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Two normality criteria and the converse of the Bloch principle
J. Math. Anal. Appl. 353 (2009) 43–48
Contents lists available at ScienceDirect
Journal of Mathematical Analysis and Applications www.elsevier.com/locate/jmaa
Two normality criteria and the converse of the Bloch principle K.S. Charak a , J. Rieppo b,∗,1 a b
Department of Mathematics, University of Jammu, Jammu-180 006, India Department of Physics and Mathematics, University of Joensuu, P.O. Box 111, FIN-80101 Joensuu, Finland
a r t i c l e
i n f o
a b s t r a c t
Article history: Received 15 July 2008 Available online 28 November 2008 Submitted by J. Xiao
We prove two normality criteria for a family of meromorphic functions satisfying a certain differential condition and provide a counterexample to the converse of the Bloch principle. © 2008 Elsevier Inc. All rights reserved.
Keywords: Meromorphic function Normal family Bloch principle Differential polynomial Value distribution
1. Introduction and results More than three decades ago, Lawrence Zalcman [13] proved a heuristic lemma characterizing normal families of analytic and meromorphic functions on plane domains. Over the years, the lemma has continuously given an impact to a wide variety of topics in function theory and related areas. With renewed interest in normal families of analytic and meromorphic functions in plane domains, mainly because of their role in complex dynamics, it has become quite interesting to talk about normal families in their own right. Another valuable heuristic tool in the study of normal families is the Bloch principle stated as follows: Let us look at the statements (a) If a meromorphic function satisfies a condition P in the complex plane, then it must be a constant function. (b) If a family of meromorphic functions satisfies the condition P in an arbitrary complex domain, then the family is normal. In common, what is understood to be the Bloch principle is when (a) implies (b) and the converse of the Bloch principle is then the statement when (b) implies (a). In this article we focus on normality criteria that are connected to the value distribution of differential polynomials. For example, Theorem A. (See [4].) Let n 5 be an integer, a, b ∈ C and a = 0. If, for a meromorphic function f , f + af n = b for all z ∈ C, then f must be a constant.
* 1
Corresponding author. E-mail addresses: [email protected] (K.S. Charak), jarkko.rieppo@jns.fi (J. Rieppo). The author has been partially supported by the Academy of Finland grant 210245.
0022-247X/$ – see front matter doi:10.1016/j.jmaa.2008.11.066
© 2008 Elsevier Inc.
All rights reserved.
44
K.S. Charak, J. Rieppo / J. Math. Anal. Appl. 353 (2009) 43–48
Theorem B. (See [8,9].) Let n 3 be an integer, a, b ∈ C, a = 0 and F be a family of meromorphic functions in a domain D. If f + af n = b for all f ∈ F , then F is a normal family. E. Mues [7] has given examples showing that the claim of Theorem A is not valid when n = 3 and n = 4. That means that the converse of the Bloch principle is not true in these cases. Finding appropriate general conditions for the property P under which (a) and (b) are equivalent or generalizing the Bloch principle in some other way is an interesting problem, see e.g. [2]. From that point of view, it is important to look for a variety of examples concerning the Bloch principle. In the literature there are not too many counterexamples about the converse of the Bloch principle (the recent one is by Bao Qin Li [6]), and hence we give the one in this article. Recently I. Lahiri proved the following criterion for the normality by using Zalcman’s lemma. Theorem C. (See [5].) Let F be a family of meromorphic functions in a complex domain D. Let a, b ∈ C such that a = 0. Define
E f = z ∈ D: f ( z) +
a f ( z)
=b .
If there exists a positive constant M such that | f ( z)| M for all f ∈ F whenever z ∈ E f , then F is a normal family. Note that in Theorem C, when the set E f is empty for every f in a family, then the family is normal. Lahiri gave a counterexample to the converse of the Bloch principle by using this fact. In this paper, we prove two normality criteria of Lahiri’s type and using one of them we provide another counterexample to the general converse of the Bloch principle. Theorem 1. Let F be a family of meromorphic functions in a complex domain D. Let a, b ∈ C such that a = 0. Let m1 , m2 , n1 , n2 be nonnegative integers such that m1 n2 − m2 n1 > 0, m1 + m2 1, n1 + n2 2, and put
E f = z ∈ D:
n1
f ( z)
m1
f ( z)
+
a
( f (z))n2 ( f (z))m2
=b .
If there exists a positive constant M such that | f ( z)| M for all f ∈ F whenever z ∈ E f , then F is a normal family. To consider the case m1 n2 = m2 n1 , we need a value distribution result for corresponding differential expression. Theorem 2. Let a, b ∈ C such that a = 0 and let f be a nonconstant meromorphic function. If n1 , n2 , m1 , m2 are positive integers such that m1 n2 = m2 n1 , then a function Ψ ( z) defined by
n m Ψ (z) := f (z) 1 f (z) 1 +
a
( f (z))n2 ( f (z))m2
−b
(1)
has a finite zero. Theorem 3. Let F be a family of meromorphic functions in a complex domain D. Let a, b ∈ C such that a = 0. Let m1 , m2 , n1 , n2 be positive integers such that m1 n2 = m2 n1 , and put
E f = z ∈ D:
n1
f ( z)
m1
f ( z)
+
a
( f (z))n2 ( f (z))m2
=b .
If there exists a positive constant M such that | f ( z)| M for all f ∈ F whenever z ∈ E f , then F is a normal family. 2. Proofs of the results 2.1. Auxiliary results For the proof of Theorem 1 we need the following lemmas: Lemma 4. [12] Take nonnegative integers n, n1 , . . . , nk with n 1, n1 + n2 + · · · + nk 1 and define d = n + n1 + n2 + · · · + nk . Let f be a transcendental meromorphic function with the deficiency δ(0, f ) > 3d3+1 . Then for any nonzero value c, the function nk
f n ( f )n1 . . . ( f (k) )
− c has infinitely many zeros.
It is noted in [12] that for n 2 the deficient condition in Lemma 4 can be omitted. In our application of this result we consider n 2, n1 1 and n2 , . . . , nk are all equal to 0. The version of the Zalcman lemma given here is due to Pang [10].
K.S. Charak, J. Rieppo / J. Math. Anal. Appl. 353 (2009) 43–48
45
Lemma 5 (Zalcman–Pang lemma). Let F be a family of meromorphic functions in a domain D. Let α ∈ R : −1 < α < 1. Then F is not normal in a neighbourhood of z0 ∈ D iff there exist
• points zk ∈ D : zk → z0 as k → ∞; • positive real numbers ρk : ρk → 0 as k → ∞; and • functions f k ∈ F such that ρk α f k (zk + ρk ζ ) → g (ζ ) spherically uniformly on compact subsets of C, where g is a nonconstant meromorphic function. Lemma 6. Let f be a nonconstant rational function and m, n natural numbers. Then, the function f n ( f )m takes every finite nonzero value. Proof. Denote f ( z) = A ( z)/ B ( z), where A and B are complex polynomials. Consider a quantity defined by d( f ) := deg A − deg B. First we show that d( f n ( f )m ) = 0. To do that, suppose first d( f ) = 0. Then, it is easy to see that deg( A B − A B ) < deg A + deg B − 1 and thus
d f n ( f )m = nd( f ) + md( f ) = m deg( A B − A B ) − deg B 2 < m(deg A + deg B − 1 − 2 deg B ) = md( f ) − m = −m. Assume next d( f ) = 0. Then, deg( A B − A B ) = deg A + deg B − 1 and since d( f ) is an integer we obtain
d f n ( f )m = nd( f ) + md( f ) = nd( f ) + m(deg A − deg B − 1) = (m + n)d( f ) − m = 0. Now, if g := f n ( f )m avoids a nonzero complex value a, then 1/( g − a) is a polynomial, and thus g (∞) = a. But this contradicts the fact d( g ) = 0, that was shown above. 2 2.2. Proof of Theorem 1 Proof. Suppose on the contrary that F is not normal at z0 ∈ D. Then by Lemma 5, for α : −1 < α < 1, there exists a sequence of complex numbers { zt } in D such that zt → z0 , a sequence of positive numbers {ρt } such that ρt → 0, and a sequence of functions { f t } in F such that gt ( w ) := ρt −α f t ( zt + ρt w ) → g ( w )
(2)
spherically uniformly on compact subsets of C as t → ∞, where g is a nonconstant meromorphic function in C. To abbreviate the notions below, we shall denote f t ( zt + ρt w ) just by f t and k j := n j + m j , for j = 1, 2. Now by Lemma 4 in combination with Lemma 6, there exists w 0 ∈ C such that
n1
g(w0)
m1
g(w0)
+
a
( g ( w 0 ))n2 g ( w 0 )m2
= 0.
(3)
Since g ( w 0 ) = 0, ∞, gt ( w ) → g ( w ) uniformly in a closed disk Δ( w 0 ; δ) for some δ > 0. Now in Δ( w 0 ; δ), we have
n1
gt ( w )
m1
gt ( w )
+
a
m − ρt αk2 −m2 b ( gt ( w ))n2 g ( w ) 2
= ρt −αk1 +m1 ( f t )n1 ( f t )m1 +
a
ρt −αk2 +m2 ( f t )n2 ( f t )m2
m −α (k1 +k2 )+m2 +m1 = ρt αk2 −m2 ρt ( f t )n1 ( f t ) 1 + Taking
− ρt αk2 −m2 b a
( f t )n2 ( f t )m2
−b .
m2 α = mk11 + +k2 and using the assumption m1 n2 − n1 m2 > 0, we see that
g n1 ( g )m1 +
a
(4)
g n2 ( g )m2
is the uniform limit of m1 n2 −n1 m2 k1 +k2
ρt
( f t )n1 ( f t )m1 +
a
( f t )n2 ( f t )m2
−b
(5)
in Δ( w 0 ; δ). In view of (3) we apply Hurwitz’s theorem to get a sequence { w t } such that { w t } converges to w 0 and for all large t and ζt := zt + ρt w t ,
n1
f t (ζt )
m1
f t (ζt )
+
a
( f t (ζt ))n2 ( f t (ζt ))m2
= b.
(6)
46
K.S. Charak, J. Rieppo / J. Math. Anal. Appl. 353 (2009) 43–48
Hence, for all large t, ζt := zt + ρt w t ∈ E f and thus there exists an M > 0 such that | f t ( zt + ρt w t )| M. This further implies that m1 +m2 gt ( w t ) M ρt − k1 +k2 .
(7)
Since g is holomorphic at w 0 , | g ( w )| C for some positive constant C , and for all w ∈ Δ( w 0 ; η) for some since gt → g, for all > 0 there exists some t 0 such that for all w ∈ Δ( w 0 ; η), we have
η > 0. Again,
gt ( w ) − g ( w ) < for all t t 0 ,
and now by using (7), we find that
C g ( w t ) gt ( w t ) − gt ( w t ) − g ( w t ) M ρt
−
m1 +m2 k1 +k2
−
for all t t 0 . That is C M ρt
−
m1 +m2 k1 +k2
− for all t t 0 ,
which is not in reason since
ρt → 0 as t → ∞. 2
2.3. Proof of Theorem 2 Proof. The algebraic complex equation x+
a xn2 /n1
−b=0
has always a nonzero solution, say x0 ∈ C. By [3, Theorem 2], [1, Corollary 3], Lemma 4 and Lemma 6, the meromorphic function f n1 ( f )m1 cannot avoid it and thus there exists z0 ∈ C such that ( f ( z0 ))n1 ( f ( z0 ))m1 = x0 . By assumption, we may write m2 = (n2 /n1 )m1 and n2 = (n2 /n1 )n1 . Consequently,
Ψ ( z0 ) =
n1
f ( z0 )
m1
f ( z0 )
+
a
[( f (z0 ))n1 ( f (z0 ))m1 ]n2 /n1
−b=0
2
and we are done.
2.4. The proof of Theorem 3 Proof. Suppose on the contrary that F is not normal at z0 ∈ D. Then by Lemma 5, for α : −1 < α < 1, there exists a sequence of complex numbers { zt } in D such that zt → z0 , a sequence of positive numbers {ρt } such that ρt → 0 as, and a sequence of functions { f t } in F such that gt ( w ) := ρt −α f t ( zt + ρt w ) → g ( w )
(8)
spherically uniformly on compact subsets of C as t → ∞, where g is a nonconstant meromorphic function in C. By Theorem 2, there exists w 0 ∈ C such that
n1
g(w0)
m1
g(w0)
+
a
( g ( w 0 ))n2 g ( w 0 ))m2
− b = 0.
(9)
Since g ( w 0 ) = 0, ∞, gt ( w ) → g ( w ) uniformly in a closed disk Δ( w 0 ; δ) for some δ > 0. Taking
n1
gt ( w )
m1
gt ( w )
+
a
( gt ( w ))n2 ( g ( w ))m2
− b = ( f t )n1 ( f t )m1 +
a
( f t )n2 ( f t )m2
m2 α = mk11 + +k2 , we have
−b
in Δ( w 0 ; δ). Thus
m1
g n1 g
+
a g n2 ( g )m2
−b
is the uniform limit of a
( f t )n1 ( f t )m1 +
( f t )n2 ( f t )m2
−b
in Δ( w 0 ; δ). Since we have Eq. (9), we may apply Hurwitz’s theorem to get a sequence { w t } such that { w t } converges to w 0 and for all large t
n1
f t ( zt + ρt w t )
m1
f t ( zt + ρt w t )
+
a n t w t )) 2 ( f t ( zt
( f t (zt + ρ
+ ρt w t ))m2
= b.
From now on the proof will continue word by word similarly as the proof of Theorem 1 after Eq. (6).
2
K.S. Charak, J. Rieppo / J. Math. Anal. Appl. 353 (2009) 43–48
47
3. Discussion Considering the converse of the Bloch principle and in the view of Theorems 1, 2 and 3, one is tempted to ask: Question 7. For what values of m1 , n1 , m2 , n2 there exists a nonconstant meromorphic function f such that
Ψ (z) := f n1 ( f )m1 +
a f n2 ( f )m2
−b
has no zeros in C for some a, b ∈ C, a = 0? In particular, one may pose: Question 8. Does there exist a nonconstant meromorphic function f such that
Ψ (z) := f n ( f )m +
a f
+b
has no zeros in C for some n 1, m 1 and for some a = 0, b in C? As another counterexample to the converse of the Bloch principle, we give Example 9. Let f ( z) = tan z, then f ( z) = 1 + tan2 z = 0 for all z ∈ C. Now we see that
Ψ ( z) = f ( z) +
1
( f (z))2
+1=
(tan2 z + 1)2 tan2 z
= 0,
but Theorem 1 is true especially when E f is an empty set for every f in the family. To consider the questions above, let R ( y 0 , y 1 , . . . , ym ), m ∈ N ∪ {0}, be a rational function in variables y 0 , y 1 , . . . , ym with constant coefficients. Let D be a domain in the complex plane containing the origin. If all meromorphic functions f in D satisfying
R f , f , . . . , f (m) ≡ 0
(10)
form a normal family, then, by Marty’s criterion, any function g meromorphic in the whole plane such that
R g , g , . . . , g (m) ≡ 0
(11)
must satisfy ρ ( g ) 2, where ρ ( g ) is the order of growth of the meromorphic function g. Indeed, assume contrary to the assertion that ρ := ρ ( g ) > 2. Then there exists a sequence ( zk ) such that | zk | → ∞ and g # ( zk )
| zk |ρ /2−1
→∞
as k → ∞. By Marty’s criterion, the family { gk }, where gk ( z) = g ( zk + z), is not normal at the origin. But since each gk satisfies
(m)
R gk , gk , . . . , gk
≡ 0,
(12)
we have a contradiction with the assumptions. We note that we may replace the identities (10), (11) and (12) by R ( f , f , . . . , f (m) ) = 0, R ( g , g , . . . , g (m) ) = 0 and R ( gk , gk , . . . , gkm ) = 0 for all z ∈ C and the conclusion remains the same. In the light of the preceding discussion, f in Question 7 (as well as in Question 8) satisfies ρ ( f ) 2. If Ψ ( z) in Question 8 has no zeros, then we must have n = 2 or n = 1 according to Satz 3 in [11]. Acknowledgments The authors would like to thank the anonymous reviewer for suggesting necessary corrections to our manuscript.
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K.S. Charak, J. Rieppo / J. Math. Anal. Appl. 353 (2009) 43–48
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13]
A. Alotaibi, On the zeros of af ( f ( k))n − 1 for n 2, Comput. Methods Funct. Theory 4 (2004) 227–235. W. Bergweiler, Bloch’s principle, Comput. Methods Funct. Theory 6 (2006) 77–108. W. Bergweiler, A. Eremenko, On the singularities of the inverse of meromorphic function of finite order, Rev. Mat. Iberoamericana 11 (1995) 355–373. W.K. Hayman, Picard values of meromorphic functions and their derivatives, Ann. of Math. 70 (1959) 9–42. I. Lahiri, A simple normality criterion leading to a counterexample to the converse of the Bloch principle, New Zealand J. Math. 34 (2005) 61–65. B.Q. Li, On the Bloch heuristic principle, normality, and totally ramified values, Arch. Math. 87 (2006) 417–426. E. Mues, Über ein Problem von Hayman, Math. Z. 164 (1979) 239–259. X.C. Pang, Bloch’s principle and normal criterion, Sci. China Ser. A 32 (1989) 782–791. X.C. Pang, On normal criterion of meromorphic function, Sci. China Ser. A 33 (1990) 521–527. X.C. Pang, Criteria for normality about differential polynomial, Chinese Sci. Bull. 22 (1988) 1690–1693. N. Steinmetz, Über die Nullstellen von Differentialpolynomen, Math. Z. 176 (1981) 255–264. C.C. Yang, P.C. Hu, On the value distribution of f f (k) , Kodai Math. J. 19 (1996) 157–167. L. Zalcman, A heurisitic principle in complex function theory, Amer. Math. Monthly 82 (1975) 813–817.
Contents lists available at ScienceDirect
Journal of Mathematical Analysis and Applications www.elsevier.com/locate/jmaa
Two normality criteria and the converse of the Bloch principle K.S. Charak a , J. Rieppo b,∗,1 a b
Department of Mathematics, University of Jammu, Jammu-180 006, India Department of Physics and Mathematics, University of Joensuu, P.O. Box 111, FIN-80101 Joensuu, Finland
a r t i c l e
i n f o
a b s t r a c t
Article history: Received 15 July 2008 Available online 28 November 2008 Submitted by J. Xiao
We prove two normality criteria for a family of meromorphic functions satisfying a certain differential condition and provide a counterexample to the converse of the Bloch principle. © 2008 Elsevier Inc. All rights reserved.
Keywords: Meromorphic function Normal family Bloch principle Differential polynomial Value distribution
1. Introduction and results More than three decades ago, Lawrence Zalcman [13] proved a heuristic lemma characterizing normal families of analytic and meromorphic functions on plane domains. Over the years, the lemma has continuously given an impact to a wide variety of topics in function theory and related areas. With renewed interest in normal families of analytic and meromorphic functions in plane domains, mainly because of their role in complex dynamics, it has become quite interesting to talk about normal families in their own right. Another valuable heuristic tool in the study of normal families is the Bloch principle stated as follows: Let us look at the statements (a) If a meromorphic function satisfies a condition P in the complex plane, then it must be a constant function. (b) If a family of meromorphic functions satisfies the condition P in an arbitrary complex domain, then the family is normal. In common, what is understood to be the Bloch principle is when (a) implies (b) and the converse of the Bloch principle is then the statement when (b) implies (a). In this article we focus on normality criteria that are connected to the value distribution of differential polynomials. For example, Theorem A. (See [4].) Let n 5 be an integer, a, b ∈ C and a = 0. If, for a meromorphic function f , f + af n = b for all z ∈ C, then f must be a constant.
* 1
Corresponding author. E-mail addresses: [email protected] (K.S. Charak), jarkko.rieppo@jns.fi (J. Rieppo). The author has been partially supported by the Academy of Finland grant 210245.
0022-247X/$ – see front matter doi:10.1016/j.jmaa.2008.11.066
© 2008 Elsevier Inc.
All rights reserved.
44
K.S. Charak, J. Rieppo / J. Math. Anal. Appl. 353 (2009) 43–48
Theorem B. (See [8,9].) Let n 3 be an integer, a, b ∈ C, a = 0 and F be a family of meromorphic functions in a domain D. If f + af n = b for all f ∈ F , then F is a normal family. E. Mues [7] has given examples showing that the claim of Theorem A is not valid when n = 3 and n = 4. That means that the converse of the Bloch principle is not true in these cases. Finding appropriate general conditions for the property P under which (a) and (b) are equivalent or generalizing the Bloch principle in some other way is an interesting problem, see e.g. [2]. From that point of view, it is important to look for a variety of examples concerning the Bloch principle. In the literature there are not too many counterexamples about the converse of the Bloch principle (the recent one is by Bao Qin Li [6]), and hence we give the one in this article. Recently I. Lahiri proved the following criterion for the normality by using Zalcman’s lemma. Theorem C. (See [5].) Let F be a family of meromorphic functions in a complex domain D. Let a, b ∈ C such that a = 0. Define
E f = z ∈ D: f ( z) +
a f ( z)
=b .
If there exists a positive constant M such that | f ( z)| M for all f ∈ F whenever z ∈ E f , then F is a normal family. Note that in Theorem C, when the set E f is empty for every f in a family, then the family is normal. Lahiri gave a counterexample to the converse of the Bloch principle by using this fact. In this paper, we prove two normality criteria of Lahiri’s type and using one of them we provide another counterexample to the general converse of the Bloch principle. Theorem 1. Let F be a family of meromorphic functions in a complex domain D. Let a, b ∈ C such that a = 0. Let m1 , m2 , n1 , n2 be nonnegative integers such that m1 n2 − m2 n1 > 0, m1 + m2 1, n1 + n2 2, and put
E f = z ∈ D:
n1
f ( z)
m1
f ( z)
+
a
( f (z))n2 ( f (z))m2
=b .
If there exists a positive constant M such that | f ( z)| M for all f ∈ F whenever z ∈ E f , then F is a normal family. To consider the case m1 n2 = m2 n1 , we need a value distribution result for corresponding differential expression. Theorem 2. Let a, b ∈ C such that a = 0 and let f be a nonconstant meromorphic function. If n1 , n2 , m1 , m2 are positive integers such that m1 n2 = m2 n1 , then a function Ψ ( z) defined by
n m Ψ (z) := f (z) 1 f (z) 1 +
a
( f (z))n2 ( f (z))m2
−b
(1)
has a finite zero. Theorem 3. Let F be a family of meromorphic functions in a complex domain D. Let a, b ∈ C such that a = 0. Let m1 , m2 , n1 , n2 be positive integers such that m1 n2 = m2 n1 , and put
E f = z ∈ D:
n1
f ( z)
m1
f ( z)
+
a
( f (z))n2 ( f (z))m2
=b .
If there exists a positive constant M such that | f ( z)| M for all f ∈ F whenever z ∈ E f , then F is a normal family. 2. Proofs of the results 2.1. Auxiliary results For the proof of Theorem 1 we need the following lemmas: Lemma 4. [12] Take nonnegative integers n, n1 , . . . , nk with n 1, n1 + n2 + · · · + nk 1 and define d = n + n1 + n2 + · · · + nk . Let f be a transcendental meromorphic function with the deficiency δ(0, f ) > 3d3+1 . Then for any nonzero value c, the function nk
f n ( f )n1 . . . ( f (k) )
− c has infinitely many zeros.
It is noted in [12] that for n 2 the deficient condition in Lemma 4 can be omitted. In our application of this result we consider n 2, n1 1 and n2 , . . . , nk are all equal to 0. The version of the Zalcman lemma given here is due to Pang [10].
K.S. Charak, J. Rieppo / J. Math. Anal. Appl. 353 (2009) 43–48
45
Lemma 5 (Zalcman–Pang lemma). Let F be a family of meromorphic functions in a domain D. Let α ∈ R : −1 < α < 1. Then F is not normal in a neighbourhood of z0 ∈ D iff there exist
• points zk ∈ D : zk → z0 as k → ∞; • positive real numbers ρk : ρk → 0 as k → ∞; and • functions f k ∈ F such that ρk α f k (zk + ρk ζ ) → g (ζ ) spherically uniformly on compact subsets of C, where g is a nonconstant meromorphic function. Lemma 6. Let f be a nonconstant rational function and m, n natural numbers. Then, the function f n ( f )m takes every finite nonzero value. Proof. Denote f ( z) = A ( z)/ B ( z), where A and B are complex polynomials. Consider a quantity defined by d( f ) := deg A − deg B. First we show that d( f n ( f )m ) = 0. To do that, suppose first d( f ) = 0. Then, it is easy to see that deg( A B − A B ) < deg A + deg B − 1 and thus
d f n ( f )m = nd( f ) + md( f ) = m deg( A B − A B ) − deg B 2 < m(deg A + deg B − 1 − 2 deg B ) = md( f ) − m = −m. Assume next d( f ) = 0. Then, deg( A B − A B ) = deg A + deg B − 1 and since d( f ) is an integer we obtain
d f n ( f )m = nd( f ) + md( f ) = nd( f ) + m(deg A − deg B − 1) = (m + n)d( f ) − m = 0. Now, if g := f n ( f )m avoids a nonzero complex value a, then 1/( g − a) is a polynomial, and thus g (∞) = a. But this contradicts the fact d( g ) = 0, that was shown above. 2 2.2. Proof of Theorem 1 Proof. Suppose on the contrary that F is not normal at z0 ∈ D. Then by Lemma 5, for α : −1 < α < 1, there exists a sequence of complex numbers { zt } in D such that zt → z0 , a sequence of positive numbers {ρt } such that ρt → 0, and a sequence of functions { f t } in F such that gt ( w ) := ρt −α f t ( zt + ρt w ) → g ( w )
(2)
spherically uniformly on compact subsets of C as t → ∞, where g is a nonconstant meromorphic function in C. To abbreviate the notions below, we shall denote f t ( zt + ρt w ) just by f t and k j := n j + m j , for j = 1, 2. Now by Lemma 4 in combination with Lemma 6, there exists w 0 ∈ C such that
n1
g(w0)
m1
g(w0)
+
a
( g ( w 0 ))n2 g ( w 0 )m2
= 0.
(3)
Since g ( w 0 ) = 0, ∞, gt ( w ) → g ( w ) uniformly in a closed disk Δ( w 0 ; δ) for some δ > 0. Now in Δ( w 0 ; δ), we have
n1
gt ( w )
m1
gt ( w )
+
a
m − ρt αk2 −m2 b ( gt ( w ))n2 g ( w ) 2
= ρt −αk1 +m1 ( f t )n1 ( f t )m1 +
a
ρt −αk2 +m2 ( f t )n2 ( f t )m2
m −α (k1 +k2 )+m2 +m1 = ρt αk2 −m2 ρt ( f t )n1 ( f t ) 1 + Taking
− ρt αk2 −m2 b a
( f t )n2 ( f t )m2
−b .
m2 α = mk11 + +k2 and using the assumption m1 n2 − n1 m2 > 0, we see that
g n1 ( g )m1 +
a
(4)
g n2 ( g )m2
is the uniform limit of m1 n2 −n1 m2 k1 +k2
ρt
( f t )n1 ( f t )m1 +
a
( f t )n2 ( f t )m2
−b
(5)
in Δ( w 0 ; δ). In view of (3) we apply Hurwitz’s theorem to get a sequence { w t } such that { w t } converges to w 0 and for all large t and ζt := zt + ρt w t ,
n1
f t (ζt )
m1
f t (ζt )
+
a
( f t (ζt ))n2 ( f t (ζt ))m2
= b.
(6)
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K.S. Charak, J. Rieppo / J. Math. Anal. Appl. 353 (2009) 43–48
Hence, for all large t, ζt := zt + ρt w t ∈ E f and thus there exists an M > 0 such that | f t ( zt + ρt w t )| M. This further implies that m1 +m2 gt ( w t ) M ρt − k1 +k2 .
(7)
Since g is holomorphic at w 0 , | g ( w )| C for some positive constant C , and for all w ∈ Δ( w 0 ; η) for some since gt → g, for all > 0 there exists some t 0 such that for all w ∈ Δ( w 0 ; η), we have
η > 0. Again,
gt ( w ) − g ( w ) < for all t t 0 ,
and now by using (7), we find that
C g ( w t ) gt ( w t ) − gt ( w t ) − g ( w t ) M ρt
−
m1 +m2 k1 +k2
−
for all t t 0 . That is C M ρt
−
m1 +m2 k1 +k2
− for all t t 0 ,
which is not in reason since
ρt → 0 as t → ∞. 2
2.3. Proof of Theorem 2 Proof. The algebraic complex equation x+
a xn2 /n1
−b=0
has always a nonzero solution, say x0 ∈ C. By [3, Theorem 2], [1, Corollary 3], Lemma 4 and Lemma 6, the meromorphic function f n1 ( f )m1 cannot avoid it and thus there exists z0 ∈ C such that ( f ( z0 ))n1 ( f ( z0 ))m1 = x0 . By assumption, we may write m2 = (n2 /n1 )m1 and n2 = (n2 /n1 )n1 . Consequently,
Ψ ( z0 ) =
n1
f ( z0 )
m1
f ( z0 )
+
a
[( f (z0 ))n1 ( f (z0 ))m1 ]n2 /n1
−b=0
2
and we are done.
2.4. The proof of Theorem 3 Proof. Suppose on the contrary that F is not normal at z0 ∈ D. Then by Lemma 5, for α : −1 < α < 1, there exists a sequence of complex numbers { zt } in D such that zt → z0 , a sequence of positive numbers {ρt } such that ρt → 0 as, and a sequence of functions { f t } in F such that gt ( w ) := ρt −α f t ( zt + ρt w ) → g ( w )
(8)
spherically uniformly on compact subsets of C as t → ∞, where g is a nonconstant meromorphic function in C. By Theorem 2, there exists w 0 ∈ C such that
n1
g(w0)
m1
g(w0)
+
a
( g ( w 0 ))n2 g ( w 0 ))m2
− b = 0.
(9)
Since g ( w 0 ) = 0, ∞, gt ( w ) → g ( w ) uniformly in a closed disk Δ( w 0 ; δ) for some δ > 0. Taking
n1
gt ( w )
m1
gt ( w )
+
a
( gt ( w ))n2 ( g ( w ))m2
− b = ( f t )n1 ( f t )m1 +
a
( f t )n2 ( f t )m2
m2 α = mk11 + +k2 , we have
−b
in Δ( w 0 ; δ). Thus
m1
g n1 g
+
a g n2 ( g )m2
−b
is the uniform limit of a
( f t )n1 ( f t )m1 +
( f t )n2 ( f t )m2
−b
in Δ( w 0 ; δ). Since we have Eq. (9), we may apply Hurwitz’s theorem to get a sequence { w t } such that { w t } converges to w 0 and for all large t
n1
f t ( zt + ρt w t )
m1
f t ( zt + ρt w t )
+
a n t w t )) 2 ( f t ( zt
( f t (zt + ρ
+ ρt w t ))m2
= b.
From now on the proof will continue word by word similarly as the proof of Theorem 1 after Eq. (6).
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K.S. Charak, J. Rieppo / J. Math. Anal. Appl. 353 (2009) 43–48
47
3. Discussion Considering the converse of the Bloch principle and in the view of Theorems 1, 2 and 3, one is tempted to ask: Question 7. For what values of m1 , n1 , m2 , n2 there exists a nonconstant meromorphic function f such that
Ψ (z) := f n1 ( f )m1 +
a f n2 ( f )m2
−b
has no zeros in C for some a, b ∈ C, a = 0? In particular, one may pose: Question 8. Does there exist a nonconstant meromorphic function f such that
Ψ (z) := f n ( f )m +
a f
+b
has no zeros in C for some n 1, m 1 and for some a = 0, b in C? As another counterexample to the converse of the Bloch principle, we give Example 9. Let f ( z) = tan z, then f ( z) = 1 + tan2 z = 0 for all z ∈ C. Now we see that
Ψ ( z) = f ( z) +
1
( f (z))2
+1=
(tan2 z + 1)2 tan2 z
= 0,
but Theorem 1 is true especially when E f is an empty set for every f in the family. To consider the questions above, let R ( y 0 , y 1 , . . . , ym ), m ∈ N ∪ {0}, be a rational function in variables y 0 , y 1 , . . . , ym with constant coefficients. Let D be a domain in the complex plane containing the origin. If all meromorphic functions f in D satisfying
R f , f , . . . , f (m) ≡ 0
(10)
form a normal family, then, by Marty’s criterion, any function g meromorphic in the whole plane such that
R g , g , . . . , g (m) ≡ 0
(11)
must satisfy ρ ( g ) 2, where ρ ( g ) is the order of growth of the meromorphic function g. Indeed, assume contrary to the assertion that ρ := ρ ( g ) > 2. Then there exists a sequence ( zk ) such that | zk | → ∞ and g # ( zk )
| zk |ρ /2−1
→∞
as k → ∞. By Marty’s criterion, the family { gk }, where gk ( z) = g ( zk + z), is not normal at the origin. But since each gk satisfies
(m)
R gk , gk , . . . , gk
≡ 0,
(12)
we have a contradiction with the assumptions. We note that we may replace the identities (10), (11) and (12) by R ( f , f , . . . , f (m) ) = 0, R ( g , g , . . . , g (m) ) = 0 and R ( gk , gk , . . . , gkm ) = 0 for all z ∈ C and the conclusion remains the same. In the light of the preceding discussion, f in Question 7 (as well as in Question 8) satisfies ρ ( f ) 2. If Ψ ( z) in Question 8 has no zeros, then we must have n = 2 or n = 1 according to Satz 3 in [11]. Acknowledgments The authors would like to thank the anonymous reviewer for suggesting necessary corrections to our manuscript.
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