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Value distribution of difference and q-difference polynomials

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Abstract

In this paper, we investigate the value distribution of difference polynomial and obtain the following result, which improves a recent result of K. Liu and L.Z. Yang: Let f be a transcendental meromorphic function of finite order σ, c be a nonzero constant, and α(z)0 be a small function of f, and let

P(z)= a n z n + a n 1 z n 1 ++ a 1 z+ a 0

be a polynomial with a multiple zero. If λ(1/f)<σ, then P(f)f(z+c)α(z) has infinitely many zeros. We also obtain a result concerning the value distribution of q-difference polynomial.

MSC:30D35, 39A05.

1 Introduction and main results

Throughout the paper, we assume that the reader is familiar with the standard symbols and fundamental results of Nevanlinna theory as found in [13]. A function f(z) is called the meromorphic function, if it is analytic in the complex plane except at isolated poles. For any non-constant meromorphic function f, we denote by S(r,f) any quantity satisfying

lim r S ( r , f ) T ( r , f ) =0,

possibly outside of a set of finite linear measure in R + . A meromorphic function a(z) is called a small function of f(z) provided that T(r,a)=S(r,f). As usual, we denote by σ(f) the order of a meromorphic function f(z), and denote by λ(f) (λ(1/f)) the exponent of convergence of the zeros (poles) of f(z).

Recently, a number of papers concerning the complex difference products and the differences analogues of Nevanlinna’s theory have been published (see [412] for example), and many excellent results have been obtained. In 2007, Laine and Yang [10] investigated the value distribution of difference products of entire functions, and obtained the following result.

Theorem A Let f(z) be a transcendental entire function of finite order, and c be a non-zero complex constant. Then for n2, f ( z ) n f(z+c) assumes every non-zero value aC infinitely often.

Liu and Yang [11] improved Theorem A, and proved the next result.

Theorem B Let f(z) be a transcendental entire function of finite order, and c be a non-zero complex constant. Then for n2, f ( z ) n f(z+c)p(z) has infinitely many zeros, where p(z)0 is a polynomial in z.

The purpose of this paper is to investigate the value distribution of difference polynomial P(f)f(z+c)α(z) and q-difference polynomial P(f)f(qz)α(z), where P(z)= a n z n + a n 1 z n 1 ++ a 1 z+ a 0 with constant coefficients a n (0), a n 1 ,, a 0 , and α(z) is a mall function of f(z).

For the sake of simplicity, we denote by s(P) and m(P) the number of the simple zeros and the number of multiple zeros of a polynomial

P(z)= a n z n + a n 1 z n 1 ++ a 1 z+ a 0

respectively.

We obtain the following result which improves Theorem A and Theorem B.

Theorem 1.1 Let f be a transcendental meromorphic function of finite order σ(f)=σ, and c be a non-zero constant, and let

P(z)= a n z n + a n 1 z n 1 ++ a 1 z+ a 0

be a polynomial with constant coefficients a n (0), a n 1 ,, a 0 and m(P)>0. If λ( 1 f )<σ, then P(f)f(z+c)α(z) has infinitely many zeros, where α(z)0 is a small function of f.

Remark 1 The result of Theorem 1.1 may be false if α(z)0, for example, f(z)= e z 2 z , it is obvious that f 2 f(z+1) has no zeros. The following example shows that the assumption λ( 1 f )<σ in Theorem 1.1 cannot be deleted. In fact, let f(z)= 1 e z 1 + e z , c=πi, α(z)=1, and P(z)= z 2 . Then λ( 1 f )=σ(f)=1 and P(f)f(z+c)α(z)= 2 1 + e z has no zeros. Also, let f(z)=i+ e z , c=πi, α(z)=1, and P(z)=z(zi+1)(zi1). Then P(f)f(z+c)α(z)= e 4 z has no zeros. This shows that the restriction in Theorem 1.1 to the multiple zero case is essential.

Considering the value distribution of q-differences polynomials, we obtain the following result.

Theorem 1.2 Let f(z) be a transcendental entire function of zero order, and α(z)S(r,f). Suppose that q is a non-zero complex constant and n is an integer. If m(P)>0, then P(f)f(qz)α(z) has infinitely many zeros.

2 Some lemmas

Lemma 2.1 [6]

Given two distinct complex constants η 1 , η 2 , let f be a meromorphic function of finite order σ. Then, for each ε>0, we have

m ( r , f ( z + η 1 ) f ( z + η 2 ) ) =O ( r σ 1 + ε ) .

Lemma 2.2 [6]

Let f be a transcendental meromorphic function of finite order σ, c be a complex number. Then, for each ε>0, we have

T ( r , f ( z + c ) ) =T ( r , f ( z ) ) +O ( r σ 1 + ε ) +O(logr).

The following lemma is a revised form of Lemma 2.4.2 in [2].

Lemma 2.3 Let f(z) be a transcendental meromorphic solution of

f n A(z,f)=B(z,f),

where A(z,f), B(z,f) are differential polynomials in f and its derivatives with meromorphic coefficients, say { a λ λI}, n be a positive integer. If the total degree of B(z,f) as a polynomial in f and its derivatives is less than or equal to n, then

m ( r , A ( z , f ) ) λ I m(r, a λ )+S(r,f).

Lemma 2.4 [12]

Let f(z) be a non-constant meromorphic function of finite order, cC. Then

N ( r , 1 f ( z + c ) ) N ( r , 1 f ( z ) ) + S ( r , f ) , N ( r , f ( z + c ) ) N ( r , f ) + S ( r , f ) , N ¯ ( r , 1 f ( z + c ) ) N ¯ ( r , 1 f ( z ) ) + S ( r , f ) , N ¯ ( r , f ( z + c ) ) N ¯ ( r , f ) + S ( r , f ) ,

outside of a possible exceptional set E with finite logarithmic measure.

Lemma 2.5 [4]

Let f be a non-constant zero-order meromorphic function, and qC{0}. Then

m ( r , f ( q z ) f ( z ) ) =o ( T ( r , f ) )

on a set of logarithmic density 1.

Remark 2 For the similar reason in Theorem 1.1 in [4], we can easily deduce that

m ( r , f ( z ) f ( q z ) ) =o ( T ( r , f ) )

also holds on a set of logarithmic density 1.

Proof

Using the identity

ρ 2 r 2 ρ 2 2 ρ r cos ( φ θ ) + r 2 =Re ( ρ e i θ + z ρ e i θ z ) ,

and let Poisson-Jensen formula with R=ρ, we see

log | f ( z ) f ( q z ) | = 0 2 π log | f ( ρ e i θ ) | Re ( ρ e i θ + z ρ e i θ z ρ e i θ + q z ρ e i θ q z ) d θ 2 π + | a n | < ρ log | ( z a n ) ( ρ 2 a ¯ n q z ) ( q z a n ) ( ρ 2 a ¯ n z ) | | b m | < ρ log | ( z b m ) ( ρ 2 b ¯ m q z ) ( q z b m ) ( ρ 2 b ¯ m z ) | = S 1 ( z ) + S 1 ( z ) S 3 ( z ) ,

where { a n } and { b m } are the zeros and poles of f, respectively. Integration on the set E:={φ[0,2π]:| f ( r e i φ ) f ( q r e i φ ) |1} gives us the proximity function,

m ( r , f ( z ) f ( q z ) ) = E log | f ( z ) f ( q z ) | d ψ 2 π = E ( S 1 ( r e i ψ ) + S 2 ( r e i ψ ) S 3 ( r e i ψ ) ) d ψ 2 π 0 2 π ( | S 1 ( r e i ψ ) | + | S 2 ( r e i ψ ) | + | S 3 ( r e i ψ ) | ) d ψ 2 π .

Since S i = S i (i=1,2,3) in [4], we get | S i |=| S i | (i=1,2,3).

Following the similar method in the proof of Theorem 1.1 in [4], we get the result. □

Lemma 2.6 Let f be a non-constant zero-order entire function, and qC{0}. Then

T ( r , P ( f ) f ( q z ) ) =T ( r , P ( f ) f ( z ) ) +S(r,f)

on a set of logarithmic density 1.

Proof Since f is an entire function of zero-order, we deduce from Lemma 2.5 that

T ( r , P ( f ) f ( q z ) ) = m ( r , P ( f ) f ( q z ) ) m ( r , P ( f ) f ( z ) ) + m ( r , f ( q z ) f ( z ) ) m ( r , P ( f ) f ( z ) ) + S ( r , f ) = T ( r , P ( f ) f ( z ) ) + S ( r , f ) ,

that is

T ( r , P ( f ) f ( q z ) ) T ( r , P ( f ) f ( z ) ) +S(r,f).
(2.1)

On the other hand, using Remark 2, we get

T ( r , P ( f ) f ( z ) ) = m ( r , P ( f ) f ( z ) ) m ( r , P ( f ) f ( q z ) ) + m ( r , f ( z ) f ( q z ) ) m ( r , P ( f ) f ( q z ) ) + S ( r , f ) = T ( r , P ( f ) f ( q z ) ) + S ( r , f ) ,

that is

T ( r , P ( f ) f ( z ) ) T ( r , P ( f ) f ( q z ) ) +S(r,f).
(2.2)

The assertion follows from (2.1) and (2.2). □

3 Proof of Theorem 1.1

Let β(z) be the canonical products of the nonzero poles of P(f)f(z+c)α(z). Since λ(1/f)<σ and α(z) is a small function of f(z), we know that σ(β)=λ(β)<σ(f). Suppose on contrary to the assertion that P(f)f(z+c)α(z) has finitely many zeros. Then we have

P(f)f(z+c)α(z)=R(z) e Q ( z ) /β(z),

where Q(z) is a polynomial, and R(z)0 is a rational function. Set H(z)=R(z)/β(z). Then

σ(H)<σ(f)=σ,
(3.1)

and

P(f)f(z+c)α(z)=H(z) e Q ( z ) .
(3.2)

Differentiating (3.2) and eliminating e Q ( z ) , we obtain

P ( f ) f ( z ) f ( z + c ) H ( z ) + P ( f ) f ( z + c ) H ( z ) P ( f ) f ( z + c ) H ( z ) P ( f ) f ( z + c ) Q ( z ) H ( z ) = α ( z ) H ( z ) α ( z ) H ( z ) α ( z ) Q ( z ) H ( z ) .
(3.3)

Let α 1 , α 2 ,, α t be the distinct zeros of P(z). Then

P(f)= a n ( f α 1 ) n 1 ( f α 2 ) n 2 ( f α t ) n t .

Substituting this into (3.3), we have

a n j = 1 t ( f α j ) n j 1 { ( n 1 j 1 ( f α j ) + n 2 j 2 ( f α j ) + + n t j t ( f α j ) ) × f ( z + c ) H ( z ) f ( z ) + f ( z + c ) H ( z ) × j = 1 t ( f α j ) f ( z + c ) ( H ( z ) + Q ( z ) H ( z ) ) j = 1 t ( f α j ) } = α ( z ) H ( z ) α ( z ) H ( z ) α ( z ) Q ( z ) H ( z ) .

Note that P(z) has at least one multiple zero, we may assume that n 1 >1 without loss of generality, and we have

a n ( f α 1 ) n 1 1 F(z,f)= α (z)H(z)α(z) H (z)α(z) Q (z)H(z),
(3.4)

where

F ( z , f ) = j = 2 t ( f α j ) n j 1 { ( n 1 j 1 ( f α j ) + n 2 j 2 ( f α j ) + + n t j t ( f α j ) ) × f ( z + c ) H ( z ) f ( z ) + f ( z + c ) H ( z ) j = 1 t ( f α j ) f ( z + c ) ( H ( z ) + Q ( z ) H ( z ) ) j = 1 t ( f α j ) } .

Now we distinguish two cases.

Case 1. F(z,f)0. In this case, we obtain from (3.4) that

α (z)H(z)α(z) H (z)α(z) Q (z)H(z)0.

Since α(z)0 and H(z)0, by integrating, we have

α ( z ) H ( z ) =k e Q ( z ) ,
(3.5)

where k is a non-zero constant. From (3.2) and (3.5), we have

P(f)f(z+c)= ( 1 k + 1 ) α(z).

By Lemma 2.2, we have

n T ( r , f ( z ) ) = T ( r , P ( f ) ) + O ( 1 ) T ( r , f ( z + c ) ) + T ( r , α ( z ) ) + O ( 1 ) = T ( r , f ( z ) ) + O ( r σ 1 + ε ) + S ( r , f ) .

Since n n 1 2, and f(z) is a transcendental, this is impossible.

Case 2. F(z,f)0. In this case, we set

F ( z , f ) = F ( z , f ) f α 1 = j = 2 t ( f α j ) n j 1 { ( n 1 j 1 ( f α j ) + n 2 j 2 ( f α j ) + + n t j t ( f α j ) ) × f ( z + c ) f ( z ) f ( z ) H ( z ) f ( z ) f α 1 + f ( z + c ) f ( z + c ) f ( z + c ) f ( z ) f ( z ) H ( z ) j = 2 t ( f α j ) f ( z + c ) f ( z ) f ( z ) ( H ( z ) + Q ( z ) H ( z ) ) j = 2 t ( f α j ) } .

Since f(z)=(f(z) α 1 )+ α 1 and f ( k ) = ( f α 1 ) ( k ) , we know that F (z,f) is a differential polynomial of f(z) α 1 with meromorphic coefficients, and

a n ( f α 1 ) n 1 F (z,f)= α (z)H(z)α(z) H (z)α(z) Q (z)H(z).
(3.6)

By Lemma 2.3, we have

m ( r , ( f α 1 ) k F ( z , f ) ) 3 m ( r , f ( z + c ) f ( z ) ) + m ( r , f ( z + c ) f ( z + c ) ) + m ( r , f ( z ) f α 1 ) + 5 T ( r , H ) + S ( r , f )
(3.7)

for k=0 and k=1.

Now for any given ε (0<ε<1), we obtain from Lemma 2.1, Lemma 2.2 and (3.1) that

m ( r , f ( z + c ) f ( z ) ) =O ( r σ ε ) ,T(r,H)=O ( r σ ε ) ,
(3.8)
m ( r , f ( z + c ) f ( z + c ) ) =O ( r σ ε ) +S(r,f).
(3.9)

The lemma of logarithmic derivative implies that

m ( r , f ( z ) f α 1 ) =S(r,f).
(3.10)

It follows from (3.7) to (3.10) that

m ( r , F ( z , f ) ) =O ( r σ ε ) +S(r,f),
(3.11)
m ( r , ( f α 1 ) F ( z , f ) ) =O ( r σ ε ) +S(r,f).
(3.12)

Since (f α 1 ) F (z,f)=F(z,f), we obtain from the definition of F(z,f) that

N ( r , F ( z , f ) ) =O ( N ( r , H ( z ) ) + N ( r , f ) ) =O ( r σ ε ) +S(r,f).

Thus,

T ( r , ( f α 1 ) F ( z , f ) ) =O ( r σ ε ) +S(r,f).
(3.13)

Note that, a zero of f(z) α 1 which is not a pole of f(z+c) and H(z), is a pole of F (z,f) with the multiplicity at most 1, we know from (3.6), (3.1), Lemma 2.4 and λ(1/f)<σ that

( n 1 1 ) N ( r , 1 f ( z ) α 1 ) N ( r , 1 α ( z ) H ( z ) α ( z ) H ( z ) α ( z ) Q ( z ) H ( z ) ) + O ( N ( r , f ( z + c ) ) ) + O ( N ( r , H ) ) = O ( r σ ε )
(3.14)

for the positive ε sufficiently small. Hence (see the definition of F (z,f)),

N ( r , F ( z , f ) ) = O ( N ( r , 1 f α 1 ) + N ( r , f ) + N ( r , H ) ) = O ( r σ ε ) + S ( r , f ) .
(3.15)

It follows from (3.15) and (3.11) that

T ( r , F ( z , f ) ) =O ( r σ ε ) +S(r,f).
(3.16)

Thus, we deduce from (3.16) and (3.13) that

T ( r , f ( z ) ) = T ( r , f ( z ) α 1 ) + O ( 1 ) = T ( r , ( f α 1 ) F ( z , f ) F ( z , f ) ) = O ( r σ ε ) + S ( r , f ) .

This contradicts that f is of order σ. Theorem 1.1 is proved.

4 Proof of Theorem 1.2

Denote F(z)=P(f)f(qz). From Lemma 2.6 and the standard Valiron-Mohon’ko theorem, we deduce

T ( r , F ( z ) ) = T ( r , P ( f ) f ( z ) ) + S ( r , f ) = ( n + 1 ) T ( r , f ( z ) ) + S ( r , f ) .

Since f is a entire function, then by the second main theorem and Lemma 2.5, we have

T ( r , F ( z ) ) N ¯ ( r , F ( z ) ) + N ¯ ( r , 1 F ( z ) ) + N ¯ ( r , 1 F ( z ) α ( z ) ) + S ( r , f ) N ¯ ( r , 1 P ( f ) ) + N ¯ ( r , 1 f ( q z ) ) + N ¯ ( r , 1 F ( z ) α ( z ) ) + S ( r , f ) ( s ( P ) + m ( P ) ) T ( r , f ( z ) ) + T ( r , f ( q z ) ) + N ¯ ( r , 1 F ( z ) α ( z ) ) + S ( r , f ) ( s ( P ) + m ( P ) ) T ( r , f ( z ) ) + m ( r , f ( q z ) f ( z ) ) + m ( r , f ( z ) ) + N ¯ ( r , 1 F ( z ) α ( z ) ) + S ( r , f ) ( s ( P ) + m ( P ) + 1 ) T ( r , f ( z ) ) + N ¯ ( r , 1 F ( z ) α ( z ) ) + S ( r , f ) ,

that is,

N ¯ ( r , 1 F ( z ) α ( z ) ) ( n s ( P ) m ( P ) ) T ( r , f ( z ) ) +S ( r , f ( z ) ) .

Since f is a transcendental entire function with m(P)>0, we deduce that P(f)f(qz)α(z) has infinitely many zeros.

References

  1. 1.

    Hayman WK: Meromorphic Functions. Clarendon, Oxford; 1964.

  2. 2.

    Laine I: Nevanlinna Theory and Complex Differential Equations. de Gruyter, Berlin; 1993.

  3. 3.

    Yi HX, Yang CC: Uniqueness Theory of Meromorphic Functions. Kluwer Academic, Dordrecht; 2003.

  4. 4.

    Barnett DC, Halburd RG, Korhonen RJ, Morgan W: Nevanlinna theory for the q -difference operator and meromorphic solutions of q -difference equations. Proc. R. Soc. Edinb. A 2007, 137: 457–474.

  5. 5.

    Bergweiler W, Langley JK: Zeros of difference of meromorphic functions. Math. Proc. Camb. Philos. Soc. 2007, 142: 133–147. 10.1017/S0305004106009777

  6. 6.

    Chiang YM, Feng SJ:On the Nevanlinna characteristic f(z+η) and difference equations in the complex plane. Ramanujan J. 2008, 16: 105–129. 10.1007/s11139-007-9101-1

  7. 7.

    Chiang YM, Feng SJ: On the growth of logarithmic differences, difference quotients and logarithmic derivatices of meromorphic functions. Trans. Am. Math. Soc. 2009, 361(7):3767–3791. 10.1090/S0002-9947-09-04663-7

  8. 8.

    Halburd RG, Korhonen RJ: Nevanlinna theory for the difference operator. Ann. Acad. Sci. Fenn. Math. 2006, 31: 463–478.

  9. 9.

    Halburd RG, Korhonen RJ: Difference analogue of the lemma on the logarithmic derivative with applications to difference equations. J. Math. Anal. Appl. 2006, 314: 477–487. 10.1016/j.jmaa.2005.04.010

  10. 10.

    Laine I, Yang CC: Value distribution of difference polynomials. Proc. Jpn. Acad., Ser. A, Math. Sci. 2007, 83: 148–151. 10.3792/pjaa.83.148

  11. 11.

    Liu K, Yang LZ: Value distribution of the difference operator. Arch. Math. 2009, 92: 270–278. 10.1007/s00013-009-2895-x

  12. 12.

    Qi XG, Yang LZ, Liu K: Uniqueness and periodicity of meromorphic functions concerning difference operator. Comput. Math. Appl. 2010, 60(6):1739–1746. 10.1016/j.camwa.2010.07.004

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Acknowledgements

This work was supported by the NSF of Shandong Province, P.R. China (No. ZR2010AM030) and the NNSF of China (No. 11171013 and No. 11041005).

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Correspondence to Lianzhong Yang.

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Keywords

  • meromorphic functions
  • difference polynomials
  • uniqueness