Journal of Inequalities in Pure and Applied Mathematics

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1 Journl o Inequlities in Pure nd Applied Mthemtics Volume 6, Issue 4, Article 6, 2005 MROMORPHIC UNCTION THAT SHARS ON SMALL UNCTION WITH ITS DRIVATIV QINCAI ZHAN SCHOOL O INORMATION RNMIN UNIVRSITY O CHINA BIJIN 00872, P.R. CHINA qingcizhng@yhoo.com.cn Received 4 Mrch, 2005; ccepted 03 Septembe 2005 Communicted by H.M. Srivstv ABSTRACT. In this pper we study the problem o meromorphic unction shring one smll unction with its derivtive nd improve the results o K.-W. Yu nd I. Lhiri nd nswer the open questions posed by K.-W. Yu. Key words nd phrses: Meromorphic unction; Shred vlue; Smll unction Mthemtics Subject Clssiiction. 30D35.. INTRODUCTION AND MAIN RSULTS By meromorphic unction we shll lwys men unction tht is meromorphic in the open complex plne C. It is ssumed tht the reder is milir with the nottions o Nevnlinn theory such s T, m, N, N, S nd on, tht cn be ound, or instnce, in [2], [5]. Let nd g be two non-constnt meromorphic unctions, C { }, we sy tht nd g shre the vlue IM ignoring multiplicities i nd g hve the sme zeros, they shre the vlue CM counting multiplicities i nd g hve the sme zeros with the sme multiplicities. When = the zeros o mens the poles o see [5]. Let l be non-negtive integer or ininite. or ny C { }, we denote by l, the set o ll -points o where n -point o multiplicity m is counted m times i m l nd l + times i m > l. I l, = l, g, we sy nd g shre the vlue with weight l see [3], [4]. nd g shre vlue with weight l mens tht z 0 is zero o with multiplicity m l i nd only i it is zero o g with the multiplicity m l, nd z 0 is zero o with multiplicity m> l i nd only i it is zero o g with the multiplicity n> l, where m is not necessrily equl to n. ISSN electronic: c 2005 Victori University. All rights reserved

2 2 QINCAI ZHAN We write nd g shre, l to men tht nd g shre the vlue with weight l. Clerly, i nd g shre, l, then nd g shre, p or ll integers p, 0 p l. Al we note tht nd g shre vlue IM or CM i nd only i nd g shre, 0 or, respectively see [3], [4]. A unction z is sid to be smll unction o i z is meromorphic unction stisying T = S, T = ot s r + possibly outside set o inite liner mesure. Similrly, we deine tht nd g shre smll unction IM or CM or with weight l by nd g shring the vlue 0 IM or CM or with weight l respectively. Brück [] irst considered the uniqueness problems o n entire unction shring one vlue with its derivtive nd proved the ollowing result. Theorem A. Let be n entire unction which is not constnt. I nd shre the vlue CM nd i N = S, then c or me constnt c C\{0}. Brück [] urther posed the ollowing conjecture. Conjecture.. Let be n entire unction which is not constnt, ρ be the irst iterted order o. I ρ < + nd ρ is not positive intege nd i nd shre one vlue CM, then c or me constnt c C\{0}. Yng [7] proved tht the conjecture is true i is n entire unction o inite order. Zhng [9] extended Theorem A to meromorphic unctions. Yu [8] recently considered the problem o n entire or meromorphic unction shring one smll unction with its derivtive nd proved the ollowing two theorems. Theorem B [8]. Let be non-constnt entire unction nd z be meromorphic unction such tht 0, nd T = ot s r +. I nd k shre the vlue 0 CM nd δ0, > 3 4, then k. Theorem C [8]. Let be non-constnt, non-entire meromorphic unction nd z be meromorphic unction such tht 0, nd T = ot s r +. I i nd hve no common poles, ii nd k shre the vlue 0 CM, iii 4δ0, + 2Θ, > 9 + 2k, then k, where k is positive integer. In the sme pper Yu [8] urther posed the ollowing open questions: Cn CM shred be replced by n IM shred vlue? 2 Cn the condition δ0, > 3 o Theorem B be urther relxed? 4 3 Cn the condition iii o Theorem C be urther relxed? 4 Cn, in generl, the condition i o Theorem C be dropped? Let p be positive integer nd C { }. We use N p to denote the counting unction o the zeros o counted with proper multiplicities whose multiplicities re not greter thn p, N p+ to denote the counting unction o the zeros o whose multiplicities re not less thn p +. And N p nd N p+ denote their corresponding reduced counting unctions ignoring multiplicities respectively. We l use N p to denote the counting unction o the zeros o where zero o multiplicity m is counted m J. Inequl. Pure nd Appl. Mth., 64 Art. 6,

3 MROMORPHIC UNCTIONS 3 times i m p nd p times i m > p. Clerly N = N N p δ p, = lim sup. r + T. Deine Obviously δ p, δ,. Lhiri [4] improved the results o Zhng [9] with weighted shred vlue nd obtined the ollowing two theorems Theorem D [4]. Let be non-constnt meromorphic unction nd k be positive integer. I nd k shre, 2 nd 2N 2 k 2 < λ + ot k or r I, where 0 < λ < nd I is set o ininite liner mesure, then k constnt c C\{0}. c or me Theorem [4]. Let be non-constnt meromorphic unction nd k be positive integer. I nd k shre, nd 2N 2 + 2N < λ + ot k k or r I, where 0 < λ < nd I is set o ininite liner mesure, then k constnt c C\{0}. c or me In the sme pper Lhiri [4] l obtined the ollowing result which is n improvement o Theorem C. Theorem [4]. Let be non-constnt meromorphic unction nd k be positive integer. Al, let z 0, be meromorphic unction such tht T = S. I i hs no zero pole which is l zero pole o or k with the sme multiplicity. ii nd k shre 0, 2 CM, iii 2δ 2+k 0, kθ, > 5 + k, then k. In this ppe we still study the problem o meromorphic or entire unction shring one smll unction with its derivtive nd obtin the ollowing two results which re the improvement nd complement o the results o Yu [8] nd Lhiri [4] nd nswer the our open questions o Yu in [8]. Theorem.2. Let be non-constnt meromorphic unction nd k, l 0 be integers. Al, let z 0, be meromorphic unction such tht T = S. Suppose tht nd k shre 0, l. I l 2 nd. 2N 2 or l = nd.2 2N 2 k 2 / < λ + ot k, + 2N < λ + ot k, k / J. Inequl. Pure nd Appl. Mth., 64 Art. 6,

4 4 QINCAI ZHAN or l = 0, nd k shre the vlue 0 IM nd.3 4N + 3N 2 + 2N < λ + ot k, k / or r I, where 0 < λ < nd I is set o ininite liner mesure, then k constnt c C\{0}. c or me Theorem.3. Let be non-constnt meromorphic unction nd k, l 0 be integers. Al, let z 0, be meromorphic unction such tht T = S. Suppose tht nd k shre 0, l. I l 2 nd kθ, + 2δ 2+k 0, > k + 4, or l = nd kθ, + 3δ 2+k 0, > k + 6, or l = 0, nd k shre the vlue 0 IM nd kΘ, + 5δ 2+k 0, > 2k + 0, then k. Clerly Theorem.2 extends the results o Lhiri Theorem D nd to smll unctions. Theorem.3 gives the improvements o Theorem C nd, which removes the restrictions on the zeros poles o z nd z nd relxes other conditions, which l includes result o meromorphic unction shring one vlue or smll unction IM with its derivtive, it nswers the our open questions o Yu [8]. rom Theorem.2 we hve the ollowing corollry which is the improvement o Theorem A. Corollry.4. Let be n entire unction which is not constnt. I nd shre the vlue IM nd i N = S, then c or me constnt c C\{0}. rom Theorem.3 we hve Corollry.5. Let be non-constnt entire unction nd z 0, be meromorphic unction such tht T = S. I nd k shre the vlue 0 CM nd δ0, > 2, or i nd k shre the vlue 0 IM nd δ0, > 4 5, then k. Clerly Corollry.5 is n improvement nd complement o Theorem B. 2. MAIN LMMAS Lemm 2. see [4]. Let be non-constnt meromorphic unction, k be positive intege then N p N k p+k + kn + S. This lemm cn be obtined immeditely rom the proo o Lemm 2.3 in [4] which is the specil cse p = 2. Lemm 2.2 see [5]. Let be non-constnt meromorphic unction, n be positive integer. P = n n + n n + + where i is meromorphic unction such tht T i = S i =, 2,..., n. Then T P = nt + S. J. Inequl. Pure nd Appl. Mth., 64 Art. 6,

5 MROMORPHIC UNCTIONS 5 3. PROO O THORM.2 Let =, = k, then =, = k. Since nd k shre 0, l, nd shre, l except the zeros nd poles o z. Deine 3. H = 2 2, We hve the ollowing two cses to investigte. Cse. H 0. Integrtion yields 3.2 C + D, where C nd D re constnts nd C 0. I there exists pole z 0 o with multiplicity p which is not the pole nd zero o z, then z 0 is the pole o with multiplicity p nd the pole o with multiplicity p + k. This contrdicts with 3.2. So N N 3.3 = S, N = S, N = S. 3.2 l shows nd shre the vlue CM. Next we prove D = 0. We irst ssume tht D 0, then 3.4 D + C D. So 3.5 N = N = S. + C D I C D hve, by the second undmentl theorem nd 3.3, 3.5 nd S = S, we T N + S + C D N + S T + S. So 3.6 T = N T k = N + S, k + S, this contrdicts with conditions.,.2 nd.3 o this theorem. I C =, rom 3.4 we know D C, then C C. J. Inequl. Pure nd Appl. Mth., 64 Art. 6,

6 6 QINCAI ZHAN Noticing tht =, = k, we hve C C 2 k. By Lemm 2.2 nd 3.3 nd 3.7, then 2T = T S C = T + + S C = T k + S N + kn + S T + S. So T = S, this is impossible. Hence D = 0, nd C, k is just the conclusion o this theorem. C. This Cse 2. H 0. rom 3. it is esy to see tht m H = S. Subcse 2. l. rom 3. we hve 3.9 N H N l , where N 0 denotes the counting unction o the zeros o which re not the zeros o nd, nd N 0 denotes its reduced orm. In the sme wy, we cn deine N0 nd N 0. Let z0 be simple zero o but z 0 0,, then z 0 is l the simple zero o. By clculting z 0 is the zero o H, 3.0 N N N H + S. H Noticing tht N = N + S, we hve 3. N = N 2 N l S. J. Inequl. Pure nd Appl. Mth., 64 Art. 6,

7 MROMORPHIC UNCTIONS 7 By the second undmentl theorem nd 3. nd noting N = N + S, then T N N S 2N 2 2 l S. While l 2, 3.3 N 2 l+ 2 0 N 2, T 2N S, T k 2N 2 This contrdicts with.. While l =, 3.3 turns into N 2 l+ Similrly s bove, we hve 2 T k 2N 2 This contrdicts with.2. k 2 + S. / 0 2N. + 2N + S. k / Subcse 2.2 l = 0. In this cse, nd shre IM except the zeros nd poles o z. Let z 0 be the zero o with multiplicity p nd the zero o with multiplicity q. We denote by N the counting unction o the zeros o where p = q = ; by N 2 the counting unction o the zeros o where p = q 2; by N L the counting unction o the zeros o where p > q, ech point in these counting unctions is counted only once. In the sme wy, we cn deine N is esy to see tht 3.4 N N N 2 = N = N = N = N 2 L, N 2 + S, + S, + S 2 L nd N L. It + S. J. Inequl. Pure nd Appl. Mth., 64 Art. 6,

8 8 QINCAI ZHAN rom 3. we hve now 3.5 N H N 2 2 L L S. In this cse, 3.0 is replced by 3.6 N N H + S. rom 3.4, 3.5 nd 3.6, we hve N N N L + 2N L S N + 2N + 2N L S. By the second undmentl theorem, then T N N 0 + S 2N + 2N 2 + 2N L + S 2N + 2N + 2N + S. rom Lemm 2. or p =, k = we know N N 2 + S. So T 4N + 3N 2 + 2N + S, T k 4N + 3N 2 This contrdicts with.3. The proo is complete. + 2N + S. k / J. Inequl. Pure nd Appl. Mth., 64 Art. 6,

9 MROMORPHIC UNCTIONS 9 4. PROO O THORM.3 The proo is similr to tht o Theorem.2. We deine nd nd 3. s bove, nd we l distinguish two cses to discuss. Cse 3. H 0. We l hve 3.2. rom 3.3 we know tht Θ, =, nd rom.4,.5 nd.6, we urther know δ 2+k 0, >. Assume tht D 0, then 2 D D C, N = N = S. D I D, using the second undmentl theorem or, similrly s 3.6 we hve T = N + S, T = N + S. Hence Θ0, = 0, this contrdicts with Θ0, δ 2+k 0, >. 2 I D =, then N = S, N = S, nd Then nd thus, C. C C 4. k k + C C 2 k. As sme s 3.8, by Lemm 2.2 nd 3.3 nd N = S, rom 4. we hve 2T k = T k + S = N k + S kn + kn + S = S. So T k = S nd T k = S. Hence T T + T k + O k = T k + T k + O = S, this is impossible. Thereore D = 0, nd rom 3.2 then. C J. Inequl. Pure nd Appl. Mth., 64 Art. 6,

10 0 QINCAI ZHAN I C, then nd N + C, C = N. + C By the second undmentl theorem nd 3.3 we hve T N + S + C N + S. By Lemm 2. or p = nd 3.3, we hve T N + S k N +k + S 2N +k + S. Hence δ +k 0,. This is contrdiction with δ 2 +k0, δ 2+k 0, >. So C = nd 2, k. This is just the conclusion o this theorem. Cse 4. H 0. Subcse 4. l. As similr s Subcse 2., rom 3.9 nd 3.0 we hve N = N 2 N 2 2 l S. While l 2, N l+ 2 N T + O, J. Inequl. Pure nd Appl. Mth., 64 Art. 6,

11 MROMORPHIC UNCTIONS N N 2 By the second undmentl theorem, we hve T + S. T + T N N 0 N 0 + S + S 3N T + S, T 3N S, T 3N S. k By Lemm 2. or p = 2 we hve This contrdicts with.4. While l =, N l+ T 3 + kn + 2N 2+k 3 + kθ, + 2δ 2+k 0, k + 4. by Lemm 2. or p =, k =, we hve N N 2 N + S, 2 2 T + O, T + S N T + S 2N T + S J. Inequl. Pure nd Appl. Mth., 64 Art. 6,

12 2 QINCAI ZHAN As sme s bove, by the second undmentl theorem we hve T + T 4N + 2N T + S, T 4N + 2N S, T 4N + 2N S. k By Lemm 2. or p = 2 we hve This contrdicts with.5. Subcse 4.2 l = 0. then N N 2 T 4 + kn + 3N 2+k 4 + kθ, + 3δ 2+k 0, k S, rom 3.4, 3.5 nd 3.6 nd Lemm 2. or p =, k =, noticing L N T + S, = N 2 L L N N L L 2 L S N + 2N + T + S 4N + 2N T + S. As sme s bove, by the second undmentl theorem, we cn obtin T + T 6N + 3N 2 + 2N 2 + T + S, T 6N + 3N 2 + 2N 2 + S, J. Inequl. Pure nd Appl. Mth., 64 Art. 6,

13 MROMORPHIC UNCTIONS 3 T 6N + 3N 2 + 2N 2 + S. k By Lemm 2. or p = 2 we hve T 6 + 2kN + 5N 2+k + S, 6 + 2kΘ, + 5δ 2+k 0, 2k + 0. This contrdicts with.6. Now the proo hs been completed. RRNCS [] R. BRÜCK, On entire unctions which shre one vlue CM with their irst derivtive, Results in Mth., , [2] W.K. HAYMAN, Meromorphic unction, Oxord, Clrendon Press, 964. [3] I. LAHIRI, Weighted shring nd uniqueness o meromorphic unctions, Ngoy Mth. J., 6 200, [4] I. LAHIRI, Uniqueness o meromorphic unction nd its derivtive, J. Inequl. Pure Appl. Mth., , Art. 20. [ONLIN [5] C.C. YAN, On deiciencies o dierentil polynomils II, Mth. Z., , [6] C.C. YAN AND H.Y. YI, Uniqueness Theory o Meromorphic unctions, Beijing/New York, Science Press/Kluwer Acdemic Publishers, [7] L.Z. YAN, Solution o dierentil eqution nd its pplictions, Kodi Mth. J., , [8] K.-W. YU, On entire nd meromorphic unctions tht shre smll unctions with their derivtives, J. Inequl. Pure Appl. Mth., , Art. 2. [ONLIN rticle.php?sid=257]. [9] Q.C. ZHAN, The uniqueness o meromorphic unctions with their derivtives, Kodi Mth. J., 2 998, J. Inequl. Pure nd Appl. Mth., 64 Art. 6,