Ä D C Ã F D {f n } F,

2016, 37A(2):233 242 DOI: 1016205/jcnkicama20160020 Í Æ ß È Õ Ä Ü È Ø Ó Đ * 1 2 3 Ð Ã µ½ ¹Ï ½» ÒÄà µ½ Í ÞÞ Ï Å ¹Ï µ½ MR (2010) Î 30D35, 30D45 Ð ÌÎ O17452 Ñ A ÛÁ 1000-8314(2016)02-0233-10 1 Ú Ö Ä D C à F D { n } F, ±± { nv } ± D Á «Å Í º, ² F ± D Î Ø F ± D Î F ± D Î [1] Ä, g D ¾ à a, b ÃÀ (z) = a È g(z) = b, Ó Ó (z) = a g(z) = b (z) = a g(z) = b, g(z) = b (z) = a, (z) = a g(z) = b a = b È, g IM a Ð σ(x, y) ÆË x y ¾ «Ë «¾ ØÐ [2] 2000 µ Zalcman ±Ð [3] º Ý Ê A Ä F ¾ a, b à À c À À F, ² F ± Î (z) = 0 (z) = a, (z) = c (z) = b, 2004 µ Singh Singh ±Ð [4] Ú A ¾ a, b Ì Æ ¾ º Ñ 2013 8 27 λ 2014 5 8 λ Á 1 ßÛÇ ³ Ó ±½ 830054 E-mail: chenwei198841@126com 2 µ ßÛÇ ³ Ó ±½ 830054 E-mail: tianhg@xjnueducn 3 Í ³ Õ Í 510006 E-mail: wjyuan1957@126com ÑÏ» Õ ß (No 11461070, No 11271090), Í Æ ß (No S2012010010121) ßÛ ÐÅ ßܲ (No XJGRI2013131)

234 37 A Ê B Ä F ¾ F, ±± À b, c, (i) b c (ii) min{σ(0, b ), σ(0, c ), σ(b, c )} m, ¼ m > 0, (iii) (z) = 0 (z) = 0 (z) = c (z) = b, ² F ± Î 1967 µ Hayman ±Ð [5] Ã Ë Î ²¾ Ú Ê C Ä n a 0, b à À F D ( ) F, (z) a n (z) b, ¼ n 3 (n 2), ² F ± D Î 2008 µ ³ [6] ¼¾ ÚÚ C Ì ¼º Ý Ê D Ä n ( 4) Ì Ã a ( 0), b à À Ä F D, g F, (z) a n (z) g (z) ag n (z) ± D IM b, ² F ± D Î B ¾Ú Ò Ú D ̼º Ê 11 Ä F D n ( 4) F, ±± À b, c, (i) b c (ii) min{σ(0, b ), σ(0, c ), σ(b, c )} m, ¼ m > 0, (iii), g F, 1 ² F ± D Î n = c g 1 2005 µ Lahiri ±Ð [7] º g g n = c g, Ê E Ä F D a ( 0), b à À E = { z : z D, (z) + a } (z) = b ±± à M, ɼ F, z E È (z) M, ² F ± D Î 2011 µ Ð ±Ð [8] º Ê F Ä F D n ( 2) a ( 0), b à À, g F, a n g ag n IM b, ² F ± D Î Å 11 ¾Ú Ò º Ê 12 Ä F D n ( 4) F, ±± À b, c, (i) b c (ii) min{σ(0, b ), σ(0, c ), σ(b, c )} m, ¼ m > 0,

2» À Æ Ñ Ã µ½ ¹Ï Þ Ô 235 (iii), g F, bn+1 ² n F ± D Î = c g bn+1 g g = c n g, 2009 µ Charak Rieppo ±Ð [9] Ì D, ¼º à Π² Ê G Ä F D a ( 0), b à À m 1, m 2, n 1, n 2 m 1 n 2 m 2 n 1 > 0, m 1 + m 2 1 n 1 + n 2 2 { E = z : z D, n1 ( a } ) m1 + n2 ( ) = b m2 ±± M, ɼ F, z E È (z) M, ² F ± D Î Ê H Ä F D a ( 0), b à À m 1, m 2, n 1, n 2 m 1 n 2 = m 2 n 1 > 0 E = { z : z D, (z) + a } (z) = b ±± M, ɼ F, z E È (z) M, ² F ± D ΠЫÚÁ Ã Ì ¼º Ê 13 Ä F D m 1, m 2, n 1, n 2 m 1 n 2 m 2 n 1 > 0, m 1 + m 2 1 n 1 + n 2 2 F, ±± À b, c, (i) b c (ii) min{σ(0, b ), σ(0, c ), σ(b, c )} m, ¼ m > 0, (iii), g F, n1 ( ) m1 + bs+t n 2( ) m 2 = c s gn1 (g ) m1 + bs+t g n 2( ) m 2 = c s g, ¼ s = n 1 + m 1, t = n 2 + m 2, ² F ± D Î Ê 14 Ä F D m 1, m 2, n 1, n 2 m 1 n 2 = m 2 n 1 > 0 F, ±± À b, c, (i) b c (ii) min{σ(0, b ), σ(0, c ), σ(b, c )} m, ¼ m > 0, (iii), g F, n1 ( ) m1 + s = n 1 + m 1, t = n 2 + m 2, ² F ± D Î 2 Ú bs+t n 2( ) m 2 = c s gn1 (g ) m1 + bs+t g n 2( ) m 2 = c s g, ¼ º Ð¾Ý Ò ĐÅ Ù 21 [10] Ä F ¾ à  p,  q p < α < q, F ± D z 0 Î ±± (1) D z n z 0 (n ), (2) ρ n 0 (n ), (3) F n, ɼ g n (ζ) = ρ α n n(z n + ρ n ζ) ±À¹ C Á «Í º à ¾ g(ζ), ¼  p,  q, ¾Ñ 2

236 37 A Ù 22 [2] ÄÌ Ã m a, b, c 3 à Möbius g σ(g(a), g(b)) m, σ(g(b), g(c)) m, σ(g(c), g(a)) m, ² g ÆºÖ Ù Ð σ(g(z), g(w)) k m σ(z, w), k n Ì m ˾ Ù 23 [11] Ä À¹ C Á à n b n b, ² È n À Ô Ù 24 [4] Ä m, n ³ n ( ) m º À a Ù 25 Ä n ( 2), m à a 0 À Ä ² n ( ) m a Â Ã Ý º ÖÄ n ( ) m a ¾ à 24» n ( ) m a Ã Ô I ÖÄ ÛÊ ² n ( ) m = A(z z 0 ) l + a, (21) ¼ A l l 2 Ø (21) «¾ Ì (21) «±± Ô II ÖÄ ÛÊ (z) = A(z α 1) m1 (z α 2 ) m2 (z α s ) ms (z β 1 ) n1 (z β 2 ) n2 (z β t ) nt, (22) ¼ A m i 1 (i = 1, 2,, s), n j 1 (j = 1, 2,, t) ¾Ø Ó (22)» M = m 1 + m 2 + + m s s, (23) N = n 1 + n 2 + + n t t (24) (z) = A(z α 1) m1 1 (z α 2 ) m2 1 (z α s ) ms 1 g(z) (z β 1 ) n1+1 (z β 2 ) n2+1 (z β t ) nt+1, (25) ¼ g(z) ÛÊ (22) (25) ¼ deg g (s + t 1) n ( ) m (z) = An (z α 1 ) (n+m)m1 m (z α s ) (n+m)ms m g m (z) (z β 1 ) (n+m)n1+m (z β t ) (n+m)nt+m = P(z) Q(z), (26) ¼ P(z) Q(z) à ¾ ÛÊ 24, Ä n ( ) m a à z 0, (26) ¼ n ( ) m (z) = a + B(z z 0 ) l P(z) = (z β 1 ) (n+m)n1+m (z β t )(n+m)nt+m Q(z), (27)

2» À Æ Ñ Ã µ½ ¹Ï Þ Ô 237 ¼ B Ø a 0» z 0 α i (i = 1, 2,, s) (26) ¹ ¼ ( n ( ) m (z)) = (z α 1) (n+m)m1 m 1 (z α s ) (n+m)ms m 1 g 1 (z) (z β 1 ) (n+m)n1+m+1 (z β t ) (n+m)nt+m+1, (28) ¼ g 1 (z) deg g 1 (m + 1)(s + t 1) ¾ ÛÊ (27) ¹ ¼ ( n ( (z)) m ) = (z z 0 ) (l 1) g 2 (z), (29) (z β 1 ) (n+m)n1+m+1 (z β t )(n+m)nt+m+1 ¼ g 2 (z) = b[l (n + m)n mt]z t + a t 1 z t 1 + + a 0, b, a 0, a 1,, a t 1 Ô 1 l (n + m)n + mt, (27)» deg P deg Q (26), (n + m)n + mt (n + m)m ms + deg g m (n + m)m ms + m(s + t 1) ² (n + m)(m N) m, M > N z 0 α i (i = 1, 2,, s), Ý (28) (29),» M s N t, ² (n + m)m (m + 1)s deg g 2 t (n + m)m (m + 1)s + t (m + 1)M + N < (m + 2)M, n 2, ØÁʱ± Ô 2 l = (n + m)n + mt È Ð Ô 21 M > N È Å Ð 1 ¾ ¼ Ô 21 M N È (28) (29), l 1 deg g 1 (m + 1)(s + t 1) º l = (n + m)n + mkt, ² M s, N t M N, ¼ (n + m)n = l mt (m + 1)(s + t 1) mt + 1 (n + m)n (m + 1)s + t m (m + 1)M + N 1 (m + 2)N 1 n 2, ÁÊؼ É 25 ¼º Ù 26 [12] Ä n ( 2), m ( 1) à ³ À c, n ( ) m c Ô Ã Ù 27 Ä a ( 0), b à À m 1, m 2, n 1, n 2 m 1 n 2 = m 2 n 1, n 1 2, Ó ² Φ(z) Â Ã Ý Ø À Φ(z) = ((z)) n1 ( (z)) m1 + a ((z)) n2 ( b, m2 (z)) x + a x n 2 n 1 b = 0 ±± Ã Ä x 0 C 25 26, ±± z 1, z 2 C, ɼ ((z)) n1 ((z) ) m1 = x 0

238 37 A ¼ 3 Ë É Þ Φ(z 1 ) = ((z 1 )) n1 ( a (z 1 )) m1 + ((z 1 )) n2 ( b = 0, m2 (z 1 )) Φ(z 2 ) = ((z 2 )) n1 ( a (z 2 )) m1 + ((z 2 )) n2 ( b = 0 m2 (z 2 )) Ê 11 ÇÝ ÖÄ Ä M = b c Ø º à À b, c, M = b c ³ F, à Möbius g : g = c z c, ² g 1 = cz c Ü º G = {(g 1 ) F} ± D Î º ÖÄ G ± z 0 D Î 21, ±± g j G, z j z 0 ρ j 0 +, ɼ T j (ξ) = g j (z j + ρ j ξ)ρ 1 n 1 ³ «Å Í º à T(ξ) À T (ξ) 1 T n (ξ) 0, 1 1 n 1 T n 1 bn 1 ξ + c, ¼ c n 4, Ø T à j À T (ξ) 1 T n (ξ) 0, ² T T n 1 T = 1 ϕ, ³ ϕn 2 ϕ 1 23 ¼ ϕ à T «Ì à ؼº T (ξ) 1 b T n (ξ)  ±± à n 1 Úº T (ξ) 1 T n (ξ) ±± à À Öı± ξ 0, ξ 0 T (ξ) 1 T n (ξ) ¾ à Çݾ δ > 0, ɼ ¼ º D(ξ 0, δ) = {ξ : ξ ξ 0 < δ}, D(ξ 0, δ) D(ξ 0, δ) =, D(ξ 0, δ) = {ξ : ξ ξ 0 < δ} T j(ξ) 1 T j n (ξ) ρ n n 1 j c T (ξ) 1 T n (ξ), T j (ξ 1 j) Tj n(ξ j) ρ n n 1 j c ( = ρ n n 1 j g j(z 1 ) j + ρ j ξ j ) gj n(z j + ρ j ξ j ) c Hurwitz» ±± ξ j D(ξ 0, δ) ξj D(ξ 0, δ), ɼ Dz¾ j, j(z j + ρ j ξ j ) 1 j n j (z j + ρ j ξ j ) = c j, j(z j + ρ j ξ j ) 1 j n j (z j + ρ j ξ j ) = c j

2» À Æ Ñ Ã µ½ ¹Ï Þ Ô 239 m, ¾ÖÄ, g F, 1 n = c g 1 g m (z j + ρ j ξ j ) 1 m n m (z j + ρ j ξ j ) = c m, m (z j + ρ j ξ j ) 1 m n m (z j + ρ j ξ j ) = c m Ê m, j, º z j + ρ j ξ j z 0, z j + ρ j ξ j z 0, ² m 1 m m (z 0) 1 m n m (z 0) = c m n m c m, ÌÈ ¾ z j + ρ j ξ j = z 0, z j + ρ j ξ j = z 0 ξ j = z 0 z j ρ j, ξ j = z 0 z j ρ j g n = c g,» Ø ξ j D(ξ 0, δ), ξ j D(ξ 0, δ) D(ξ 0, δ) D(ξ 0, δ) = T (ξ) 1 T n (ξ) Ã Ó ξ 0 Å Ð [6] 1 ¾º Ò ¼ ± º G = {(g 1 ) F} ± D Î ³ ( ε k m 22 ¾ ±± δ > 0, ɼ «σ(x, y) < δ, F, 22, ¼ σ((g 1 )(x), (g 1 σ((x), (y)) = σ((g g 1 = k m σ((g 1 F ± D Î 11 ¼º )(y)) < ε k m )(x), (g g 1 )(y)) )(x), (g 1 )(y)) < ε k m > 0), ¼ Ê 12 ÇÝ ÖÄ Ä M = b c Ø º à À b, c, M = b c ³ F, à Möbius g : g = c z c, ² g 1 = cz c Úº G = {(g 1 ) F} ± D Î º ÖÄ G ± z 0 D Î 21, ±± g j G, z j z 0 ρ j 0 +, ɼ T j (ξ) = gj(zj+ρjξ) 1 «n+1 ρj Å Í º à T(ξ) T n (ξ)t (ξ) b n+1 0, ² T n = (n + 1)(b n+1 ξ + c), ¼ c Ø T Ì Ã T n (ξ)t (ξ) b n+1 0, 23» T à T 23 25, T n (ξ)t (ξ) b n+1 ±± à ÖÄ ξ 0, ξ 0 T n (ξ)t (ξ) b n+1 à Çݾ δ > 0, ɼ D(ξ 0, δ) D(ξ 0, δ) =,

240 37 A ¼ ³ º D(ξ 0, δ) = {ξ : ξ ξ 0 < δ}, T j bn+1 (ξ) Tj n(ξ) ρ n j n+1 T j(ξ j ) bn+1 Tj n(ξ j) ρ n n+1 j c = ρ n j c T (ξ) bn+1 T n (ξ), n+1 D(ξ 0, δ) = {ξ : ξ ξ 0 < δ} ( g j(z b n+1 ) j + ρ j ξ j ) Tj n(z j + ρ j ξ j ) c Hurwitz ±± ξ j D(ξ 0, δ) ξj D(ξ 0, δ), ɼ Dz¾ j, j(z j + ρ j ξ j ) b n+1 j n j (z j + ρ j ξ j ) = c j, j (z j + ρ j ξj ) b n+1 j j n(z j + ρ j ξj ) = c j ÅĐ ¾ÖÄ ¾, g F, bn+1 n m (z j + ρ j ξ j ) m(z j + ρ j ξ j ) = c g bn+1 g g n b n+1 m n m (z j + ρ j ξ j ) = c m, b n+1 m n m(z j + ρ j ξ j ) = c m Ê m, j, º z j + ρ j ξ j z 0, z j + ρ j ξ j z 0, ² m bn+1 m m n c m m(z 0 ) bn+1 m m n (z 0 ) = c m z j + ρ j ξ j = z 0, z j + ρ j ξ j = z 0 ξ j = z 0 z j ρ j, ξ j = z 0 z j ρ j = c g, Ø ξ j D(ξ 0, δ), ξj D(ξ 0, δ) D(ξ 0, δ) D(ξ0, δ) = G = {(g 1 ) F} ± D Î ³ ( ε k m > 0), ¼ k m 22 ¾ ±± δ > 0, ɼ «σ(x, y) < δ, F, 22, ¼ σ((g 1 )(x), (g 1 σ((x), (y)) = σ((g g 1 = k m σ((g 1 F ± D Î 12 ¼º )(y)) < ε k m, )(x), (g g 1 )(y)) )(x), (g 1 )(y)) < ε Ê 13 ÇÝ ÖÄ Ä M = b c Ø º à À b, c, M = b c ³ F, à Möbius g : g = c z c, ² g 1 = cz c

2» À Æ Ñ Ã µ½ ¹Ï Þ Ô 241 º G = {(g 1 ) F} ± D Î º ÖÄ G ± z 0 D Î 21, ±± g j G, z j z 0 ρ j 0 +, ɼ T j (ξ) = g j (z j + ρ j ξ)ρ m1+m 2 s+t «Å Í º à T(ξ) j (T(ξ)) n1+n2 (T (ξ)) m1+m2 b s+t, ²» T T 0, T ¾Ñ 1 T(ξ) = e cξ+d, ¼ c 0, d ³ Ø (T(ξ)) m1+m2 T (ξ)) n1+n2 = c m1+m2 e c(s+t)ξ+(s+t)d, (T(ξ)) n1+n2 (T (ξ)) m1+m2 b s+t Ý 25 26, Å 12 ¾º ¼ 11 12 ¾º Ù º F ± D Î 13 ¼º 26 27, Ú º 14, ± Ã Ð Ò [1] Hayman W K Meromorphic unctions [M] Oxord: Clarendon Press, 1964 [2] Beardon A F Iteration o rational unctions [M] New York: Springer-Verlag, 1991 [3] Pang X C, Zalcman L Normality and shared values [J] Arkiv Math, 2000, 38:171 182 [4] Singh A P, Singh A Sharing values and normality o meromorphic unctions [J] Complex Variables Theory Appl, 2004, 49(6):417 425 [5] Hayman W K Research problems in unction theory [M] London: The Athlone Press o University o London, 1967 [6] Zhang Q C Normal amilies o meromorphic unctions [J] J Math Anal Appl, 2008, 338:545 551 [7] Lahiri I A simple normality criterion leading to a counterexample to the converse o the Bloch principle [J] New Zealand Journal o Mathematics, 2005, 34:61 65 [8] Yuan W J, Zhu B, Lin J M Normal criteria o unction amilies concerning shared values [J] Journal o Applied Mathematics, 2011, 2011 [9] Charak K S, Rieppo J Two normality criteria and the converse o the Bloch principle [J] Journal o Mathematical Analysis and Applications, 2009, 353:43 48 [10] Zalcman L Normal amilies: new perspectives [J] Bull Amer Math Soc, 1998, 35:215 230 [11] Chen H H, Fang M L On the value distribution o n [J] Sci China, 1995, 38:789 798 [12] Yang C C, Hu P C On the value distribution o (k) [J] 1996, 19:157 167

242 37 A Some Notes on Normal Families o Meromorphic Functions Concerning Shared Values CHEN Wei 1 TIAN Honggen 2 YUAN Wenjun 3 1 School o Mathematical Sciences, Xinjiang Normal University, Urumqi 830054, China E-mail: chenwei198841@126com 2 Corresponding author School o Mathematical Sciences, Xinjiang Normal University, Urumqi 830054, China E-mail: tianhg@xjnueducn 3 School o Mathematics and Inormation Sciences, Guangzhou University, Guangzhou 510006, China E-mail: wjyuan1957@126com Abstract This paper deals with the normal amilies o meromorphic unctions concerning shared values The authors get some theorems concerning shared values, which improve some earlier results Keywords Meromorphic unction, Normality, Shared value 2010 MR Subject Classiication 30D35, 30D45