Jump to ContentJump to Main Navigation
Show Summary Details
More options …

Open Mathematics

formerly Central European Journal of Mathematics

Editor-in-Chief: Gianazza, Ugo / Vespri, Vincenzo

1 Issue per year


IMPACT FACTOR 2016 (Open Mathematics): 0.682
IMPACT FACTOR 2016 (Central European Journal of Mathematics): 0.489

CiteScore 2016: 0.62

SCImago Journal Rank (SJR) 2016: 0.454
Source Normalized Impact per Paper (SNIP) 2016: 0.850

Mathematical Citation Quotient (MCQ) 2016: 0.23

Open Access
Online
ISSN
2391-5455
See all formats and pricing
More options …
Volume 14, Issue 1 (Jan 2016)

Issues

Results on the deficiencies of some differential-difference polynomials of meromorphic functions

Xiu-Min Zheng
  • Corresponding author
  • Institute of Mathematics and Information Science, Jiangxi Normal University, Nanchang, Jiangxi, 330022, China
  • Email
  • Other articles by this author:
  • De Gruyter OnlineGoogle Scholar
/ Hong-Yan Xu
Published Online: 2016-02-24 | DOI: https://doi.org/10.1515/math-2016-0009

Abstract

In this paper, we study the relation between the deficiencies concerning a meromorphic function f(z), its derivative f′(z) and differential-difference monomials f(z)mf(z+c)f′(z), f(z+c)nf′(z), f(z)mf(z+c). The main results of this paper are listed as follows: Let f(z) be a meromorphic function of finite order satisfying limsupr+T(r,f)T(r,f)<+, and c be a non-zero complex constant, then δ(∞, f(z)m f(z+c)f′(z))≥δ(∞, f′) and δ(∞,f(z+c)nf′(z))≥ δ(∞, f′). We also investigate the value distribution of some differential-difference polynomials taking small function a(z) with respect to f(z).

Keywords: Meromorphic function; Differential-difference polynomial; Deficiency

MSC 2010: 30D35; 39A10

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 Hayman [1], Yang [2] and Yi-Yang [3]). In addition, for a meromorphic function f(z), we use δ (a, f) to denote the Nevanlinna deficiency of a ∊ ℂ̃ = ℂ⋃{} where δ(a,f)=liminfr+m(r,1fa)T(r,f)=1limsupr+m(r,1fa)T(r,f). We also 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 finite logarithmic measure limr[1,r)Edtt<

Throughout this paper, we assume m; n; k; t are positive integers.

Many people were interested in the value distribution of different expressions of meromorphic functions and obtained lots of important theorems (see [1, 46]).

Recently, the topic of complex differences has attracted the interest of many mathematicians, and a number of papers have focused on the value distribution of complex differences and difference analogues of Nevanlinna theory (including [711]). By combining complex differentiates and complex differences, we proceed in this way in this paper.

Firstly, we study the Nevanlinna deficiencies related to a meromorphic function f(z), its derivative f′(z) and its differential-difference monomials F1(z)=f(z)mf(z+c)f(z)andF2(z)=f(z+c)nf)f(z), and obtain the following theorems.

Let f(z) be a meromorphic function of finite order satisfying limsupr+T(r,f)T(r,f)<+,(1) and c be a non-zero complex constant. Then δ(,F1)δ(,f).

Let f(z) be a meromorphic function of finite order satisfying (1), and c be a non-zero complex constant. Then δ(,F2)δ(,f).

It is easy to find meromorphic functions to make the inequalities in Theorems 1.1 and 1.2 hold. For example, let f1(z) = ez, then δ(,f1')=δ(,F1)=δ(,F2)=δ(,f1)=1, showing the equalities in Theorems 1.1 and 1.2 may hold.

Let f2(z)=z3+2z and c = 1 = m = n, then f2'(z)=2z32z2andf2(z+1)=z3+3z2+3z+3z+1.

It follows that F1(z)=f2(z)f2(z+1)f2'(z)=2z9P1(z)z4+z3,F2(z)=f2(z+1)f2'(z)=2z6P2(z)z3+z2, where P1(z); P2(z) are polynomials in z with deg P1(z) ≥ 8 and deg P2(z) ≥ 5. Clearly, f2(z) satisfies limsupr+T(r,f2)T(r,f2')=limsupr+3logr3logr=1<+.

Thus, we have δ(,F1)=59>δ(,f2')=13, and δ(,F2)=12>δ(,f2')=13, showing the inequalities in Theorems 1.1 and 1.2 may hold.

Thus, Theorems 1.1 and 1.2 are sharp.

In addition, from the above examples, we can find that δ(,F1)=59<δ(,f2)=23 and δ(,F2)=12<δ(,f2)=23.. So, we give the following question: Under the conditions of Theorems 1.1 and 1.2, do F1; F2satisfy δ(,f)δ(,F1)δ(,f), and δ(,f)δ(,F2)δ(,f)?

We also get the following relations between δ(∞, f) and δ(∞, Fi), i = 1, 2.

Let f(z) be a meromorphic function of finite order, and c be a non-zero complex constant. If δ=δ(,f)>8m+6,, then δ(∞, F1) > 0.

Let f(z) be a meromorphic function of finite order, and c be a non-zero complex constant. If δ=δ(,f)>8m+5,, then δ(∞, F2) > 0.

Let f(z) be a meromorphic function of finite order, and c be a non-zero complex constant. Set F3(z)=f(z)mf(z+c).

If δ=δ(,f)>4m+3,, then δ(∞, F)> 0.

From the conclusions of Theorems 1.1, 1.4, 1.5 and 1.6, we see that there may exist some meromorphic function f(z) satisfying δ(∞, Fi) = 0, i = 1, 2, 3 as δ(∞, f) > 0. An interesting problem arises naturally: How can we find some meromorphic function f(z) to satisfy δ(∞, Fi) = 0, i = 1, 2, 3 as δ(∞, f) > 0?

The following ideas derive from Hayman [5], Laine-Yang [12], Zheng-Chen [13]. In 1959, Hayman [5] studied the value distribution of meromorphic functions and their derivatives, and obtained the following famous theorems.

([5]). Let f(z) be a transcendental entire function. Then

  1. for n = 3 and a ≠ 0, Ψ(z) = f′(z) − af(z)n assumes all finite values infinitely often.

  2. for n = 2, Φ(z) = f′(z) f(z)nassumes all finite values except possibly zero infinitely often.

Recently, some authors studied the zeros of f(z+c) f(z)na and f(z+c) − af(z)nb, where a(≠0), b are complex constants or small functions. Some related results can be found in [1217]. Especially, Laine-Yang [12] and Zheng-Chen [13] proved the following result, which is regarded as a difference counterpart of Theorem1. 8.

([12, 13]). Let f(z) be a transcendental entire function of finite order, and c be a non-zero complex constant. Then

  1. for n ≥ 2, Φ1(z) = f(z+c) f(z)n assumes every a ∈ ℂ\{0} infinitely often.

  2. for n ≥ 3, a ≠0, Ψ1(z) = f(z+c)− af(z)n assumes every b ∈ ℂ infinitely often.

In the following, we investigate the zeros of some differential-difference polynomials of a meromorphic function f(z) taking small function a(z) with respect to f(z), where and in the following a(z) is a non-zero small function of growth S(r, f), and obtain some theorems as follows.

Let f(z) be a transcendental meromorphic function of finite order, and c be a non-zero complex constant. Set G1(z)=f(z)mf(z+c)nf(z).

If mn + 8 or nm + 8, then G1(z) − a(z) has infinitely many zeros.

An example shows that the conclusion can not hold if f(z) is of infinite order. Let f(z) = 2eez, a(z) = ez, m = 9, n = 1 and ec = −10, then G1(z)a(z)=(2m+n+1e[(m+1)+nec]ez1)ez=(2111)ez, then G1(z) − a(z) has finite many zeros.

Let f(z) be a transcendental meromorphic function of finite order, and c1, c2, … cn be non-zero complex constants. Set G2(z)=f(z)mj=1nf(z+cj)sji=1kf(i)(z).

If m ≥ σ1 + 2n + k(k + 3) + 4, where σ1 = s1 + s2 + … + sn, then G2(z) − a(z) has infinitely many zeros.

Let Pn(z) = anzn + an−1zn−1 + … + a1z + a0 be a non-zero polynomial, where a0 a1an are complex constants and t is the number of the distinct zeros of Pn(z). Then we further obtain the following results.

Let f(z) be a transcendental meromorphic function of finite order, and c be a non-zero complex constant. Set G3(z)=f(z)mPn(f(z+c))i=1kf(i)(z).

If mn + t + k(k + 3) + 4, then G3(z) − a(z) has infinitely many zeros.

Let f(z) be a transcendental meromorphic function of finite order, and c be a non-zero complex constant. Set G4(z)=Pm(f(z))f(z+c)ni=1kf(i)(z).

If mn + t + k(k + 3) + 4, then G4(z) − a(z) has infinitely many zeros.

2 Some lemmas

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

([7, 10]).Let f(z) be a meromorphic function of finite order ρ and c be a fixed non-zero complex number, then we have m(r,f(z+c)f(z))+m(r,f(z)f(z+c))=S(r,f).

By [18], [19, p.66] and [20], we immediately deduce the following lemma.

Let f(z) be a meromorphic function of finite order, and c be a non-zero complex constant. Then T(r,f(z+c))=T(r,f)+S(r,f),N(r,f(z+c))=N(r,f)+S(r,f),N(r,1f(z+c))=N(r,1f)+S(r,f).

([3, p.37]).Let f(z) be a nonconstant meromorphic function in the complex plane and l be a positive integer. Then N(r,f(i))=N(r,f)+lN¯(r,f),T(r,f(l))T(r,f)+lN¯(r,f)+S(r,f).

Let f(z) be a transcendental meromorphic function of finite order, and G1(z) = f(z)mf(z + c)n .f(z). Then we have (|mn|1)T(r,f)+S(r,f)T(r,G1)(n+m+2)T(r,f)+S(r,f).(2)

From Lemmas 2.2 and 2.3, we have T(r,G1)T(r,fm)+T(r,f(z+c)n)+T(r,f)(n+m+2)T(r,f)+S(r,f).

On the other hand, from Lemma 2.2 again, we have (n+m+1)T(r,f)=T(r,fn+m+1)=T(r,f(z)n+1G1(z)f(z+c)nf(z))T(r,G1)+T(r,ff)+T(r,f(z)nf(z+c)n)T(r,G1)+(2n+2)T(r,f)+S(r,f), where we assume mn without loss of generality. Thus, (2) is proved. □

Using the similar method as in Lemma 2.4, we get the following lemmas.

Let f(z) be a transcendental meromorphic function of finite order, and G2(z)=f(z)mj=1nf(z+cj)Sji=1kf(i)(z), then we have (mσ1k(k+3)2)T(r,f)+S(r,f)(r,G2)(m+σ1+k(k+3)2)T(r,f)+S(r,f).

Let f(z)be a transcendental meromorphic function of finite order, and G3(z)=f(z)mPn(f(z+c))i=1kf(i)(z), then we have (mnk(k+3)2)T(r,f)+S(r,f)T(r,G3)(m+n+k(k+3)2)T(r,f)+S(r,f).

Let f(z) be a transcendental meromorphic function of finite order, and G4(z)=Pm(f(z))f(z+c)ni=1kf(i)(z), then we have (mnk(k+3)2)T(r,f)+S(r,f)T(r,G4)(m+n+k(k+3)2)T(r,f)+S(r,f).

3 Proofs of Theorems 1.1 and 1.2

3.1 Proof of Theorem 1.1

We firstly give the following elementary inequalities αα+βα1α1+β,αα+βα+γa+β+γ,(3) for α, β, γ ≥ 0 and αα1.

Since f(z)m+2=F1(z)f(z)m+1f(z)m+1f(z)f(z+c), it follows that (m+2)m(r,f)m(r,F1)+(m+1)m(r,ff)+m(r,f(z)f(z+c),

Then by Lemma 2.1, we have m(r,F1)(m+2)m(r,f)+S(r,f).(4)

Since N(r, f′) = N(r, f) = N̅ (r, f), it follows by Lemma 2.2 that N(r,F1)(m+2)N(r,f)+N¯(r,f)(m+2)N(r,f).(5)

From (1), we have lim sup limr+S(r,f)T(r,f)=limr+S(r,f)T(r,f)T(r,f)T(r,f)=0(6)

Then, from (3)- (6), we have N(r,F1)T(r,F1)(m+2)N(r,f)(m+2)N(r,f)+(m+2)m(r,f)+S(r,f)N(r,f)T(r,f)+S(r,f)=N(r,f)(1+ο(1))T(r,f).

It follows that δ(∞, f′) ≤ δ(∞, F′).

Thus, we complete the proof of Theorem 1.1.

3.2 Proof of Theorem 1.2

Since f(z)n+1=F2f(z)nf(z)nf(z)nf(z+c)n, then by using the similar method as in the proof of Theorem 1.1, we can prove Theorem 1.2 easily.

4 Proofs of Theorems 1.4, 1.5 and 1.6

4.1 Proof of Theorem 1.4

Let F4(z) = f(z)m + 2 then we have N(r,F4)=(m+2)N(r,f)andT(r,F4)=(m+2)T(r,f)

It follows that δ(∞, F4) = δ(∞, f′) = δ. Since F4(z) = f(z)m+2, we have N¯(r,f)N(r,f)1m+2N(r,F4)1δm+2T(r,F4)+S(r,F4),(7) N¯(r,1f)N(r,1f)1m+2N(r,1F4)1m+2T(r,F4)+ο(1).(8)

From (7), (8) and Lemmas 2.1 and 2.2, we have T(r,F4)=T(r,F1(z)f(z)f(z+c)f(z)f(z))T(r,F1)+N(r,1f)+N¯(r,1f)+N(r,f)+N¯(r,f)+S(r,f)T(r,F1)+42δm+2T(r,F4)+S(r,f), that is T(r,F1)(m2(1δ)m+2+ο(1))T(r,F4).(9)

From (7) and (8) again, we have N(r,F1)N(r,F4)+N(r,f(z+c)f(z)+N(r,ff)N(r,F4)+N(r,1f)+N¯(r,1f)+N(r,f)+N¯(r,f)+S(r,f)(m+4)(1δ)+2m+2T(r,F4)+S(r,f), that is, N(r,F1)((m+4)(1δ)+2m+2+ο(1))T(r,F4).(10)

Thus, from (9), (10) and δ=δ(,f)=δ(,F4)>8m+6, it follows that that is, limr+N(r,F1)T(r,F1)(m+4)(1δ)+2m2(1δ)<1,δ(,F4)=1limr+supN(r,F1)T(r,F1)>0.

This completes the proof of Theorem 1.4.

4.2 Proofs of Theorems 1.5 and 1.6

Using the similar method as in the proof of Theorem 1.4, we can prove Theorems 1.5 and 1.6 easily.

5 Proofs of Theorems 1.10, 1.12, 1.13 and 1.14

5.1 Proof of Theorem 1.10

Suppose that f(z) is a transcendental meromorphic function of finite order. Since mn + 8 or nm + 8, then by Lemma 2.4, we have S(r, f) = S(r, G1). Thus, by using the second fundamental theorem and Lemmas 2.2 and 2.4, we have (|mn|1)T(r,f)T(r,G1)+S(r,f)N¯(r,G1)+N¯(r,1G1)+N¯(r,1G1(z)a(z))+S(r,G1)+N¯(r,1f(z+c))+N¯(r,1f)+N¯(r,1G1(z)a(z))+S(r,G1)6T(r,f)+N¯(r,1G1(z)a(z))+S(r,G1), that is |mn|7n+m+2T(r,G1)+S(r,G1)(|mn|7)T(r,f)N¯(r,1G1(z)a(z))+S(r,G1).

Thus, we have from mn + 8 or nm + 8 that δ(a,G1)1|mn|7n+m+2<1.

Consequently, G1(z)− a(z) has infinitely many zeros.

This completes the proof of Theorem 1.10.

5.2 Proof of Theorem 1.14

If f(z) is a transcendental meromorphic function of finite order, then by Lemma 2.7, we have S(r, f) = S(r. G4). Thus, by using the second fundamental theorem and Lemmas 2.2 and 2.7 again, (mnk(k+3)2)T(r,f)T(r,G4)+S(r,f)N¯(r,G4)+N¯(r,1G4)+N¯(r,1G4(z)a(z))+S(r,G4)N¯(r,f)+N¯(r,f(z+c))+i=1tN¯(r,1fγi)+N¯(r,1f(z+c))+j=1kN¯(r,1f(j))+N¯(r,1G4(z)a(z))+S(r,G4)(3+t+k(k+1)2+k)T(r,f)+N¯(r,1G4(z)a(z))+S(r,G4), that is, mntk(k+3)3m+n+k(k+3)2T(r,G4)+S(r,G4)(mntk(k+3)3)T(r,f)N¯(r,1G4(z)a(z))+S(r,G4),

where γ1, γ2,.....,γt are distinct zeros of Pm(z). Since mn + t + k(k + 3) + 4, we have δ(a,G4)1mntk(k+3)m+n+k(k+3)2<1.

Consequently, G4(z) − a(z) has infinitely many zeros.

This completes the proof of Theorem 1.14.

5.3 Proofs of Theorems 1.12 and 1.13

Using the similar method as in the proofs of Theorems 1.10 and 1.14 and combining Lemmas 2.5 and 2.6, we can prove Theorems 1.12 and 1.13 easily.

Acknowledgement

The authors are grateful to the referees and editors for their valuable comments which lead to the improvement of this paper.

References

  • 1

    Hayman W.K., Meromorphic Functions, The Clarendon Press, Oxford, 1964. Google Scholar

  • 2

    Yang L., Value distribution theory, Springer-Verlag, Berlin, 1993. Google Scholar

  • 3

    Yi H.X., Yang C.C., Uniqueness theory of meromorphic functions, Kluwer Academic Publishers, Dordrecht, 2003; Chinese original: Science Press, Beijing, 1995. Google Scholar

  • 4

    Gross F., On the distribution of values of meromorphic functions, Trans. Amer. Math. Soc. 131, 1968, 199-214. Google Scholar

  • 5

    Hayman W.K., Picard values of meromorphic functions and their derivatives, Ann. of Math. 70(2), 1959, 9-42. Google Scholar

  • 6

    Mues E., Über ein Problem von Hayman, Math. Zeit. 164(3), 1979, 239-259. Google Scholar

  • 7

    Chiang Y.M., Feng S.J., On the Nevanlinna characteristic of f .z C [DC1]/ and difference equations in the complex plane, Ramanujan J. 16, 2008, 105-129. Google Scholar

  • 8

    Halburd R.G., Korhonen R.J., Difference analogue of the lemma on the logarithmic derivative with applications to difference equations, J. Math. Anal. Appl. 314, 2006, 477-487.Google Scholar

  • 9

    Halburd R.G., Korhonen R.J., Finite-order meromorphic solutions and the discrete Painlevé equations, Proc. London Math. Soc. 94, 2007, 443-474.Google Scholar

  • 10

    Halburd R.G., Korhonen R.J., Nevanlinna theory for the difference operator, Ann. Acad. Sci. Fenn. 31, 2006, 463-478. Google Scholar

  • 11

    Liu K., Yang L.Z., Value distribution of the difference operator, Arch. Math. 92, 2009, 270-278. Google Scholar

  • 12

    Laine I., Yang C.C., Value distribution of difference polynomials, Proc. Japan Acad. Ser. A 83, 2007, 148-151. Google Scholar

  • 13

    Zheng X.M., Chen Z.X., On the value distribution of some difference polynomials, J. Math. Anal. Appl. 397(2), 2013, 814-821. Google Scholar

  • 14

    Chen Z.X., On value distribution of difference polynomials of meromorphic functions, Abstr. Appl. Anal. 2011, 2011, Art. 239853, 9 pages.Google Scholar

  • 15

    Liu K., Liu X.L., Cao T.B., Value distributions and uniqueness of difference polynomials, Adv. Difference Equ. 2011, 2011, Art. 234215, 12 pages.Google Scholar

  • 16

    Xu H.Y., On the value distribution and uniqueness of difference polynomials of meromorphic functions, Adv. Difference Equ. 2013, 2013, Art. 90, 15 pages.Google Scholar

  • 17

    Xu J.F., Zhang X.B., The zeros of difference polynomials of meromorphic functions, Abstr. Appl. Anal. 2012, 2012, Art. 357203, 13 pages. Google Scholar

  • 18

    Ablowitz M.J., Halburd R.G., Herbst B., On the extension of the Painlevé property to difference equations, Nonlinearity, 13, 2000, 889-905. Google Scholar

  • 19

    Gol’dberg A.A., Ostrovskii I.V., The distribution of values of meromorphic functions, Nauka, Moscow, 1970. (in Russian) Google Scholar

  • 20

    Halburd R.G., Korhonen R.J., Tohge K., Holomorphic curves with shift-invariant hyper-plane preimages, Trans. Amer. Math. Soc. 366, 2014, 4267-4298. Google Scholar

About the article

This project was supported by the National Natural Science Foundation of China (11301233, 11561033), the Natural Science Foundation of Jiangxi Province in China (20151BAB201008, 20151BAB201004), and the Youth Science Foundation of Education Bureau of Jiangxi Province in China (GJJ14644, GJJ14271).


Received: 2015-08-28

Accepted: 2015-11-24

Published Online: 2016-02-24

Published in Print: 2016-01-01


Citation Information: Open Mathematics, ISSN (Online) 2391-5455, DOI: https://doi.org/10.1515/math-2016-0009.

Export Citation

© 2016 Zheng and Xu, published by De Gruyter Open.. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License. BY-NC-ND 3.0

Comments (0)

Please log in or register to comment.
Log in