In this paper, we study the uniqueness question of meromorphic functions whose certain differential polynomials having two pseudo common values, and obtain some results which improve and generalize the related results due to S.S. Bhoosnurmath and R.S. Davanal [1], P. Sahoo [5], J. Xia and Y. Xu [8] and C. Wu, C. Mu and J. Li [6]

Tanuja, A.

Waghamore, H.P.

ePrints@Bangalore University

Southeast AsianBulletin ofMathematicsc⃝SEAMS. 2013Southeast Asian Bulletin of Mathematics (2013) 37: 941–949Uniqueness of Meromorphic Functions Whose CertainDiﬀerential Polynomials Have Two Pseudo CommonValuesHarina P. Waghamore and A. TanujaDepartment of Mathematics, Central College Campus, Bangalore University, Bangalore-560 001, IndiaEmail: pree.tam@rediﬀmail.com; a.tanuja1@gmail.comReceived 15 February 2012Accepted 8 October 2012Communicated by C.C. YangAMS Mathematics Subject Classiﬁcation(2000): 30D35Abstract. In this paper, we study the uniqueness question of meromorphic functionswhose certain diﬀerential polynomials having two pseudo common values, and obtainsome results which improve and generalize the related results due to S.S. Bhoosnurmathand R.S. Davanal [1], P. Sahoo [5], J. Xia and Y. Xu [8] and C. Wu, C. Mu and J. Li[6].Keywords:Meromorphic functions; Uniqueness; Diﬀerential polynomials; Sharing value.1. IntroductionIn this paper, by meromorphic function we will always mean meromorphic func-tion in complex plane. We adopt the standard notations of Nevanlinna theoryof meromorphic function as explained in [2], [9] and [10]. It will be convenientto let 퐸 denote any set of positive real numbers of ﬁnite linear measure, not nec-essarily the same at each occurrence. For a non-constant meromorphic functionℎ, we denote by 푇 (푟, ℎ) the Nevanlinna characteristic of ℎ and by 푆(푟, ℎ) anyquantity satisfying 푆(푟, ℎ) = 표{푇 (푟, ℎ)}, as 푟 →∞ and 푟 ∕ 휖퐸.Let 푓 and 푔 be two non-constant meromorphic functions, and let 푎 be a valuein the extended plane. We say that 푓 and 푔 share the value 푎 CM, providedthat 푓 and 푔 have the same 푎-points with the same multiplicities. We say that푓 and 푔 share the value 푎 IM, provided that 푓 and 푔 have the same 푎-points942 H.P. Waghamore and A. Tanujaignoring multiplicities (see [10]). We say that 푎 is a small function of 푓 , if 푎 is ameromorphic function satisfying 푇 (푟, 푎) = 푆(푟, 푓) (see [10]). Let 푙 be a positiveinteger or ∞. Next we denote by 퐸푙)(푎; 푓) the set of those 푎-points of 푓 in thecomplex plane, where each point is of multiplicity ≤ 푙 and counted accordingto its multiplicity. By 퐸푙)(푎; 푓) we denote the reduced form of 퐸푙)(푎; 푓). If퐸푙)(푎; 푓) = 퐸푙)(푎; 푔), we say that 푎 is a 푙-order pseudo common value of 푓 and푔 (see [3]). Obviously, if 퐸∞)(푎; 푓) = 퐸∞)(푎; 푔)(퐸∞)(푎; 푓) = 퐸∞)(푎; 푔)푟푒푠푝.),then 푓 and 푔 share 푎 CM (IM, resp.).Recall that S.S. Bhoosmurmath and R.S. Davanal [2]in 2007 proved the fol-lowing two theorems. Also, it is noted that the problem of meromphic functionhaving three weighted values and some examples of best possible of the aboveresults were given in [4].Theorem 1.1. [2] Let 푓 and 푔 be two non-constant meromorphic functions, andlet 푛, 푘 be two positive integers with 푛 > 3푘+8. If [푓푛](푘) and [푔푛](푘) share 1 CM,then either 푓(푧) = 푐1푒푐푧, 푔(푧) = 푐2푒−푐푧 where 푐1, 푐2 and 푐 are three constantssatisfying (−1)푘(푐1푐2)푛(푛푐)2푘 = 1 or 푓 = 푡푔 for a constant t such that 푡푛 = 1.Theorem 1.2. [2] Let 푓 and 푔 be two non-constant meromorphic functions satis-fying Θ(∞, 푓) > 3푛+1 and let 푛, 푘 be two positive integers with 푛 > 3푘 + 13. If[푓푛(푓 − 1)](푘) and [푔푛(푔 − 1)](푘) share 1 CM, then 푓 ≡ 푔.In 2010, P. Sahoo [5] proved the following theorem.Theorem 1.3. [5] Let 푓 and 푔 be two transcendental meromorphic functions, andlet 푛 ≥ 1, 푘 ≥ 1 and 푚 ≥ 0 be three positive integers. Let [푓푛(푓 − 1)푚](푘) and[푔푛(푔 − 1)푚](푘) share 1 IM. Then one of the following holds:(i) when 푚 = 0, if 푓 ∕= ∞, 푔 ∕= ∞ and 푛 > 9푘 + 14, then either 푓(푧) =푐1푒푐푧 and 푔(푧) = 푐2푒−푐푧, where 푐1, 푐2 and 푐 are three constants satisfying(−1)푘(푐1푐2)푛(푛푐)2푘 = 1 or 푓 ≡ 푡푔 for a constant 푡 such that 푡푛 = 1;(ii) when 푚 = 1, 푛 > 9푘 + 20 and Θ(∞, 푓) > 2푛, then either [푓푛(푓 −1)푚](푘)[푔푛(푔 − 1)푚](푘) ≡ 1 or 푓 ≡ 푔;(iii) when 푚 ≥ 2 and 푛 > 9푘 + 4푚 + 16, then either [푓푛(푓 − 1)푚](푘)[푔푛(푔 −1)푚](푘) ≡ 1 or 푓 ≡ 푔 or 푓 and 푔 satisfy the algebraic equation 푅(푓, 푔) =0, where 푅(푥, 푦) = 푥푛(푥 − 1)푚 − 푦푛(푦 − 1)푚. The possibility [푓푛(푓 −1)푚](푘)[푔푛(푔 − 1)푚](푘) ≡ 1 does not arise for 푘 = 1.In 2011, J.Xia and Y. Xu [8] proved the following three theorems.Theorem 1.4. [8] Let 푛, 푘 and 푚 be three positive integers, and 푓 and 푔 betwo non-constant meromorphic functions such that [푓푛(푓 − 1)푚](푘) and [푔푛(푔 −1)푚](푘) share 1 CM. If 푚 > 푘 and 푛 > 3푘 +푚 + 8, and Θ(∞, 푓) > 2푚(푚 +푛)/[(푛+푚)2 − 4푘2]or Θ(∞, 푔) > 2푚(푚 + 푛)/[(푛+푚)2 − 4푘2], then eitherUniqueness of Meromorphic Functions 943푓 ≡ 푔, or 푓 and 푔 satisfy the algebraic equation 푅(푓, 푔) = 0, where 푅(푥, 푦) =푥푛(푥− 1)푚 − 푦푛(푦 − 1)푚.Theorem 1.5. [8] Let 푛, 푘 and 푚 be three positive integers, and 푓 and 푔 be twonon-constant meromorphic functions such that [푓푛(푓−1)푚](푘) and [푔푛(푔−1)푚](푘)share 1 CM. If 푚 ≤ 푘 and 푛 > 3푘 +푚+ 8, andΘ(∞, 푓) + Θ[ 푘푚])(1, 푓) > 1 + 2푚(푚+ 푛)/[(푛+푚)2 − 4푘2]orΘ(∞, 푔) + Θ[ 푘푚])(1, 푔) > 1 + 2푚(푚+ 푛)/[(푛+푚)2 − 4푘2]then the conclusion of Theorem 1.4 holds.Theorem 1.6. [8] Let 푛, 푘 and 푚 be three positive integers such that 푛 > 3푘+푚+8, and 푓 and 푔 be two non-constant meromorphic functions such that [푓푛(푓 −1)푚](푘) and [푔푛(푔 − 1)푚](푘) share 1 CM. If 푓 and 푔 have the same poles (notnecessary with the same multiplicity) then the conclusion of Theorem 1.4 holds.In 2011, C. Wu, C.Mu and J.Li [6] proved the following theorem.Theorem 1.7. [6] Let 푓 and 푔 be two non-constant meromorphic functions, andlet 푛 ≥ 1, 푘 ≥ 1 and 푚 ≥ 0 be three positive integers. Let [푓푛(푓 − 1)푚](푘) and[푔푛(푔 − 1)푚](푘) share 1 IM. Then one of the following holds:(i) when 푚 = 0 and 푛 > 9푘+14, then either 푓(푧) = 푐1푒푐푧 and 푔(푧) = 푐2푒−푐푧,where 푐1, 푐2 and 푐 are constants satisfying (−1)푘(푐1푐2)푛(푛푐)2푘 = 1 or푓 ≡ 푡푔 for a constant 푡 with 푡푛 = 1;(ii) when 푚 = 1, 푛 > 9푘 + 18 and Θ(∞, 푓) > 2푛, then 푓 ≡ 푔;(iii) when 푚 ≥ 2 and 푛 > 9푘 + 4푚 + 14, then 푓 ≡ 푔 or 푓 and 푔 satisfy thealgebraic equation 푅(푥, 푦) = 푥푛(푥− 1)푚 − 푦푛(푦 − 1)푚 = 0.One may ask the following question which is the motivation of the paper: Isit possible to relax the nature of the sharing value in Theorem 1.7 ?In this paper, we give positive answers to the above question by establishingthe following two theorems, which improves Theorems 1.1–1.7.Theorem 1.8. Let 푓 and 푔 be two non-constant meromorphic functions, and let푛 ≥ 1, 푘 ≥ 1 and 푚 ≥ 0 be three positive integers. If 퐸푙)(1; [푓푛(푓 − 1)푚](푘)) =퐸푙)(1; [푔푛(푔−1)푚](푘)) and 퐸1)(1; [푓푛(푓−1)푚](푘)) = 퐸1)(1; [푔푛(푔−1)푚](푘)), where푙 ≥ 3 is an integer. Then one of the following holds:(i) If 푚 = 0, if 푓 ∕= ∞, 푔 ∕= ∞ and 푛 > 13푘+283 , then either 푓(푧) =푐1푒푐푧 and 푔(푧) = 푐2푒−푐푧, where 푐1, 푐2 and 푐 are constants satisfying(−1)푘(푐1푐2)푛(푛푐)2푘 = 1 or 푓 ≡ 푡푔 for a constant 푡 such that 푡푛 = 1;944 H.P. Waghamore and A. Tanuja(ii) If 푚 = 1, 푛 > 13푘+413 and Θ(∞, 푓) >2푛, then 푓 ≡ 푔;(iii) If 푚 ≥ 2 and 푛 > 13푘+5푚+363 , then either 푓 ≡ 푔 or 푓 and 푔 satisfy thealgebraic equation 푅(푥, 푦) = 푥푛(푥− 1)푚 − 푦푛(푦 − 1)푚 = 0.Theorem 1.9. Let 푓 and 푔 be two non-constant meromorphic functions, and let푛 ≥ 푘 + 1, 푘 ≥ 1 and 푚 ≥ 푘 + 1 be three positive integers. If 퐸푙)(1; [푓푛(푓 −1)푚](푘)) = 퐸푙)(1; [푔푛(푔 − 1)푚](푘)) and 퐸2)(1; [푓푛(푓 − 1)푚](푘)) = 퐸2)(1; [푔푛(푔 −1)푚](푘)), where 푙 ≥ 4 is an integer. Then one of the following holds:(i) If 푚 = 0, if 푓 ∕= ∞, 푔 ∕= ∞ and 푛 > 3푘+83 , then either 푓(푧) =푐1푒푐푧 and 푔(푧) = 푐2푒−푐푧, where 푐1, 푐2 and 푐 are constants satisfying(−1)푘(푐1푐2)푛(푛푐)2푘 = 1 or 푓 ≡ 푡푔 for a constant 푡 such that 푡푛 = 1;(ii) If 푚 = 1, 푛 > 3푘+93 and Θ(∞, 푓) >2푛, then 푓 ≡ 푔;(iii) If 푚 ≥ 2 and 푛 > 3푘−푚+103 , then either 푓 ≡ 푔 or 푓 and 푔 satisfy thealgebraic equation 푅(푥, 푦) = 푥푛(푥− 1)푚 − 푦푛(푦 − 1)푚 = 0.Remark 1.10. Theorem 1.8 and Theorem 1.9 extend Theorem 1.3 and Theorem1.7.Remark 1.11. Theorem 1.9 extends Theorem 1.1 for 푚 = 0 and Theorem 1.2 for푚 = 1.Remark 1.12. Theorem 1.9 extends Theorem 1.4, Theorem 1.5 and Theorem1.6.2. LemmasIn this section, we present some lemmas which are needed in the sequel.Lemma 2.1. [9] Let 푓 be a nonconstant meromorphic function and 푃 (푓) = 푎0 +푎1푓+ ...+푎푛푓푛,where 푎0, 푎1,...,푎푛 are constants and 푎푛 ∕= 0. Then 푇 (푟, 푃 (푓)) =푛푇 (푟, 푓) + 푆(푟, 푓).Lemma 2.2. [7] Let 퐸푙)(1; [퐹∗](푘)) = 퐸푙)(1; [퐺∗](푘)), 퐸1)(1; [퐹∗](푘)) = 퐸1)(1;[퐺∗](푘)) and 퐻∗ ∕= 0, where 푙 ≥ 3. Then푇 (푟, 퐹 ∗) ≤(83+23푘)푁(푟,∞;퐹 ∗) +53푁(푟, 0;퐹 ∗) +23푁푘(푟, 0;퐹∗)+ 푁푘+1(푟, 0;퐹∗) + (푘 + 2)푁(푟,∞;퐺∗) +푁(푟, 0;퐺∗)+ 푁푘+1(푟, 0;퐺∗) + 푆(푟, 퐹 ∗) + 푆(푟,퐺∗)Uniqueness of Meromorphic Functions 945where퐻∗ ≡[(퐹 ∗)(푘+2)(퐹 ∗)(푘+1)−2(퐹 ∗)(푘+1)(퐹 ∗)(푘) − 1]−[(퐺∗)(푘+2)(퐺∗)(푘+1)−2(퐺∗)(푘+1)(퐺∗)(푘) − 1].Lemma 2.3. [7] Let 퐸푙)(1; [퐹∗](푘)) = 퐸푙)(1; [퐺∗](푘)), 퐸1)(1; [퐹∗](푘)) = 퐸1)(1;[퐺∗](푘)), where 푙 ≥ 3. IfΔ1푙 =(83+23푘)Θ(∞, 퐹 ∗) + (푘 + 2)Θ(∞, 퐺∗) +53Θ(0, 퐹 ∗) + Θ(0, 퐺∗)+ 훿푘+1(0, 퐹∗) + 훿푘+1(0, 퐺∗) +23훿푘(0, 퐹∗)>53푘 + 9,then either [퐹 ∗](푘)[퐺∗](푘) ≡ 1 or 퐹 ∗ ≡ 퐺∗.Lemma 2.4. [7] Let 퐸푙)(1; [퐹∗](푘)) = 퐸푙)(1; [퐺∗](푘)), 퐸2)(1; [퐹∗](푘)) = 퐸2)(1;[퐺∗](푘)) and 퐻∗ ∕= 0, where 푙 ≥ 4. Then푇 (푟, 퐹 ∗) + 푇 (푟,퐺∗) ≤ (푘 + 4)푁(푟,∞;퐹 ∗) + 2푁(푟, 0;퐹 ∗)+ 2푁푘+1(푟, 0;퐹∗) + (푘 + 4)푁(푟,∞;퐺∗) + 2푁(푟, 0;퐺∗)+ 2푁푘+1(푟, 0;퐺∗) + 푆(푟, 퐹 ∗) + 푆(푟,퐺∗)where 퐻∗ is deﬁned as Lemma 2.2.Lemma 2.5. [7] Let 퐸푙)(1; [퐹∗](푘)) = 퐸푙)(1; [퐺∗](푘)), 퐸2)(1; [퐹∗](푘)) = 퐸2)(1;[퐺∗](푘)), where 푙 ≥ 4. IfΔ2푙 =(12푘 + 2)[Θ(∞, 퐹 ∗) + Θ(∞, 퐺∗)] + Θ(0, 퐹 ∗) + Θ(0, 퐺∗)+ 훿푘+1(0, 퐹∗) + 훿푘+1(0, 퐺∗)>푘 + 5,then either [퐹 ∗](푘)[퐺∗](푘) ≡ 1 or 퐹 ∗ ≡ 퐺∗.Lemma 2.6. Let 푓 and 푔 be two non-constant meromorphic functions, and let 푛 ≥푘+1, 푘 ≥ 1 and 푚 ≥ 푘+1 be a integers. Then [푓푛(푓−1)푚](푘)[푔푛(푔−1)푚](푘) ∕≡ 1.Proof. Let[푓푛(푓 − 1)푚](푘)[푔푛(푔 − 1)푚](푘) ≡ 1. (2.1)Let 푧0 be a zero of 푓 of order 푝0. From (2.1) we get 푧0 is a pole of 푔. Suppose that푧0 is a pole of 푔 of order 푞0. Again by (2.1), we obtain 푛푝0− 푘 = 푛푞0+푚푞0+ 푘,946 H.P. Waghamore and A. Tanujai.e., 푛(푝0 − 푞0) = 푚푞0 + 2푘. which implies that 푞0 ≥푛−2푘푚and so we have푝0 ≥푛+푚−2푘푚.Let 푧1 be a zero of 푓 − 1 of order 푝1, then 푧1 is a zero of [푓푛(푓 − 1)푚](푘)of order 푝1 − 푘. Therefore from (2.1) we obtain 푝1 − 푘 = 푛푞1 + 푚푞1 + 푘 i.e.,푝1 ≥ 푛+푚+ 2푘.Let 푧2 be a zero of 푓′ of order 푝2 that is not a zero of 푓(푓−1), then from (2.1)푧2 is a pole of 푔 of order 푞2. Again by (2.1) we get 푝2 − (푘− 1) = 푛푞2 +푚푞2+ 푘i.e., 푝2 ≥ 푛+푚+ 2푘 − 1.In the same manner as above, we have similar results for the zeros of [푔푛(푔−1)푚](푘).On other hand, suppose that 푧3 is a pole of 푓 . From (2.1), we get that 푧3 isthe zero of [푔푛(푔 − 1)푚](푘).Thus푁(푟, 푓) ≤푁(푟,1푔)+푁(푟,1푔 − 1)+푁(푟,1푔′)≤1푝0푁(푟,1푔)+1푝1푁(푟,1푔 − 1)+1푝2푁(푟,1푔′)≤[푚푛+푚− 2푘+1푛+푚+ 2푘+2푛+푚+ 2푘 − 1]푇 (푟, 푔)+ 푆(푟, 푔).(2.2)By second fundamental theorem and equation (2.2), we have푇 (푟, 푓) ≤푁(푟,1푓)+푁(푟,1푓 − 1)+푁(푟, 푓)≤푚푛+푚− 2푘푁(푟,1푓)+1푛+푚+ 2푘푁(푟,1푓 − 1)+[푚푛+푚− 2푘+1푛+푚+ 2푘+2푛+푚+ 2푘 − 1]푇 (푟, 푔)+ 푆(푟, 푔) + 푆(푟, 푓).푇 (푟, 푓) ≤[푚푛+푚− 2푘+1푛+푚+ 2푘]푇 (푟, 푓)+[푚푛+푚− 2푘+1푛+푚+ 2푘+2푛+푚+ 2푘 − 1]푇 (푟, 푔)+ 푆(푟, 푔) + 푆(푟, 푓).(2.3)Similarly, we have푇 (푟, 푔) ≤[푚푛+푚− 2푘+1푛+푚+ 2푘]푇 (푟, 푔)+[푚푛+푚− 2푘+1푛+푚+ 2푘+2푛+푚+ 2푘 − 1]푇 (푟, 푓)+ 푆(푟, 푔) + 푆(푟, 푓).(2.4)Uniqueness of Meromorphic Functions 947Adding (2.3) and (2.4) we get푇 (푟, 푓) + 푇 (푟, 푔)≤[2푚푛+푚− 2푘+2푛+푚+ 2푘+2푛+푚+ 2푘 − 1]{푇 (푟, 푓) + 푇 (푟, 푔)}+ 푆(푟, 푔) + 푆(푟, 푓).which is a contradiction. Thus Lemma proved.3. Proof of the TheoremProof of Theorem 1.8. Let 퐹 ∗ = 푓푛(푓 − 1)푚, 퐺∗ = 푔푛(푔 − 1)푚.By Lemma 2.1, we getΘ(0, 퐹 ∗) = 1− lim sup푟→∞푁(푟, 0;퐹 ∗)푇 (푟, 퐹 ∗)≥푛+푚−푚∗ − 1푛+푚(3.1)where 푚∗ = {0 푖푓 푚=01 푖푓 푚≥1SimilarlyΘ(0, 퐺∗) ≥푛+푚−푚∗ − 1푛+푚(3.2)Θ(∞, 퐹 ∗) = 1− lim sup푟→∞푁(푟,∞;퐹 ∗)푇 (푟, 퐹 ∗)≥푛+푚− 1푛+푚(3.3)SimilarlyΘ(∞, 퐺∗) ≥푛+푚− 1푛+푚(3.4)훿푘+1(0, 퐹∗) = 1− lim sup푟→∞푁푘+1(푟, 0;퐹∗)푇 (푟, 퐹 )≥푛− 푘 − 1푛+푚(3.5)Similarly훿푘+1(0, 퐺∗) ≥ 1−푛− 푘 − 1푛+푚, 훿푘(0, 퐹∗) ≥푛− 푘푛+푚, 훿푘(0, 퐺∗) ≥푛− 푘푛+푚(3.6)From the assumptions of Theorem 1.8, we have 퐸푙)(1; [푓푛(푓 − 1)푚](푘)) = 퐸푙)(1;[푔푛(푔−1)푚](푘)) and 퐸1)(1; [푓푛(푓−1)푚](푘)) = 퐸1)(1; [푔푛(푔−1)푚](푘)), where 푙 ≥ 3.From (3.1)-(3.6) and Lemma 2.3, we haveΔ1푙 ≥(143+53푘)푛+푚− 1푛+푚+83푛+푚−푚∗ − 1푛+푚+23푛− 푘푛+푚+ 2푛− 푘 − 1푛+푚.It is easily veriﬁed that if 푛 > 13푘+5푚+8푚∗+283 , then Δ1푙 >53푘 + 9. Since13푘 + 5푚+ 8푚∗ + 283=13푘 + 283if 푚 = 0=13 + 413if 푚 = 1=13푘 + 5푚+ 363if 푚 ≥ 2948 H.P. Waghamore and A. Tanujaby Lemma 2.3, we have 퐹 ∗ ≡ 퐺∗ or (퐹 ∗)(푘)(퐺∗)(푘) ≡ 1. If (퐹 ∗)(푘)(퐺∗)(푘) ≡ 1,i.e.,[푓푛(푓 − 1)푚](푘)[푔푛(푔 − 1)푚](푘) ≡ 1then by Lemma 2.6 we can get a contradiction. Hence, we deduce that 퐹 ∗ ≡ 퐺∗,i.e.,푓푛(푓 − 1)푚 ≡ 푔푛(푔 − 1)푚. (3.7)Now we consider following three cases.Case(i) Let 푚 = 0. Then from (3.7) we get 푓 ≡ 푡푔 for a constant 푡 such that푡푛 = 1Case (ii) Let 푚 = 1 then from (3.7) we have푓푛(푓 − 1) ≡ 푔푛(푔 − 1). (3.8)Suppose 푓 ∕≡ 푔. Let ℎ = 푓푔be a constant. Then from (3.8) it follows that ℎ ∕= 1,ℎ푛 ∕= 1, ℎ푛+1 ∕= 1 and 푔 = 1−ℎ푛1−ℎ푛+1 = 푐표푛푠푡푎푛푡, a contradiction. So we supposethat ℎ is not a constant. Since 푓 ∕≡ 푔, we have ℎ ∕≡ 1. From (3.8) we obtain 푔 =1−ℎ푛1−ℎ푛+1 and 푓 =(1−ℎ푛1−ℎ푛+1)ℎ. Hence it follows that 푇 (푟, 푓) = 푛푇 (푟, ℎ) + 푆(푟, 푓).Again by second fundamental theorem of Nevanlinna, we have 푁(푟,∞; 푓) =∑푛푗=1푁(푟, 훼푗 ;ℎ) ≥ (푛 − 2)푇 (푟, ℎ) + 푆(푟, 푓), where 훼푗(∕= 1)(푗 = 1, 2, ..., 푛) aredistinct roots of the equation ℎ푛+1 = 1. So we obtainΘ(∞, 푓) = 1− lim sup푟→∞푁(푟,∞; 푓)푇 (푟, 푓)≤2푛which contradicts the assumption Θ(∞, 푓) > 2푛. Thus 푓 ≡ 푔.Case(iii) Let 푚 ≥ 2. Then from (3.7) we obtain푓푛[푓푚 + ...+ (−1)푖퐶푚푚−푖푓푚−푖 + ...+ (−1)푚]=푔푛[푔푚 + ...+ (−1)푖퐶푚푚−푖푔푚−푖 + ...+ (−1)푚].(3.9)Let ℎ = 푓푔. If h is a constant, then substituting 푓 = 푔ℎ in (3.9) we obtain푔푛+푚(ℎ푛+푚−1)+...+(−1)푖퐶푚푚−푖푔푛+푚−푖(ℎ푛+푚−푖−1)+...+(−1)푚푔푛(ℎ푛−1) = 0,which imply ℎ = 1. Hence 푓 ≡ 푔. If ℎ is not a constant, then from (3.9) wecan say that 푓 and 푔 satisfy the algebraic equation 푅(푓, 푔) = 0, where 푅(푥, 푦) =푥푛(푥− 1)푚 − 푦푛(푦 − 1)푚. ■Proof of Theorem 1.9. From the condition of Theorem 1.9, we have퐸푙)(1; [푓푛(푓 −1)푚](푘)) = 퐸푙)(1; [푔푛(푔−1)푚](푘)) and 퐸2)(1; [푓푛(푓 −1)푚](푘)) =퐸2)(1; [푔푛(푔 − 1)푚](푘)), where 푙 ≥ 4.From (3.1)-(3.6) we haveΔ2푙 ≥ (푘 + 4)푛+푚− 1푛+푚+ 2푛+푚−푚∗ − 1푛+푚+ 2푛− 푘 − 1푛+푚.Uniqueness of Meromorphic Functions 949It is easily veriﬁed that if 푛 > 3푘−푚+2푚∗+83 , then Δ2푙 > 푘 + 5. Since3푘 −푚+ 2푚∗ + 83=3푘 + 83if 푚 = 0=3푘 + 93if 푚 = 1=3푘 −푚+ 103if 푚 ≥ 2by Lemma 2.5, we have 퐹 ∗ ≡ 퐺∗ or (퐹 ∗)(푘)(퐺∗)(푘) ≡ 1. If (퐹 ∗)(푘)(퐺∗)(푘) ≡ 1,i.e.,[푓푛(푓 − 1)푚](푘)[푔푛(푔 − 1)푚](푘) ≡ 1then by Lemma 2.6 we can get a contradiction. Hence, we deduce that 퐹 ∗ ≡ 퐺∗,i.e.,푓푛(푓 − 1)푚 ≡ 푔푛(푔 − 1)푚.Proceeding as in the proof of Theorem 1.8, we can get the conclusion of Theorem1.9. ■Acknowledgement. The authors is grateful to the referee for his/her valuablesuggestions and comments towards the improvement of the paper.References[1] S.S. Bhoosnurmath, R.S. Dyavanal, Uniqueness and value-sharing of meromorphicfunctions, Comput. Math. Appl. 53 (2007) 1191–1205.[2] W.K. Hayman, Meromorphic Functions, The Clarendon Press, Oxford, 1964.[3] I. Lahiri, A. Sarkar, Uniqueness of meromorphic functions and its derivative, J.Inequal. Pure Appl. Math. 5 (2004) 2–21.[4] X.M. Li, H.X. Yi, A result of Terglane concerning three weighted sharing values,Southeast Asian Bull. Math. 32 (2008) 1101–1114.[5] P. Sahoo, Uniqueness of meromorphic functions when two diﬀerential polynomialsshare one value IM, Mat. Bech. 62 (2010) 169–182.[6] C. Wu, C. Mu, J. Li, Uniqueness of meromorphic functions concerning diﬀerentialpolynomials share one value, J. Inequal. Appl., doi:10.1186/1029-242X-2011-133.[7] X.Y. Xu, T.B. Cao, S. Liu, Uniqueness results of meromorphic functions whosenonlinear diﬀerential polynomials have one nonzero pseudo value, Matemat.Bech.62 (2012) 1–16.[8] J. Xia, Y. Xu, Uniqueness and diﬀerential polynomial of meromorphic functionssharing one value, Filomat 25 (2011) 185–194.[9] L. Yang, Value Distribution Theory, Springer Verlag, Berlin, 1993.[10] H.X. Yi, C.C. Yang, Uniqueness Theory of Meromorphic Functions, Science Press,Beijing, 1995.

Uniqueness of Meromorphic Functions whose Certain Differential Polynomials have Two Pseudo Common Values

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Uniqueness of Meromorphic Functions whose Certain Differential Polynomials have Two Pseudo Common Values

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