, g

[parenrightbigg]

+ N

r, g

[parenrightbigg]

+ S(r, g)

(tn + )T(r, f ) + (tn + )T(r, g) + S(r, f ) + S(r, g), ()

where , , . . . , tn are the distinct zeros of Pn(z). Similarly, we have

T(r, G) (tn + )T(r, f ) + (tn + )T(r, g) + S(r, f ) + S(r, g). ()

Then (), (), (), and () result in

(m n)

T(r, f ) + T(r, g)

[parenrightbig] (tn + )

T(r, f ) + T(r, g)

[parenrightbig]+ S(r, f ) + S(r, g),

which contradicts m n + tn + .

This completes the proof of Theorem ..

Competing interests

The authors declare that they have no competing interests.

Authors contributions

All authors contributed equally to the writing of this paper. All authors drafted the manuscript, read, and approved the nal manuscript.

Author details

1Institute of Mathematics and Information Science, Jiangxi Normal University, Nanchang, 330022, China. 2Department of Informatics and Engineering, Jingdezhen Ceramic Institute, Jingdezhen, 333403, China.

Acknowledgements

This work was supported by the National Natural Science Foundation of China (11301233, 61202313, 11171119), the Natural Science Foundation of Jiangxi Province in China (20132BAB211001) and Sponsored Program for Cultivating Youths of Outstanding Ability in Jiangxi Normal University of China, and the Foundation of Education Department of Jiangxi (GJJ14271, GJJ14644) of China.

Received: 12 April 2014 Accepted: 30 June 2014 Published: 19 August 2014

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doi:10.1186/1029-242X-2014-295Cite this article as: Zheng and Xu: On value distribution and uniqueness of meromorphic function with nite logarithmic order concerning its derivative and q-shift difference. Journal of Inequalities and Applications 2014 2014:295.