Research Article Fixed Points of Difference Operator of Meromorphic Functions

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1 e Scientiic World Journal, Article ID 03249, 4 pages Research Article Fixed Points o Dierence Operator o Meromorphic Functions Zhaojun Wu and Hongyan Xu 2 School o Mathematics and Statistics, Hubei University o Science and Technology, Xianning 43700, China 2 Department o Inormatics and Engineering, Jingdezhen Ceramic Institute, Jingdezhen , China Correspondence should be addressed to Zhaojun Wu; wuzj52@hotmail.com Received 28 August 203; Accepted 24 October 203; Published 9 January 204 Academic Editors: M. Gandarias and Y. C. Shiah Copyright 204 Z. Wu and H. Xu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Let be a transcendental meromorphic unction o order less than one. The authors prove that the exact dierence Δ = (z + ) (z) has ininitely many ixed points, i a C and are Borel exceptional values (or Nevanlinna deiciency values) o.these results extend the related results obtained by Chen and Shon.. Introduction and Main Results In this paper, we assume that the reader is amiliar with the notations o requency use in Nevanlinna theory (see [ 3]). Let (z) be a meromorphic unction in the complex plane C and a C. Weusethenotationsσ() to denote the order o (z), λ(, a), and λ(/), respectively, to denote the exponent o convergence o zeros o (z) a and poles o (z). Especially,ia=0,wedenoteλ(, 0) = λ(). Apoint z C is called as a ixed point o (z) i (z) = z. There is a considerable number o results on the ixed points or meromorphic unctions in the plane; we reer the reader to Chuang and Yang [4]. It ollows Chen and Shon [5]; we use the notation τ() to denote the exponent o convergence o ixed points o that is deined as τ()=lim sup r log N (r, / ( z)). () log r Let be a transcendental meromorphic unction in the complex plane C. The exact dierences Δ are deined by Δ = (z + ) (z). Recently, there are a number o papers (including [6 6]) ocusing on the dierences analogues o Nevanlinna s theory and its application on the complex dierence equations. For theixedpointsothedierenceoperatorδ,chenandshon have proved the ollowing. Theorem A (see [7]). Let be a transcendental entire unction o order o growth σ() = and have ininitely many zeros with the exponent o convergence o zeros λ() <.Then Δ has ininitely many zeros and ininitely many ixed points. When the order o is less than, Chen and Shon have proved the ollowing. Theorem B (see [5]). Let be a transcendental meromorphic unction o order o growth σ(). Supposethat satisies λ(/) < λ() < or has ininitely many zeros (with λ() = 0) and initely many poles.then Δ has ininitely many ixed points and satisies the exponent o convergence o ixed points τ(δ) = σ(). A natural question is, letting be a transcendental meromorphic unction o order o growth σ() <,isthere a similar result as that in Theorem B i λ(/) λ() or has ininitely many zeros (with λ() = 0)and ininitely many poles? In this paper, we will prove the ollowing theorem to answer the question. Theorem (main). Let be a transcendental meromorphic unction o order o growth σ() < and a C. Suppose that satisies λ(/) < σ() and λ(, a) < σ(). ThenΔ has ininitely many ixed points and satisies the exponent o convergence o ixed points τ(δ) = σ().

2 2 The Scientiic World Journal From Theorem,we can get the ollowing corollary. Corollary 2. Let be a transcendental meromorphic unction o order o growth σ() <. Supposethat satisies λ() λ(/) < σ(). ThenΔ has ininitely many ixed points and satisies the exponent o convergence o ixed points τ(δ) = σ(). In Theorem, wesupposethat satisies λ(/) < σ() and λ(, a) < σ(). Thatistosay and a are Borel exceptional values o. Iwesupposethat and a are Nevanlinna deiciency values o, is there a similar result as that in Theorem B? In the ollowing, we give Theorem 3 to answer this question. Let (z) be a meromorphic unction in the complex plane C and a C = C { }. Nevanlinna s deiciency o with respect to a is deined by δ(a,)= lim sup r N(r,/( a)). (2) T(r,) I a=, then one should replace N(r, /( a)) in the above ormula by N(r, ). Iδ(a, ) > 0, thena is called a Nevanlinna deiciency value o. Theorem 3 (main). Let be a transcendental meromorphic unction o order o growth σ() < and a C.Supposethat satisies δ(, ) = and a isanevanlinnadeiciencyvalue o.then Δ has ininitely many ixed points. Corollary 4. Let be a transcendental entire unction o order o growth σ() < and a C.Supposethatδ(a, ) > 0.Then Δ has ininitely many ixed points. 2. Some Lemmas Lemma (lemma on the logarithmic derivative). Let (z) be a meromorphic unction. I the unction (z) has inite order, then holds or any positive integer k. m(r, (k) )=O(log r) (3) Lemma 2 (see [8]). Let (z) be a meromorphic unction with the exponent o convergence o poles λ(/)=λ<+ and let c be a nonzero complex number. Then or each ε>0,wehave N(r,(z+c)) =N(r,)+O(r λ +ε )+O(log r). (4) Lemma 3. Let be a transcendental meromorphic unction o order o growth σ() < and let c be a nonzero complex number. Then N (r, (z+c)) =N(r, ) +O(log r). (5) Proo. Since the order σ() := σ <,thenλ(/)=λ σ<. Thereore, or any 0<ε< σ,itollowsromlemma 2 that N(r,(z+c)) =N(r,)+O(r λ +ε )+O(log r) That is, =N(r,)+O() +O(log r). (6) N(r,(z+c)) =N(r,)+O(log r). (7) Lemma 4 (see [6]). Let be a unction transcendental and meromorphic in the plane which satisies T(r,) lim in =0. (8) r r Then Δ is transcendental. Lemma 5. Let be a transcendental meromorphic unction o order o growth σ() = σ <.ThenΔ is transcendental. Proo. Since the order σ() := σ <, then,oranypositive ε(0<ε< σ), there exists R>0such that or any r>rwe have Thereore, T(r,) r σ+ε. (9) T(r,) lim in =0. (0) r r Lemma5 ollows Lemma4. Lemma 6 (see [7]). Let (z) be a meromorphic unction o inite order, then σ(δ) σ(). Lemma 7 (see [7]). Let be a transcendental meromorphic unction o order o growth σ() <. Thenoranyε > 0 and any positive integer k, thereexistsasete (, ) that depends on and has inite logarithmic measure, such that or all z satisying z =r E [0,]we have Δ k (z) (z) = (k) (z) (z) +O(r (k+)(σ )+ε ). () It is easy to derive the ollowing lemma rom Lemma and Lemma 7. Lemma 8. Let be a transcendental meromorphic unction o order o growth σ() <. Thenoranypositiveintegerk there exists a set E (, ) that depends on and has inite logarithmic measure, such that m(r, Δk (z) )=O(log r), r E. (2) (z)

3 The Scientiic World Journal 3 3. Proo o Theorems Proo. Since then = Δ z zδ2 z zδ 2, (3) m (r, Δ ) m (r, z )+m(r,zδ2 ) z + m (r, zδ 2 )+O() 2m (r, Δ )+m(r,δ2 ) + m (r, zδ 2 )+O(log r). Applying the irst undamental theorem, we get m (r, m (r, ) = T (r, ) N (r, )+O(), zδ 2 )=m(r,zδ2 ) +N(r, zδ2 ) N (r, Combining (4)-(5)wehave zδ 2 )+O() m(r, zδ2 ) +N(r, zδ2 )+O(). T (r, ) N (r, ) + 2m (r, Δ )+m(r,δ2 ) +m(r, zδ2 ) (4) (5) Since Δ 2 =Δ((z+) (z)) =(z+2) 2(z+) +(z), Δ(Δ z)=δ((z+) (z) z) then, we can get =(z+2) 2(z+) +(z), zδ 2 =z(z+2) 2z(z+) +z(z) (z+) +(z). zδ () () = z (z+2) 2z(z+) Thereore, zδ 2 = = (7) +z(z) (z+) +(z). (8) zδ () () zδ (), N(r,zΔ 2 ) N(r,(z+2))+N(r,(z+)) +N(r,(z)). Thus rom Lemma 3 and (20), we deduce (9) (20) N(r,zΔ 2 ) 3N(r,(z))+O(log r). (2) By Lemmas 5 and 6, weknowthat is a transcendental meromorphic unction o order o growth σ() σ() <.ItollowsromLemma 8 and (9) that there exists a set E (, ) that has inite logarithmic measure, such that or any r Ewe have m (r, Δ )=O(log r), +N(r, zδ2 )+O(log r) N (r, ) + N (r, )+N(r,zΔ2 ) + 2m (r, Δ ) +m(r, Δ2 )+m(r,zδ2 )+O(log r). (6) m(r, Δ2 )=O(log r), m(r, zδ2 )=O(log r). From (6)and(2)-(22), we have T (r, ) 3N (r, ) + N (r, )+N(r, ) +O(log r), r E. (22) (23)

4 4 The Scientiic World Journal Denotingg aby (23) we derive, T(r,) T(r,g)+O() 3N (r, g) + N (r, g ) + N (r, Δg z ) +O(log r) 3N (r, ) + N (r, +O(log r), r E. a ) + N (r, ) (24) 3.. The Rest o the Proo o Theorem. By Lemma 6,weknow that τ(δ) σ(). Iτ(Δ) < σ(), byλ(/) < σ() and λ(, a) < σ(), there exists a number η<σ(),suchthator any suicient r we have N(r,)<r η, N (r, N (r, )<rη. a )<rη, (25) Combining (24) and(25), we can get a contradiction. Thereore, we have τ(δ) = σ() The Rest o the Proo o Theorem 3. Since δ(, ) =, then N(r, ) = o(t(r, )).By(24), we can get ( o()) T (r, ) N (r, a ) + N (r, ) +O(log r), r E. (26) Since δ(a, ) > 0, then there is a positive number θ< such that N (r, )<θt(r,). (27) a I Δ hasonlyainitenumberoixedpoints,thenrom (26)and(27)wewouldhave ( o() θ) T (r, ) O(log r), r E. (28) This contradicts being transcendental. Thereore, Δ has ininitely many ixed points. Conlict o Interests The authors declare that there is no conlict o interests regarding the publication o this paper. Acknowledgments This research was partly supported by the National Natural Science Foundation o China (Grant nos and ), by the Natural Science Foundation o Jiang- Xi Province in China (Grant nos. 2032BAB200 and 2022BAB20044) and by the Science Foundation o Educational Commission o Hubei Province (Grant nos. Q and D ). Reerences [] W. K. Hayman, Meromorphic Functions. Oxord Mathematical Monographs, Clarendon Press, Oxord, UK, 964. [2] L. Yang, Value Distribution Theory, Springer, Berlin, Germany; Science Press Beijing, Beijing, China, 993. [3] J. H. Zheng, Value Distribution o Meromorphic Functions, Tsinghua University Press, Beijing, China; Springer, Heidelberg, Germany, 200. [4] C. T. Chuang and C. C. Yang, Theory o Fix Points and FactorizationoMero-MorphicFunctions,MathematicalMonogragh Series, Peking University Press, 986. [5] Z.-X. Chen and K. H. Shon, Properties o dierences o meromorphic unctions, Czechoslovak Mathematical Journal, vol. 6, no., pp , 20. [6] W. Bergweiler and J. K. Langley, Zeros o dierences o meromorphic unctions, Mathematical Proceedings o the Cambridge Philosophical Society,vol.42,no.,pp.33 47,2007. [7] Y. M. Chiang and S. J. Feng, On the growth o logarithmic dierence, dierence equations and logarithmic derivatives o meromorphic unctions, Transactions o the American Mathematical Society,vol.36,pp ,2009. [8] R.G.HalburdandR.J.Korhonen, Nevanlinnatheoryorthe dierence operator, Annales Academiae Scientiarum Fennicae Mathematica,vol.3,pp ,2006. [9] R. G. Halburd and R. J. Korhonen, Meromorphic solutions o dierence equations, integrability and the discrete Painlevé equations, Physics A,vol.40,no.6,pp.R R38,2007. [0] R. G. Halburd and R. J. Korhonen, Dierence analogue o the lemma on the logarithmic derivative with applications to dierence equations, Mathematical Analysis and Applications,vol.34,pp ,2006. [] N. Li and L. Z. Yang, Value distribution o dierence and q- dierence polynomials, Dierence Equations, vol. 203, article 98, pp. 9, 203. [2] H. F. Liu and Z. Q. Mao, On the meromorphic solutions o some linear dierence equations, Dierence Equations,vol.203,article33,pp. 2,203. [3] K. Liu and T. B. Cao, Entire solutions o Fermat type q- dierence dierential equations, Electronic Dierential Equations,vol.203,no.59,pp. 0,203. [4]K.Liu,H.Z.Cao,andT.B.Cao, EntiresolutionsoFermat type dierential dierence equations, Archiv Der Mathematik, vol. 99, pp , 202. [5] H. Y. Xu, T. B. Cao, and B. X. Liu, The growtho solutions o systems o complex q-shit dierence equations, Dierence Equations,vol.202,article26,pp. 22,202. [6] J. F. Xu and X. B. Zhang, The zeros o q-shit dierence polynomials o meromorphic unctions, Dierence Equations,vol.202,article200,pp. 0,202. [7] Z.-X. Chen and K. H. Shon, Value distribution o meromorphic solutions o certain dierence Painlevé equations, Mathematical Analysis and Applications,vol.364,no.2,pp , 200. [8] Y.-M. Chiang and S.-J. Feng, On the Nevanlinna characteristic o (z+η) and dierence equations in the complex plane, Ramanujan Journal,vol.6,no.,pp.05 29,2008.

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