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1 2016, 37A(2): DOI: /jcnkicama Í Æ ß È Õ Ä Ü È Ø Ó Đ * Ð Ã µ½ ¹Ï ½» ÒÄÃ µ½ Í ÞÞ Ï Å ¹Ï µ½ MR (2010) Î 30D35, 30D45 Ð ÌÎ O17452 ÑÂ A ÛÁ (2016) Ú Ö Ä D C Ã 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 Ã 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 Ã 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 Î» Î» ÁÂ 1 ßÛÇ ³ Ó ±½ chenwei198841@126com 2 µ ßÛÇ ³ Ó ±½ tianhg@xjnueducn 3 Í ³ Õ Í wjyuan1957@126com ÑÏ» Õ ß (No , No ), Í Æ ß (No S ) ßÛ ÐÅ ßÜ² (No XJGRI )

2 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 Ã 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 Ã 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 Ì¼º 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 µ Lahiri ±Ð [7] º g g n = c g, Ê E Ä F D a ( 0), b Ã À E = { z : z D, (z) + a } (z) = b ±± Ã M, É¼ 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 Ã 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,

3 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 Ã Î ² Ê G Ä F D a ( 0), b Ã 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, É¼ F, z E È (z) M, ² F ± D Î Ê H Ä F D a ( 0), b Ã 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, É¼ F, z E È (z) M, ² F ± D Î Ð«ÚÁ 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 Ã Â p, Â q p < α < q, F ± D z 0 Î ±± (1) D z n z 0 (n ), (2) ρ n 0 (n ), (3) F n, É¼ g n (ζ) = ρ α n n(z n + ρ n ζ) ±À¹ C Á «Í º Ã ¾ g(ζ), ¼ Â p, Â q, ¾Ñ 2

4 A Ù 22 [2] ÄÌ Ã m a, b, c 3 Ã 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 Ë¾ Ù 23 [11] Ä À¹ C Á Ã 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 Ã a 0 À Ä ² n ( ) m a Â Ã Ý º ÖÄ n ( ) m a ¾ Ã 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 m s s, (23) N = n 1 + n 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) Ã ¾ ÛÊ 24, Ä n ( ) m a Ã 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)

5 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 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, ØÁÊ±± 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, ÁÊØ¼ É 25 ¼º Ù 26 [12] Ä n ( 2), m ( 1) Ã ³ À c, n ( ) m c Ô Ã Ù 27 Ä a ( 0), b Ã 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 Â±± Ã Ä x 0 C 25 26, Â±± z 1, z 2 C, É¼ ((z)) n1 ((z) ) m1 = x 0

6 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 Ã À b, c, M = b c ³ F, Ã 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 +, É¼ T j (ξ) = g j (z j + ρ j ξ)ρ 1 n 1 ³ «Å Í º Ã 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 Ã 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 Ã T «Ì Ã Ø¼º T (ξ) 1 b T n (ξ) Â ±± Ã n 1 Úº T (ξ) 1 T n (ξ) ±± Ã À ÖÄ±± ξ 0, ξ 0 T (ξ) 1 T n (ξ) ¾ Ã ÇÝ¾ δ > 0, É¼ 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), É¼ Ç²¾ 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

7 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, É¼ «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 Ã À b, c, M = b c ³ F, Ã 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 +, É¼ T j (ξ) = gj(zj+ρjξ) 1 «n+1 ρj Å Í º Ã 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 Ã T 23 25, T n (ξ)t (ξ) b n+1 Â±± Ã ÖÄ ξ 0, ξ 0 T n (ξ)t (ξ) b n+1 Ã ÇÝ¾ δ > 0, É¼ D(ξ 0, δ) D(ξ 0, δ) =,

8 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), É¼ Ç²¾ 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, É¼ «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 Ã À b, c, M = b c ³ F, Ã Möbius g : g = c z c, ² g 1 = cz c

9 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 +, É¼ T j (ξ) = g j (z j + ρ j ξ)ρ m1+m 2 s+t «Å Í º Ã 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 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: [4] Singh A P, Singh A Sharing values and normality o meromorphic unctions [J] Complex Variables Theory Appl, 2004, 49(6): [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: [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: [11] Chen H H, Fang M L On the value distribution o n [J] Sci China, 1995, 38: [12] Yang C C, Hu P C On the value distribution o (k) [J] 1996, 19:

10 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 , China chenwei198841@126com 2 Corresponding author School o Mathematical Sciences, Xinjiang Normal University, Urumqi , China tianhg@xjnueducn 3 School o Mathematics and Inormation Sciences, Guangzhou University, Guangzhou , China 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

Journal of Inequalities in Pure and Applied Mathematics

Journal of Inequalities in Pure and Applied Mathematics SOME NORMALITY CRITERIA INDRAJIT LAHIRI AND SHYAMALI DEWAN Department of Mathematics University of Kalyani West Bengal 741235, India. EMail: indrajit@cal2.vsnl.net.in