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On the normality criteria of Montel and Bergweiler–Langley
Accepted Manuscript On the normality criteria of Montel and Bergweiler-Langley
Tran Van Tan, Nguyen Van Thin, Vu Van Truong
PII: DOI: Reference:
S0022-247X(16)30691-6 http://dx.doi.org/10.1016/j.jmaa.2016.11.008 YJMAA 20861
To appear in:
Journal of Mathematical Analysis and Applications
Received date:
22 September 2016
Please cite this article in press as: T.V. Tan et al., On the normality criteria of Montel and Bergweiler-Langley, J. Math. Anal. Appl. (2017), http://dx.doi.org/10.1016/j.jmaa.2016.11.008
This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
ON THE NORMALITY CRITERIA OF MONTEL AND BERGWEILER-LANGLEY TRAN VAN TAN, NGUYEN VAN THIN, AND VU VAN TRUONG Abstract. A well-known result of Montel states that for a family F of meromorphic functions in a domain D ⊂ C, if there exist three distinct points and positive integers 1 , 2 , 3 such that 1 + 1 + 1 < 1 and a1 , a2 , a3 in C 1 2 3 all zeros of f − ai have multiplicity at least i for all f ∈ F and i ∈ {1, 2, 3}, then F is normal in D. Inspired by this classical result, during the past 100 years, a large number of normality criteria have been established for the case where meromorphic functions (or differential polynomials generated by the members of the family) meet some distinct points with sufficiently large multiplicities. This means that these criteria strictly apply only to the case in which derivatives of functions (differential polynomials, respectively) vanish on respective zero sets. In this paper, we generalize some normality criteria of Montel, Grahl-Nevo, Gu, and Bergweiler-Langley to the case where derivatives are bounded from above on zero sets.
1. Introduction A family F of meromorphic functions in a domain D ⊂ C is said to be normal, in the sense of Montel, if for any sequence {fv } ⊂ F, there exists a subsequence {fvi } such that {fvi } converges spherically locally uniformly in D, to a meromorphic function or ∞. Perhaps the most celebrated theorem in the theory of normal families is the following criterion of Montel [12], which can be viewed as the local counterpart of Picard’s theorem and plays an important role in complex dynamics. Theorem A. Let F be a family of meromorphic functions defined in a domain Assume that all functions D ⊂ C, and let a1 , a2 , a3 be three distinct points in C. in F omit three values a1 , a2 , a3 in D. Then F is normal in D. In 1916, Montel [13] generalized his above-mentioned result as follows:
2010 Mathematics Subject Classification. Primary 30D45, 30D35. Key words: Normal family, Meromorphic function, Nevanlinna theory. 1
2
Theorem B. Let F be a family of meromorphic functions defined in a domain and 1 , 2 , 3 be three positive D ⊂ C, and let a1 , a2 , a3 be three distinct points in C integer numbers (or possibly +∞) such that 1 1 1 + + < 1. 1 2 3 Assume that all zeros of f − ai have multiplicity at least i , for all f ∈ F, and i ∈ {1, 2, 3}. Then F is normal in D. Inspired by this classical result, during the past 100 years, a large number of normality criteria have been established. In the view of Nevanlinna theory, the above-mentioned result was extended for the case of more than three points (see Valiron [17] or Zalcman [16]) and the case of high dimension (see Fujimoto [3]). Carath´eodory [2] extended Theorem A to the case where the points a1 , a2 , a3 depend on the respective function in the family and satisfy a condition on the spherical distance. In 2014, Grahl-Nevo [4] generalized the result of Carath´eodory to the case where a1 , a2 , a3 are functions. Denote by σ the spherical metric on C, Grahl-Nevo obtained the following result. Theorem C. Let F be a family of meromorphic functions in a domain D and > 0. Assume that for each f ∈ F there exist meromorphic functions (or ∞) af , bf , cf such that f omits af , bf , cf in D and min{σ(af (z), bf (z)), σ(bf (z), cf (z)), σ(cf (z), af (z))} ≥ , for all z ∈ D. Then F is a normal family. Denote by f (k) the k th derivative of f and set f (0) := f. Note that in Theorem B, # |f (k+1) | (k) vanishes on the set i ≥ 2 and the spherical derivative f (k) := 1+|f (k) |2 of f
f −1 (ai ) for all f ∈ F and k = 0, 1, . . . , i −2. Of course, this holds also for the case of omitting values, f −1 (ai ) = ∅. On the other hand, normality criteria under a boundedness condition of the spherical derivative have been established by Marty and developed by many authors such as Lehto and Virtanen [10], Lappan [8, 9], Grahl-Nevo [5], Tan-Thin [14]. The first purpose of this paper is to generalize the above-mentioned results of Montel and Grahl-Nevo to the case where the spherical derivatives are bounded. We shall prove the following theorem. Theorem 1. Let F be a family of meromorphic functions in a domain D ⊂ C. Assume that for each compact subset K ⊂ D, there are
3
q
1 < q − 2, j ii) meromorphic functions a1f , . . . , aqf (f ∈ F) in D, positive constants and M such that σ(aif (z), ajf (z)) ≥ for all z ∈ D, 1 ≤ i, j ≤ q, i = j and # 1 (k) (k) # sup (f ) (z) ≤ M, sup (z) ≤ M, f z∈K:f (z)=ajf (z)=∞ z∈K:f (z)=ajf (z)=∞ i) positive integers (or +∞) 1 , . . . , q satisfying
j=1
for all f ∈ F, j ∈ {1, . . . , q}, and k = 0, . . . , j − 2. Then F is a normal family. It is quite easy to give examples which satisfy the assumption of Theorem 1, but not the condition in Montel’s criterion.
1 2 }n∈N in the Example 2. We consider the family F = {fn : fn (z) = z + n unit disc D = {z : |z| < 1}. We have fn# (z)
2 (z + n1 ) = 4 < 4, 1 + (z + 1 ) n
(fn )# (z) =
2 2 < 2, 1 + 4 (z + n1 )
#
f (k) (z) = 0,
for all z ∈ D, k ≥ 2. Hence, it is clear that the family F satisfies the assumption of Theorem 1. On the other hand, fn omits the value ∞; all zeros of fn (z) = 0 have multiplicity 2; all zeros of fn (z) = a (for any a ∈ C \ {0}) are simple. Then, F does not satisfy the assumption of Montel’s criterion. According to Bloch’s principle, to every “Picard-type” theorem, there should belong to a corresponding normality criterion. Montel’s normality criterion is corresponding to the Little Picard Theorem. In 1959, Hayman [7] proved a “Second Main Theorem-type” which it follows a “Picard-type” theorem that if f is a transcendent meromorphic function in the complex plane then either f takes every finite complex value infinitely often or each derivatie f (k) , k ≥ 1, takes every finite non-zero value infinitely often. A corresponding normality criterion to Hayman’s theorem was given by Gu [6] in 1979. Theorem D. Let k be a positive integer and let F be the family of meromorphic functions f on a domain D ⊂ C such that f and f (k) − 1 have no zeros in D. Then F is normal. In 2005, Bergweiler and Langley [1] established a generalization of Hayman’s theorem in which multiplicites are not taken into account, and thanks to this
4
improvement, they generalized the normality criterion of Gu to the case where all zeros of f and f (k) − 1 have sufficiently large multiplicity. Definition 3. Let k be a positive integer and let 0 < M ≤ ∞, 0 < N ≤ ∞. Let D be a plane domain. 1. Denote by F(k, M, N, D) the set of all meromorphic functions f on D such that all zeros of f have multiplicity at least M, while all zeros of f (k) − 1 have multiplicity at least N (Bergweiler-Langley [1], Definition 1.1).
M, N, c, D) the set of all 2. Let c be a positive constant. Denote by F(k, meromorphic functions f on D such that
# (i) −1 (z) : z ∈ f (0) ≤ c sup f for all i = 0, . . . , M − 2, and
# −1
(k+) (k) sup f (z) : z ∈ f (1) ≤ c for all = 0, . . . , N − 2. Theorem E (Bergweiler-Langley [1], Theorem 1.4). F(k, M, N, D) is normal if and only if F(k, M, N, C) consists of constants only. We shall prove the following generalization of Theorem E.
M, N, c, D) Theorem 4. If F(k, M, N, C) consists of constants only, then F(k, is normal.
M, N, c, D) and BergweilerWe finally remark that F(k, M, N, D) ⊂ F(k, Langley ([1], Theorem 1.5) proved that if 2k + 3 + 2/k (2 + 2/k)(k + 1) + < 1, M N then F(k, M, N, C) consists of constants only. Acknowledgements: This research was supported by Vietnam National Foundation for Science and Technology Development (NAFOSTED) under grant number 101.02-2016.17. The first named author is partially supported by the Vietnam Institute for Advanced Studies in Mathematics, and the Abdus Salam International Centre for Theoretical Physics ICTP, Trieste, Italy. The authors would like to thank the referee for a very careful reading of the manuscript, and for pointing out misprints.
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2. Proof of our theorems To prove our results, we need the following lemmas. Lemma 1 (Zalcman’s Lemma, see [15]). Let F be a family of meromorphic functions defined in the unit disc D. Then if F is not normal at a point z0 ∈ D, there exist, for each real number α satisfying −1 < α < 1, 1) a real number r, 0 < r < 1, 2) points zn , |zn | < r, zn → z0 , 3) positive numbers ρn → 0+ , 4) functions fn ∈ F such that fn (zn + ρn ξ) → g(ξ) gn (ξ) = ραn spherically locally uniformly on C, where g(ξ) is a non-constant meromorphic function and g # (ξ) g # (0) = 1. Moreover, the order of g is not greater than 2. The following result is due to Grahl-Nevo ([4], Theorem 2). Lemma 2. Let {aα , bα }α∈I be a family of pairs of meromorphic functions in a domain D ⊂ C. Assume that there is a positive constant such that σ (aα (z), bα (z)) ≥ for all α ∈ I and all z ∈ D. Then, the families {aα }α∈I and {bα }α∈I are normal in D. Proof of Theorem 1. Without loss of generality, we may assume that D is the unit disc D. Suppose that F is not normal at z0 ∈ D. By Lemma 1, for α = 0 there exist 1) a real number r, 0 < r < 1, 2) points zv , |zv | < r, zv → z0 , 3) positive numbers ρv → 0+ , 4) functions fv ∈ F such that (3.1)
gv (ξ) = fv (zv + ρv ξ) → g(ξ)
spherically locally uniformly on C, where g is a non-constant meromorphic function. Therefore, for all j ∈ N, we have (j) (j) 1 1 gv(j) → g (j) on C \ P, and (3.2) → on C \ Z gv g
6
locally uniformly with respect to the Euclidean metric, where P, Z are the pole, zero sets of g, respectively. 0| Take the compact subset K := {z : |z| ≤ 1+|z 2 } ⊂ D. Then, by the assumption, there are 1 < q − 2, i) positive integers (or +∞), 1 , . . . , q satisfying qj=1 j ii) meromorphic functions a1fv , . . . , aqfv (f ∈ F) in D, positive constants and M such that σ(aifv (z), ajfv (z)) ≥ for all z ∈ D, 1 ≤ i, j ≤ q, i = j, and # 1 (k) (k) # (fv ) (z) ≤ M, sup (z) ≤ M, sup fv z∈K:fv (z)=ajfv (z)=∞ z∈K:fv (z)=ajfv (z)=∞ for all v ≥ 1, j ∈ {1, . . . , q}, and k = 0, . . . , j − 2. By ignoring any j = 1, without loss of generality, we may assume that q ≥ 3 and j ≥ 2 for all j = 1, . . . , q. By Lemma 2, we may assume that {ajfv }v≥1 converges spherically locally uniformly on C to a meromorphic function aj (or ∞) for all j = 1, . . . , q. Then Ajv (ξ) := ajfv (zv + ρv ξ) converges spherically locally uniformly on C to the constant aj (z0 ). By the assumption on the spherical metric, a1 (z0 ), . . . , aq (z0 ) are distinct. We now prove the following claim. Claim 1: For any j ∈ {1, . . . , q}, if aj (z0 ) = ∞ then all zeros of g − aj (z0 ) have multiplicity at least j . We fix an index j. For any zero ξ0 of g(ξ) − aj (z0 ), since aj (z0 ) = ∞, we have that g is holomorphic at ξ0 . By Hurwtiz’s theorem, there are ξv → ξ0 (for all v sufficiently large) such that Ajv (ξv ) = ∞ and fv (zv + ρv ξv ) − ajfv (zv + ρξv ) = gv (ξv ) − Ajv (ξv ) = 0. ◦
Note that z0 ∈K , then zv + ρv ξv ∈ K for all v sufficiently large. Since ajfv (zv + ρξv ) → aj (z0 ) = ∞, we may assume that (3.3)
|fv (zv + ρv ξv )| = |ajfv (zv + ρξv )| ≤ 1 + |aj (z0 )|.
We have (k+1)
(3.4)
|fv 1+
(zv + ρv ξv )| (k) |fv (zv + ρv ξv )|2
≤ M,
for all k = 0, . . . , j − 2 and for all v sufficiently large. Set M1 := M · (1 + (1 + |aj (z0 )|)2 ), and Mn+1 := M · (1 + Mn2 ), for all positive integer n.
7
Using the induction method we prove the following inequality. (k) (3.5) fv (zv + ρv ξv ) ≤ Mk , for all k = 1, . . . j − 1. Indeed, for k = 1, by (3.4) we have |fv (zv + ρv ξv )| ≤ M. 1 + |fv (zv + ρv ξv )|2 Combining with (3.3), we have
|fv (zv + ρv ξv )| ≤ M · 1 + |fv (zv + ρv ξv )|2 ≤ M · 1 + (1 + |aj (z0 )|)2 = M1 . We get (3.5) for the case where k = 1. Assume that (3.5) holds for some k (k ≤ j − 2). Then, by (3.4) and by the induction hypothesis, we have
|fv(k+1) (zv + ρv ξv )| ≤ M · 1 + |fv(k) (zv + ρv ξv )|2 ≤ M · 1 + Mk2 = Mk+1 . Hence, by induction, we get (3.5). By (3.5) we have (k)
|gv (ξv )| (k−1)
1 + |gv
(ξv )|2
= ρkv · ≤
ρkv
·
(k)
|fv (zv + ρv ξv )| 2(k−1)
|fv
|fv(k) (zv
+ ρv ξv )|
≤ ρkv · Mk ,
(3.6) (k−1)
Therefore, since gv
(k−1)
1 + ρv
(zv + ρv ξv )|2
for all k = 1, . . . j − 1.
(ξv ) → g (k−1) (ξ0 ) = ∞, we have
0 ≤ |g (k) (ξ0 )| = lim |gv(k) (ξv )| ≤ lim ρkv · Mk · (1 + |gv(k−1) (ξv )|2 ) = 0. v→∞
g (k) (ξ
v→∞
Then, 0 ) = 0 for all k = 1, . . . , j − 1. Hence, the zero ξ0 of g − aj (z0 ) has multiplicity at least j . This completes the proof of Claim 1. If there exists j ∈ {1, . . . , q}, such that aj (z0 ) = ∞, then A1v (ξ) := ajf (z1v +ρv ξ) v uniformly locally converges to 0 on D \ {z : aj (z) = 0} with respect to the Euclidean metric. Therefore, by (3.2) and by an argument similar to the proof of Claim 1, we get that all zeros of g1 have multiplicity at least j . Hence, similarly to Claim 1, we have the following claim. Claim 2. If there exists j ∈ {1, . . . , q}, such that aj (z0 ) = ∞, then all poles of g have multiplicity at least j . From Claims 1 and 2 and by the First and Second Main Theorems in Nevan linna theory (for the non-constant function g and distinct points aj (z0 )’s in C),
8
we have (q − 2)T (r, g) ≤ ≤ ≤
q j=1 q j=1 q j=1
N (r,
1 ) + o(T (r, g)) g − aj (z0 )
1 1 N (r, ) + o(T (r, g)) j g − aj (z0 ) 1 T (r, g) + o(T (r, g)). j
This is impossible by the fact that q 1 < q − 2. j j=1
Thus, F is a normal family. We have completed the proof of Theorem 1.
Proof of Theorem 4. We can obtain Theorem 4 by an argument similar to the proof of Theorem 1 as follows: We may assume that D is the unit disc D. Suppose that F(k, M, N, C) consists
M, N, c, D) is not normal at z0 ∈ D. By Lemma 1, for constants only, while F(k, α = 0 there exist 1) a real number r, 0 < r < 1, 2) points zv , |zv | < r, zv → z0 , 3) positive numbers ρv → 0+ ,
M, N, c, D) 4) functions fv ∈ F(k, such that gv (ξ) = fv (zv + ρv ξ) → g(ξ) spherically locally uniformly on C, where g is a non-constant meromorphic function. Similarly to Claim 1 we also get that all zeros of g have multiplicity at least M and all zeros of g (k) − 1 have multiplicity at least N. Then, g ∈ F(k, M, N, C); this is impossible. References [1] W. Bergweiler and J. K. Langley, Multiplicities in Hayman’s Alternative, J. Australian Math. Soc. 78 (2005), 37-57 ´odory, Theory of functions of complex variable, Vol. II, Chelsea Publ. Co. New [2] C. Carathe York, 1960. [3] H. Fujimoto, On families of meromorphic maps into the complex projective space, Nagoya Math. J., 54 (1974), 21- 51.
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[4] J. Grahl and S. Nevo, Eceptional functions wandering on the sphere and normal families, Israel J. Math. 202 (2014), 21-34. [5] J. Grahl and S. Nevo, An extension of one direction in Marty’s normality criterion, Monatsh. Math. 174 (2014), 205-217. [6] Y. X. Gu, A criterion for normality of meromorphic functions (Chinese), Sci. Sinica Special Issue on Math. 1 (1979), 267-274. [7] W. K. Hayman, Picard values of meromorphic functions and their deivatives, Ann. of Math. 70 (1959), 9-42. [8] P. Lappan, A criterion for a meromorphic function to be normal, Comment. Math. Helv. 49 (1974), 492-495. [9] P. Lappan, A uniform approach to normal families, Rev. Roumaine Math. Pures Appl, 39 (1994), 691-702. [10] O. Lehto and K. L. Virtanen, Boundary behaviour and normal meromorphic functions, Acta Math. 97 (1957), 47-65. [11] X. C. Pang and L. Zalcman, On theorems of Hayman and Clunie, New Zealand J. Math. 28 (1999), 71-75. [12] P. Montel, Sur les familles de fonctions analytiques qui admettent des valeurs exception´ nelles dans un domaine, Annales scientifiques de l’Ecole Normale Sup´erieure, 29 (1912), 487-535. [13] P. Montel, Sur les familles normales de fonctions analytiques, Ann. Ecole Norm. Sup, 33 (1916), 223-302. [14] T. V. Tan and N. V. Thin, On Lappan’s five point theorem, To appear in Comput. Methods. Funct. Theory., DOI: 10.1007/s40315-016-0168-9. [15] L. Zalcman, Normal families: new perspective, Bull. Amer. Mat. Soc. 35 (1998), 215-230. [16] L. Zalcman, Variations on Montel’s theorem, Bulletin de la Soci´et´e des Sciences et des Lettres de Lod, 1 (2009) Vol. LIX, 25-36. [17] G. Valiron, Families normales et quasi-normales de fonctions meromorphes, M´em. Sci. Math., Fasc. 38 (1929). Department of Mathematics, Hanoi National University of Education, 136 Xuan Thuy Street, Cau Giay, Hanoi, Viet Nam. E-mail address: [email protected] Department of Mathematics, Thai Nguyen University of Education, Luong Ngoc Quyen Street, Thai Nguyen city, Viet Nam. E-mail address: [email protected] Department of Mathematics, Hoa Lu University, Ninh Nhat, Ninh Binh City, Viet Nam E-mail address: [email protected]
Tran Van Tan, Nguyen Van Thin, Vu Van Truong
PII: DOI: Reference:
S0022-247X(16)30691-6 http://dx.doi.org/10.1016/j.jmaa.2016.11.008 YJMAA 20861
To appear in:
Journal of Mathematical Analysis and Applications
Received date:
22 September 2016
Please cite this article in press as: T.V. Tan et al., On the normality criteria of Montel and Bergweiler-Langley, J. Math. Anal. Appl. (2017), http://dx.doi.org/10.1016/j.jmaa.2016.11.008
This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
ON THE NORMALITY CRITERIA OF MONTEL AND BERGWEILER-LANGLEY TRAN VAN TAN, NGUYEN VAN THIN, AND VU VAN TRUONG Abstract. A well-known result of Montel states that for a family F of meromorphic functions in a domain D ⊂ C, if there exist three distinct points and positive integers 1 , 2 , 3 such that 1 + 1 + 1 < 1 and a1 , a2 , a3 in C 1 2 3 all zeros of f − ai have multiplicity at least i for all f ∈ F and i ∈ {1, 2, 3}, then F is normal in D. Inspired by this classical result, during the past 100 years, a large number of normality criteria have been established for the case where meromorphic functions (or differential polynomials generated by the members of the family) meet some distinct points with sufficiently large multiplicities. This means that these criteria strictly apply only to the case in which derivatives of functions (differential polynomials, respectively) vanish on respective zero sets. In this paper, we generalize some normality criteria of Montel, Grahl-Nevo, Gu, and Bergweiler-Langley to the case where derivatives are bounded from above on zero sets.
1. Introduction A family F of meromorphic functions in a domain D ⊂ C is said to be normal, in the sense of Montel, if for any sequence {fv } ⊂ F, there exists a subsequence {fvi } such that {fvi } converges spherically locally uniformly in D, to a meromorphic function or ∞. Perhaps the most celebrated theorem in the theory of normal families is the following criterion of Montel [12], which can be viewed as the local counterpart of Picard’s theorem and plays an important role in complex dynamics. Theorem A. Let F be a family of meromorphic functions defined in a domain Assume that all functions D ⊂ C, and let a1 , a2 , a3 be three distinct points in C. in F omit three values a1 , a2 , a3 in D. Then F is normal in D. In 1916, Montel [13] generalized his above-mentioned result as follows:
2010 Mathematics Subject Classification. Primary 30D45, 30D35. Key words: Normal family, Meromorphic function, Nevanlinna theory. 1
2
Theorem B. Let F be a family of meromorphic functions defined in a domain and 1 , 2 , 3 be three positive D ⊂ C, and let a1 , a2 , a3 be three distinct points in C integer numbers (or possibly +∞) such that 1 1 1 + + < 1. 1 2 3 Assume that all zeros of f − ai have multiplicity at least i , for all f ∈ F, and i ∈ {1, 2, 3}. Then F is normal in D. Inspired by this classical result, during the past 100 years, a large number of normality criteria have been established. In the view of Nevanlinna theory, the above-mentioned result was extended for the case of more than three points (see Valiron [17] or Zalcman [16]) and the case of high dimension (see Fujimoto [3]). Carath´eodory [2] extended Theorem A to the case where the points a1 , a2 , a3 depend on the respective function in the family and satisfy a condition on the spherical distance. In 2014, Grahl-Nevo [4] generalized the result of Carath´eodory to the case where a1 , a2 , a3 are functions. Denote by σ the spherical metric on C, Grahl-Nevo obtained the following result. Theorem C. Let F be a family of meromorphic functions in a domain D and > 0. Assume that for each f ∈ F there exist meromorphic functions (or ∞) af , bf , cf such that f omits af , bf , cf in D and min{σ(af (z), bf (z)), σ(bf (z), cf (z)), σ(cf (z), af (z))} ≥ , for all z ∈ D. Then F is a normal family. Denote by f (k) the k th derivative of f and set f (0) := f. Note that in Theorem B, # |f (k+1) | (k) vanishes on the set i ≥ 2 and the spherical derivative f (k) := 1+|f (k) |2 of f
f −1 (ai ) for all f ∈ F and k = 0, 1, . . . , i −2. Of course, this holds also for the case of omitting values, f −1 (ai ) = ∅. On the other hand, normality criteria under a boundedness condition of the spherical derivative have been established by Marty and developed by many authors such as Lehto and Virtanen [10], Lappan [8, 9], Grahl-Nevo [5], Tan-Thin [14]. The first purpose of this paper is to generalize the above-mentioned results of Montel and Grahl-Nevo to the case where the spherical derivatives are bounded. We shall prove the following theorem. Theorem 1. Let F be a family of meromorphic functions in a domain D ⊂ C. Assume that for each compact subset K ⊂ D, there are
3
q
1 < q − 2, j ii) meromorphic functions a1f , . . . , aqf (f ∈ F) in D, positive constants and M such that σ(aif (z), ajf (z)) ≥ for all z ∈ D, 1 ≤ i, j ≤ q, i = j and # 1 (k) (k) # sup (f ) (z) ≤ M, sup (z) ≤ M, f z∈K:f (z)=ajf (z)=∞ z∈K:f (z)=ajf (z)=∞ i) positive integers (or +∞) 1 , . . . , q satisfying
j=1
for all f ∈ F, j ∈ {1, . . . , q}, and k = 0, . . . , j − 2. Then F is a normal family. It is quite easy to give examples which satisfy the assumption of Theorem 1, but not the condition in Montel’s criterion.
1 2 }n∈N in the Example 2. We consider the family F = {fn : fn (z) = z + n unit disc D = {z : |z| < 1}. We have fn# (z)
2 (z + n1 ) = 4 < 4, 1 + (z + 1 ) n
(fn )# (z) =
2 2 < 2, 1 + 4 (z + n1 )
#
f (k) (z) = 0,
for all z ∈ D, k ≥ 2. Hence, it is clear that the family F satisfies the assumption of Theorem 1. On the other hand, fn omits the value ∞; all zeros of fn (z) = 0 have multiplicity 2; all zeros of fn (z) = a (for any a ∈ C \ {0}) are simple. Then, F does not satisfy the assumption of Montel’s criterion. According to Bloch’s principle, to every “Picard-type” theorem, there should belong to a corresponding normality criterion. Montel’s normality criterion is corresponding to the Little Picard Theorem. In 1959, Hayman [7] proved a “Second Main Theorem-type” which it follows a “Picard-type” theorem that if f is a transcendent meromorphic function in the complex plane then either f takes every finite complex value infinitely often or each derivatie f (k) , k ≥ 1, takes every finite non-zero value infinitely often. A corresponding normality criterion to Hayman’s theorem was given by Gu [6] in 1979. Theorem D. Let k be a positive integer and let F be the family of meromorphic functions f on a domain D ⊂ C such that f and f (k) − 1 have no zeros in D. Then F is normal. In 2005, Bergweiler and Langley [1] established a generalization of Hayman’s theorem in which multiplicites are not taken into account, and thanks to this
4
improvement, they generalized the normality criterion of Gu to the case where all zeros of f and f (k) − 1 have sufficiently large multiplicity. Definition 3. Let k be a positive integer and let 0 < M ≤ ∞, 0 < N ≤ ∞. Let D be a plane domain. 1. Denote by F(k, M, N, D) the set of all meromorphic functions f on D such that all zeros of f have multiplicity at least M, while all zeros of f (k) − 1 have multiplicity at least N (Bergweiler-Langley [1], Definition 1.1).
M, N, c, D) the set of all 2. Let c be a positive constant. Denote by F(k, meromorphic functions f on D such that
# (i) −1 (z) : z ∈ f (0) ≤ c sup f for all i = 0, . . . , M − 2, and
# −1
(k+) (k) sup f (z) : z ∈ f (1) ≤ c for all = 0, . . . , N − 2. Theorem E (Bergweiler-Langley [1], Theorem 1.4). F(k, M, N, D) is normal if and only if F(k, M, N, C) consists of constants only. We shall prove the following generalization of Theorem E.
M, N, c, D) Theorem 4. If F(k, M, N, C) consists of constants only, then F(k, is normal.
M, N, c, D) and BergweilerWe finally remark that F(k, M, N, D) ⊂ F(k, Langley ([1], Theorem 1.5) proved that if 2k + 3 + 2/k (2 + 2/k)(k + 1) + < 1, M N then F(k, M, N, C) consists of constants only. Acknowledgements: This research was supported by Vietnam National Foundation for Science and Technology Development (NAFOSTED) under grant number 101.02-2016.17. The first named author is partially supported by the Vietnam Institute for Advanced Studies in Mathematics, and the Abdus Salam International Centre for Theoretical Physics ICTP, Trieste, Italy. The authors would like to thank the referee for a very careful reading of the manuscript, and for pointing out misprints.
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2. Proof of our theorems To prove our results, we need the following lemmas. Lemma 1 (Zalcman’s Lemma, see [15]). Let F be a family of meromorphic functions defined in the unit disc D. Then if F is not normal at a point z0 ∈ D, there exist, for each real number α satisfying −1 < α < 1, 1) a real number r, 0 < r < 1, 2) points zn , |zn | < r, zn → z0 , 3) positive numbers ρn → 0+ , 4) functions fn ∈ F such that fn (zn + ρn ξ) → g(ξ) gn (ξ) = ραn spherically locally uniformly on C, where g(ξ) is a non-constant meromorphic function and g # (ξ) g # (0) = 1. Moreover, the order of g is not greater than 2. The following result is due to Grahl-Nevo ([4], Theorem 2). Lemma 2. Let {aα , bα }α∈I be a family of pairs of meromorphic functions in a domain D ⊂ C. Assume that there is a positive constant such that σ (aα (z), bα (z)) ≥ for all α ∈ I and all z ∈ D. Then, the families {aα }α∈I and {bα }α∈I are normal in D. Proof of Theorem 1. Without loss of generality, we may assume that D is the unit disc D. Suppose that F is not normal at z0 ∈ D. By Lemma 1, for α = 0 there exist 1) a real number r, 0 < r < 1, 2) points zv , |zv | < r, zv → z0 , 3) positive numbers ρv → 0+ , 4) functions fv ∈ F such that (3.1)
gv (ξ) = fv (zv + ρv ξ) → g(ξ)
spherically locally uniformly on C, where g is a non-constant meromorphic function. Therefore, for all j ∈ N, we have (j) (j) 1 1 gv(j) → g (j) on C \ P, and (3.2) → on C \ Z gv g
6
locally uniformly with respect to the Euclidean metric, where P, Z are the pole, zero sets of g, respectively. 0| Take the compact subset K := {z : |z| ≤ 1+|z 2 } ⊂ D. Then, by the assumption, there are 1 < q − 2, i) positive integers (or +∞), 1 , . . . , q satisfying qj=1 j ii) meromorphic functions a1fv , . . . , aqfv (f ∈ F) in D, positive constants and M such that σ(aifv (z), ajfv (z)) ≥ for all z ∈ D, 1 ≤ i, j ≤ q, i = j, and # 1 (k) (k) # (fv ) (z) ≤ M, sup (z) ≤ M, sup fv z∈K:fv (z)=ajfv (z)=∞ z∈K:fv (z)=ajfv (z)=∞ for all v ≥ 1, j ∈ {1, . . . , q}, and k = 0, . . . , j − 2. By ignoring any j = 1, without loss of generality, we may assume that q ≥ 3 and j ≥ 2 for all j = 1, . . . , q. By Lemma 2, we may assume that {ajfv }v≥1 converges spherically locally uniformly on C to a meromorphic function aj (or ∞) for all j = 1, . . . , q. Then Ajv (ξ) := ajfv (zv + ρv ξ) converges spherically locally uniformly on C to the constant aj (z0 ). By the assumption on the spherical metric, a1 (z0 ), . . . , aq (z0 ) are distinct. We now prove the following claim. Claim 1: For any j ∈ {1, . . . , q}, if aj (z0 ) = ∞ then all zeros of g − aj (z0 ) have multiplicity at least j . We fix an index j. For any zero ξ0 of g(ξ) − aj (z0 ), since aj (z0 ) = ∞, we have that g is holomorphic at ξ0 . By Hurwtiz’s theorem, there are ξv → ξ0 (for all v sufficiently large) such that Ajv (ξv ) = ∞ and fv (zv + ρv ξv ) − ajfv (zv + ρξv ) = gv (ξv ) − Ajv (ξv ) = 0. ◦
Note that z0 ∈K , then zv + ρv ξv ∈ K for all v sufficiently large. Since ajfv (zv + ρξv ) → aj (z0 ) = ∞, we may assume that (3.3)
|fv (zv + ρv ξv )| = |ajfv (zv + ρξv )| ≤ 1 + |aj (z0 )|.
We have (k+1)
(3.4)
|fv 1+
(zv + ρv ξv )| (k) |fv (zv + ρv ξv )|2
≤ M,
for all k = 0, . . . , j − 2 and for all v sufficiently large. Set M1 := M · (1 + (1 + |aj (z0 )|)2 ), and Mn+1 := M · (1 + Mn2 ), for all positive integer n.
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Using the induction method we prove the following inequality. (k) (3.5) fv (zv + ρv ξv ) ≤ Mk , for all k = 1, . . . j − 1. Indeed, for k = 1, by (3.4) we have |fv (zv + ρv ξv )| ≤ M. 1 + |fv (zv + ρv ξv )|2 Combining with (3.3), we have
|fv (zv + ρv ξv )| ≤ M · 1 + |fv (zv + ρv ξv )|2 ≤ M · 1 + (1 + |aj (z0 )|)2 = M1 . We get (3.5) for the case where k = 1. Assume that (3.5) holds for some k (k ≤ j − 2). Then, by (3.4) and by the induction hypothesis, we have
|fv(k+1) (zv + ρv ξv )| ≤ M · 1 + |fv(k) (zv + ρv ξv )|2 ≤ M · 1 + Mk2 = Mk+1 . Hence, by induction, we get (3.5). By (3.5) we have (k)
|gv (ξv )| (k−1)
1 + |gv
(ξv )|2
= ρkv · ≤
ρkv
·
(k)
|fv (zv + ρv ξv )| 2(k−1)
|fv
|fv(k) (zv
+ ρv ξv )|
≤ ρkv · Mk ,
(3.6) (k−1)
Therefore, since gv
(k−1)
1 + ρv
(zv + ρv ξv )|2
for all k = 1, . . . j − 1.
(ξv ) → g (k−1) (ξ0 ) = ∞, we have
0 ≤ |g (k) (ξ0 )| = lim |gv(k) (ξv )| ≤ lim ρkv · Mk · (1 + |gv(k−1) (ξv )|2 ) = 0. v→∞
g (k) (ξ
v→∞
Then, 0 ) = 0 for all k = 1, . . . , j − 1. Hence, the zero ξ0 of g − aj (z0 ) has multiplicity at least j . This completes the proof of Claim 1. If there exists j ∈ {1, . . . , q}, such that aj (z0 ) = ∞, then A1v (ξ) := ajf (z1v +ρv ξ) v uniformly locally converges to 0 on D \ {z : aj (z) = 0} with respect to the Euclidean metric. Therefore, by (3.2) and by an argument similar to the proof of Claim 1, we get that all zeros of g1 have multiplicity at least j . Hence, similarly to Claim 1, we have the following claim. Claim 2. If there exists j ∈ {1, . . . , q}, such that aj (z0 ) = ∞, then all poles of g have multiplicity at least j . From Claims 1 and 2 and by the First and Second Main Theorems in Nevan linna theory (for the non-constant function g and distinct points aj (z0 )’s in C),
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we have (q − 2)T (r, g) ≤ ≤ ≤
q j=1 q j=1 q j=1
N (r,
1 ) + o(T (r, g)) g − aj (z0 )
1 1 N (r, ) + o(T (r, g)) j g − aj (z0 ) 1 T (r, g) + o(T (r, g)). j
This is impossible by the fact that q 1 < q − 2. j j=1
Thus, F is a normal family. We have completed the proof of Theorem 1.
Proof of Theorem 4. We can obtain Theorem 4 by an argument similar to the proof of Theorem 1 as follows: We may assume that D is the unit disc D. Suppose that F(k, M, N, C) consists
M, N, c, D) is not normal at z0 ∈ D. By Lemma 1, for constants only, while F(k, α = 0 there exist 1) a real number r, 0 < r < 1, 2) points zv , |zv | < r, zv → z0 , 3) positive numbers ρv → 0+ ,
M, N, c, D) 4) functions fv ∈ F(k, such that gv (ξ) = fv (zv + ρv ξ) → g(ξ) spherically locally uniformly on C, where g is a non-constant meromorphic function. Similarly to Claim 1 we also get that all zeros of g have multiplicity at least M and all zeros of g (k) − 1 have multiplicity at least N. Then, g ∈ F(k, M, N, C); this is impossible. References [1] W. Bergweiler and J. K. Langley, Multiplicities in Hayman’s Alternative, J. Australian Math. Soc. 78 (2005), 37-57 ´odory, Theory of functions of complex variable, Vol. II, Chelsea Publ. Co. New [2] C. Carathe York, 1960. [3] H. Fujimoto, On families of meromorphic maps into the complex projective space, Nagoya Math. J., 54 (1974), 21- 51.
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[4] J. Grahl and S. Nevo, Eceptional functions wandering on the sphere and normal families, Israel J. Math. 202 (2014), 21-34. [5] J. Grahl and S. Nevo, An extension of one direction in Marty’s normality criterion, Monatsh. Math. 174 (2014), 205-217. [6] Y. X. Gu, A criterion for normality of meromorphic functions (Chinese), Sci. Sinica Special Issue on Math. 1 (1979), 267-274. [7] W. K. Hayman, Picard values of meromorphic functions and their deivatives, Ann. of Math. 70 (1959), 9-42. [8] P. Lappan, A criterion for a meromorphic function to be normal, Comment. Math. Helv. 49 (1974), 492-495. [9] P. Lappan, A uniform approach to normal families, Rev. Roumaine Math. Pures Appl, 39 (1994), 691-702. [10] O. Lehto and K. L. Virtanen, Boundary behaviour and normal meromorphic functions, Acta Math. 97 (1957), 47-65. [11] X. C. Pang and L. Zalcman, On theorems of Hayman and Clunie, New Zealand J. Math. 28 (1999), 71-75. [12] P. Montel, Sur les familles de fonctions analytiques qui admettent des valeurs exception´ nelles dans un domaine, Annales scientifiques de l’Ecole Normale Sup´erieure, 29 (1912), 487-535. [13] P. Montel, Sur les familles normales de fonctions analytiques, Ann. Ecole Norm. Sup, 33 (1916), 223-302. [14] T. V. Tan and N. V. Thin, On Lappan’s five point theorem, To appear in Comput. Methods. Funct. Theory., DOI: 10.1007/s40315-016-0168-9. [15] L. Zalcman, Normal families: new perspective, Bull. Amer. Mat. Soc. 35 (1998), 215-230. [16] L. Zalcman, Variations on Montel’s theorem, Bulletin de la Soci´et´e des Sciences et des Lettres de Lod, 1 (2009) Vol. LIX, 25-36. [17] G. Valiron, Families normales et quasi-normales de fonctions meromorphes, M´em. Sci. Math., Fasc. 38 (1929). Department of Mathematics, Hanoi National University of Education, 136 Xuan Thuy Street, Cau Giay, Hanoi, Viet Nam. E-mail address: [email protected] Department of Mathematics, Thai Nguyen University of Education, Luong Ngoc Quyen Street, Thai Nguyen city, Viet Nam. E-mail address: [email protected] Department of Mathematics, Hoa Lu University, Ninh Nhat, Ninh Binh City, Viet Nam E-mail address: [email protected]