Research Article Some Normality Criteria of Meromorphic Functions

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1 Hindwi Publishing Corportion Journl of Inequlities nd Applictions Volume 2010, Article ID , 10 pges doi: /2010/ Reserch Article Some Normlity Criteri of Meromorphic Functions Junfeng Xu nd Wensheng Co Deprtment of Mthemtics, Wuyi University, Jingmen, Gungdong , Chin Correspondence should be ddressed to Junfeng Xu, Received 2 September 2009; Revised 11 Jnury 2010; Accepted 23 Mrch 2010 Acdemic Editor: Rm N. Mohptr Copyright q 2010 J. Xu nd W. Co. This is n open ccess rticle distributed under the Cretive Commons Attribution License, which permits unrestricted use, distribution, nd reproduction in ny medium, provided the originl work is properly cited. This pper studies some normlity criteri for fmily of meromorphic functions, which improve some results of Lhiri, Lu nd Gu, s well s Chrk nd Rieppo. 1. Introduction nd Min Results Let f be nonconstnt meromorphic function in the complex plne C. We shll use the stndrd nottions in Nevnlinn s vlue distribution theory of meromorphic functions such s Tr, f, Nr, f, ndmr, f see, e.g., 1, 2. The nottion Sr, f is defined to be ny quntity stisfying Sr, f otr, f s r possibly outside set of E of finite liner mesure. Let F be fmily of meromorphic functions on domin D C. We sy tht F is norml in D if every sequence of functions f n } F contins either subsequence which converges to meromorphic function f uniformly on ech compct subset of D or subsequence which converges to uniformly on ech compct subset of D. See 1, 3. The Bloch principle 3 is the hypothesis tht fmily of nlytic meromorphic functions which hve common property P in domin D will in generl be norml fmily if P reduces n nlytic meromorphic function in the open complex plne C to constnt. Unfortuntely the Bloch principle is not universlly true. But it is lso very difficult to find some counterexmples bout the converse of the Bloch principle, nd hence it is interesting to study the problem. In 2005, Lhiri 4 proved the following criterion for the normlity, nd gve counterexmple to the converse of the Bloch principle by using the result. Theorem A. Let F be fmily of meromorphic functions in domin D, nd let / 0, b be two finite constnts. Define

2 2 Journl of Inequlities nd Applictions E f z : z D,f z } fz b. 1.1 If there exists positive number M such tht for every f F, one hs fz M whenever z E f, then F is norml. In this direction, Lhiri nd Dewn 5 s well s Xu nd Zhng 6 proved the following result. Theorem B. Let F be fmily of meromorphic functions in domin D, nd let / 0, b be two finite constnts. Suppose tht E f } z : z D,f k f n b, 1.2 where k nd n re positive integers. If for every f F i ll zeros of f hve multiplicity t lest k, ii there exists positive number M such tht for every f Fone hs fz M whenever z E f, then F is norml in D so long s An 2;orBn 1 nd k 1. Here, we lso give counterexmple to the converse of the Bloch principle by considering Theorem B, which is similr to n exmple in 7. Exmple 1.1. Let fz cot z, then f z 1 cot 2 z / 0 for ll z C. Now we cn see tht f z f 2 z 1 1 cot 2 z ) 2 cot 2 z 4 sin 2 2z / 0, 1.3 but Theorem B is true especilly when E f is n empty set for every f in the fmily. In the following, we continue to study the norml fmily when n 1ndk 2in Theorem B. Theorem 1.2. Let F be fmily of meromorphic functions in domin D, nd / 0, b be two finite constnts. Suppose tht E f } z : z D,f k f 1 b, 1.4 where k 2 is positive integer. If for every f F i ll zeros of f hve multiplicity t lest k 1, ii there exists positive number M such tht for every f F, one hs fz Mwhenever z E f,thenf is norml in D.

3 Journl of Inequlities nd Applictions 3 Corollry 1.3. Let F be fmily of meromorphic functions in domin D, ll of whose zeros hve multiplicity t lest k1, nd let / 0, b be two finite constnts. Suppose tht f k f 1 / b,where k 2 is positive integer. Then F is norml in D. Recently, Lu nd Gu 8 considered two relted norml fmilies. Theorem C. Let F be fmily of meromorphic functions in domin D; ll of whose zeros hve multiplicity t lest k 2. Suppose tht, for ech f F, ff k / for z D,thenF is norml fmily on D,where is nonzero finite complex number nd k 1 is n integer number. Theorem D. Let F be fmily of meromorphic functions in domin D; ll of whose zeros hve multiplicity t lest k 1, nd ll of whose poles re multiple. Suppose tht, for ech f F, ff k / for z D,thenF is norml fmily on D, where is nonzero finite complex number nd k 1 is n integer number. In this pper, we give simple proof nd improve the bove results. Theorem 1.4. Let F be fmily of meromorphic functions in domin D; ll of whose zeros hve multiplicity t lest k 1. Suppose tht, for ech f F, ff k / for z D,thenF is norml fmily on D,where is nonzero finite complex number nd k 1 is n integer number. In 2009, Chrk nd Rieppo 7 generlized Theorem A nd obtined two normlity criteri of Lhiri s type. Theorem E. Let F be fmily of meromorphic functions in complex domin D. Let, b Csuch tht / 0.Letm 1, m 2,, n 2 be nonnegtive integers such tht m 1 n 2 m 2 > 0, m 1 m 2 1, nd n 2 2, nd put E f z D: fz ) f z ) } m 1 ) n2 fz f z ) m 2 b. 1.5 If there exists positive constnt M such tht fz M for ll f Fwhenever z E f,thenf is norml fmily. Theorem F. Let F be fmily of meromorphic functions in complex domin D. Let, b Csuch tht / 0. Letm 1, m 2,, n 2 be nonnegtive integers such tht m 1 n 2 m 2 > 0, nd put E f z D: fz ) f z ) } m 1 ) n2 fz f z ) m 2 b. 1.6 If there exists positive constnt M such tht fz M for ll f Fwhenever z E f,thenf is norml fmily. Nturlly, we sk whether the bove results re still true when f is replced by f k in Theorems E nd F. We obtin the following results.

4 4 Journl of Inequlities nd Applictions Theorem 1.5. Let F be fmily of meromorphic functions in complex domin D; ll of whose zeros hve multiplicity t lest k. Let, b Csuch tht / 0. Letm 1, m 2,, n 2 be nonnegtive integers such tht m 1 n 2 m 2 > 0, m 1 m 2 1, nd n 2 2 if n 2 1, k 5), nd put E f z D: fz ) } f z) k m1 ) n2 fz f k z ) m 2 b. 1.7 If there exists positive constnt M such tht fz M for ll f Fwhenever z E f,thenf is norml fmily. Theorem 1.6. Let F be fmily of meromorphic functions in complex domin D; ll of whose zeros hve multiplicity t lest k. Let, b Csuch tht / 0. Letm 1 2, m 2,, n 2 be nonnegtive integers such tht m 1 n 2 m 2, nd put E f z D: fz ) } f z) k m1 ) n2 fz f k z ) m 2 b. 1.8 If there exists positive constnt M such tht fz M for ll f Fwhenever z E f,thenf is norml fmily. 2. Some Lemms Lemm 2.1 see 9. Let F be fmily of functions meromorphic on the unit disc, ll of whose zeros hve multiplicity t lest k, theniff is not norml, there exist, for ech 0 α<k, number 0 <r<1, b points z n,z n < 1, c functions f n ζ, d positive number ρ n such tht ρn α f n z n ρ n ξg n ξ gξ loclly uniformly, where g is nonconstnt meromorphic on C, ll of whose zeros hve multiplicity t lest k, such tht g # ξ g # 0. Here, s usul, g # ξ g ξ /1 gξ 2 is the sphericl derivtive. Lemm 2.2. Let f be rtionl in the complex plne nd m, n positive integers. If f hs only zero with multiplicity t lest k, thenf n f k m tkes on ech nonzero vlue C. Proof. In Lemm 6 of 7, the cse of k 1 is proved. We ust consider the cse of k 2by different wy which comes from 10. If f is polynomil, obviously the conclusion holds. If f is nonpolynomil rtionl function, then we cn set f n f k) m A z α 1 m 1 z α 2 m2 z α s m s z β1 ) n1 z β2 ) n2 z β t ) nt, 2.1

5 Journl of Inequlities nd Applictions 5 where A is nonzero constnt. Since f hs only zero with multiplicity t lest k,wefindtht m i kn i 1, 2,...,s, n k 1 n 1, 2,...,t ). 2.2 For convenience, we denote M m 1 m 2 m s kns, N n 2 n t k 1 nt. 2.3 Differentiting 2.1, weobtin [f n f k) m] z α 1 m1 1 z α 2 m 2 1 z α s m s 1 ) n1 1 ) n2 1 ) nt gz, z β1 z β2 z βt where gz is polynomil with degg s t 1. Suppose tht f n f k m hs no zero, then we cn write f n f k) m B A ) n1 ) n2 ) nt, 2.5 z β1 z β2 z β t where B is nonzero constnt. Differentiting 2.5, weobtin [f n f k) m] Bg 1 z z β1 ) n1 1 z β2 ) n2 1 z βt ) nt 1, 2.6 where g 1 z is polynomil of the form BNz t 1 B t 2 z t 2 B 0, in which B 0,..., B t 2 re constnts. Compring 2.1 nd 2.5, we cn obtin M N.From2.4 nd 2.6, we hve s m i 1 M s deg g 1 z ) t 1, i1 M s t 1 M kn N k 1 n 1 M kn M k 1 n 1 ) 1 kn 1 M 1. k 1 n 2.7 It is contrdiction with n 1ndk 2. This proves the lemm.

6 6 Journl of Inequlities nd Applictions Lemm 2.3 see 11. Let f be trnscendentl meromorphic function ll of whose zeros hve multiplicity t lest t, thenff k ssumes every finite nonzero vlue infinitely often, where t k 1 if k 4, nd t 5 if k 5. Remrk 2.4. The lemm ws first proved by Wng s t 5ifk 5ndt 6ifk 6in12. Recently, the result is improved by 11. Lemm 2.5. Let f be meromorphic function ll of whose zeros hve multiplicity with t lest k 1 in the complex plne, then ff k must hve zeros for ny constnt / 0,. Proof. If f is rtionl, then by Lemm 2.2 the conclusion holds. If f is trnscendentl, supposing tht ff k hs no zeros, then by Lemm 2.3, we cn get contrdiction. This completes the proof of the lemm. Lemm 2.6. Let f be meromorphic in the complex plne, nd let / 0 be constnt, for ny positive integer k;ifff k,thenf is constnt. Proof. If f is not constnt, nd from / 0, we know tht f / 0, then with the identity ff k, we cn get tht, if r, T r, 1 ) m r, 1 ) log 1 f f m r, f ) k o T r, f )), f 2.8 nd r / E with E being set of r vlues of finite liner mesure. It is contrdiction. Lemm 2.7 see 13. Let f be trnscendentl meromorphic function, nd let n 2, n k 1 be two integers. Then for ny nonzero vlue c, the function f n f k n k c hs infinitely mny zeros. Lemm 2.8 see 14. Let f be trnscendentl meromorphic function, nd let n 2 be n integer. Then for ny nonzero vlue c, the function ff k n c hs infinitely mny zeros. Lemm 2.9. Let F be fmily of meromorphic functions in complex domin D. Let, b Csuch tht / 0. Letm 1 2, m 2,, n 2 be nonnegtive integers such tht m 1 n 2 m 2, nd put E f z D: fz ) f z) k m1 [ ) n2 fz f k z ) m 2 ] n2 b /. 2.9 hs finite zero. Proof. The lgebric complex eqution x x n 2/ b hs lwys nonzero solution; sy x 0 C.By14, Corollry 3 or 15, Lemms 2.2, 2.7, nd 2.8, the meromorphic function f f k m 1 cnnot void it nd thus there exists z 0 Csuch tht fz 0 f k z 0 m 1 x 0.

7 Journl of Inequlities nd Applictions 7 By ssumption, we my write m 2 n 2 / m 1 nd n 2 n 2 /. Consequently Ψz 0 [ fz0 ) ) f k m1 ] z 0 [ fz0 ) f k z 0 ) m 1 ] n2 b / 2.11 nd we complete the proof of the lemm. Remrk If m 1 1, we need k 5 when 1byLemm 2.3. We cn get similr result. 3. Proof of Theorems Proof of Theorem 1.2. Let α k/2 < k. Suppose tht F is not norml t z 0 D. Then by Lemm 2.1, there exist sequence of functions f F 1, 2,..., sequence of complex numbers z z 0, nd ρ >0 0 such tht g ζ ρ α f z ρ ζ ) 3.1 converges sphericlly nd loclly uniformly to nonconstnt meromorphic function gζ in C. Alsothezerosofgz re of multiplicity t lest k 1. So g k / 0. Applying Lemm 2.5 to the function gz, we know tht gζ 0 g k ζ 0 0, g k ζ 0 gζ for some ζ 0 C. Clerly ζ 0 is neither zero nor pole of g. So in some neighborhood of ζ 0, g ζ converges uniformly to gζ. Now in some neighborhood of ζ 0 we see tht g k ζ gζ 1 is the uniform limit of g k ζ 0 g ζ 0 1 ρ α b ρk/2 f k ) ) } z ρ ζ 0 f 1 z ρ ζ 0 b. 3.3 By 3.2 nd Hurwitz s theorem, there exists sequence ζ of ζ 0 such tht for ll lrge vlues f k ) ) z ρ ζ f 1 z ρ ζ b. 3.4 Therefore for ll lrge vlues of, it follows from the given condition tht g ζ f z ρ ζ /ρ α M/ρ α. Since ζ 0 is not pole of g, there exists positive number K such tht in some neighborhood of ζ 0 we get gζ K.

8 8 Journl of Inequlities nd Applictions Since g ζ converges uniformly to gζ in some neighborhood of ζ 0,wegetforll lrge vlues of nd for ll ζ in tht neighborhood of ζ 0 g ζ gζ < Since ζ ζ, we get for ll lrge vlues of K g ζ ) g ζ ) g ζ ) g ζ ) > M ρ α 1, 3.6 which is contrdiction. This proves the theorem. Proof of Theorem 1.4. If F is not norml t z 0 D. We ssume without loss of generlity tht z 0 0, then by Lemm 2.1, forα k/2, there exist sequence of points z n z 0, sequence of positive numbers ρ n 0, nd sequence of functions f n } of F such tht g ζ ρ α f z ρ ζ ) gz 3.7 sphericlly uniformly on compct subsets of C, where gz is nonconstnt meromorphic function on C; ll of whose zeros hve multiplicity k 1tlest.By3.7, g ζg k ζ f ζf k ζ / It follows tht gζg k ζ / or gζg k ζ by Hurwitz s theorem. From Lemm 2.6, we obtin tht gg k /. ByLemm 2.5, we get contrdiction. This completes the proof of the theorem. Proof of Theorem 1.5. Suppose tht F is not norml t z 0 D. Then by Lemm 2.1, for0 α< k, there exist sequence of functions f F 1, 2,..., sequence of complex number z z 0,ndρ >0 0 such tht g ζ ρ α f z ρ ζ ) 3.9 converges sphericlly nd loclly uniformly to nonconstnt meromorphic function gζ in C. Alsothezerosofgz re of multiplicity t lest k. Sog k / 0. By Lemms 2.2, 2.3, 2.7, nd 2.8, weget gζ0 ) g k m1 ζ 0 ) gζ0 ) n 2 g k ζ 0 ) m 2 0, 3.10

9 Journl of Inequlities nd Applictions 9 for some ζ 0 C. Clerly ζ 0 is neither zero nor pole of g. So in some neighborhood of ζ 0, g ζ converges uniformly to gζ. Now in some neighborhood of ζ 0 we hve g ζ ) g k ζ) m1 ρ αk 1km 1 ρ αk 2 km 2 g ζ ) n 2 g k ζ ) ) n1 f f k m1 ρ αk 2km 2 ) m2 ρ αk 2 km 2 b ) n2 f f k ρ αk 1k 2 km 1 m 2 ) ) n1 f f k m1 ) m2 ρ αk 2 km 2 b ) n2 f f k ) m2 b, 3.11 where f z ρ ζ is replced by f nd k n m, 1, 2. Tking α m 1 m 2 k/k 1 k 2 nd using the ssumption m 1 n 2 m 2 > 0, we see tht g g k) m 1 ) g k m g n 2 is the uniform limit of ρ m 1n 2 m 2 /k 1 k 2 k f f k ) m1 f n 2 f k ) m2 b 3.13 in some neighborhood of ζ 0.By3.10 nd Hurwitz s theorem, there exists sequence ζ ζ 0 such tht for ll lrge vlues of )) n1 f z ρ ζ f k ) ) m 1 z ρ ζ f z ρ ζ )) n2 f k z ρ ζ ) ) m 2 b Hence, for ll lrge, it follows from the given condition tht ) ) f g ζ z ρ ζ ρ α M ρ α In the following, we cn get contrdiction in similr wy with the proof of the lst prt of Theorem 1.2. This completes the proof of the theorem. Proof of Theorem 1.6. Suppose tht F is not norml t z 0 D. Then by Lemm 2.1, for0 α< k, there exist sequence of functions f F 1, 2,..., sequence of complex numbers z z 0,ndρ >0 0 such tht g ζ ρ α f z ρ ζ ) 3.16

10 10 Journl of Inequlities nd Applictions converges sphericlly nd loclly uniformly to nonconstnt meromorphic function gζ in C. Alsothezerosofgz re of multiplicity t lest k. Sog k / 0. By Lemm 2.9,weget gζ0 ) g k m1 ζ 0 ) gζ0 ) n 2 g k ζ 0 ) m 2 b 0, 3.17 for some ζ 0 C. In the following, we cn get contrdiction in similr wy with the proof of the lst prt of Theorem 1.5. This completes the proof of the theorem. Acknowledgments The uthors would like to thnk Professor Lhiri for supplying the electronic file of the pper 4. The uthors were supported by NSF of Chin No , No , NSFof Gungdong Province No , No , nd Deprtment of Eduction of Gungdong No. LYM References 1 L. Yng, Vlue Distribution Theory, Springer, Berlin, Germny, C.-C. Yng nd H.-X. Yi, Uniqueness Theory of Meromorphic Functions, vol. 557 of Mthemtics nd Its Applictions, Kluwer Acdemic Publishers, Dordrecht, The Netherlnds, J. F. Schiff, Norml Fmilies, Universitext, Springer, New York, NY, USA, I. Lhiri, A simple normlity criterion leding to counterexmple to the converse of the Bloch principle, New Zelnd Journl of Mthemtics, vol. 34, no. 1, pp , I. Lhiri nd S. Dewn, Some normlity criteri, Journl of Inequlities in Pure nd Applied Mthemtics, vol. 5, no. 2, rticle 35, J. F. Xu nd Z. L. Zhng, Note on the norml fmily, Journl of Inequlities in Pure nd Applied Mthemtics, vol. 7, no. 4, rticle 133, K. S. Chrk nd J. Rieppo, Two normlity criteri nd the converse of the Bloch principle, Journl of Mthemticl Anlysis nd Applictions, vol. 353, no. 1, pp , Q. Lu nd Y. Gu, Zeros of differentil polynomil fzf k z nd its normlity, Chinese Qurterly Journl of Mthemtics, vol. 24, no. 1, pp , X. C. Png nd L. Zlcmn, Norml fmilies nd shred vlues, The Bulletin of the London Mthemticl Society, vol. 32, no. 3, pp , P.-C. Hu nd D.-W. Meng, Normlity criteri of meromorphic functions with multiple zeros, Journl of Mthemticl Anlysis nd Applictions, vol. 357, no. 2, pp , W. L. Zou nd Q. D. Zhng, On the zero of ϕff k, Journl of Sichun Norml University, vol. 31, no. 6, pp , J.-P. Wng, On the vlue distribution of ff k, Kyungpook Mthemticl Journl, vol. 46, no. 2, pp , C. C. Yng nd P. C. Hu, On the vlue distribution of ff k, Kodi Mthemticl Journl, vol. 19, no. 2, pp , A. Alotibi, On the zeros of ff k n 1forn 2, Computtionl Methods nd Function Theory, vol. 4, no. 1, pp , Z. F. Zhng nd G. D. Song, On the zeros of ff k z, Chinese Annls of Mthemtics, Series A, vol. 19, no. 2, pp , 1998.