NORMALITY CRITERIA FOR A FAMILY OF MEROMORPHIC FUNCTIONS WITH MULTIPLE ZEROS AND POLES
arXiv:1305.6214v4 [math.CV] 31 Jul 2013
GOPAL DATT AND SANJAY KUMAR Abstract. In this article, we prove some normality criteria for a family of meromorphic functions having multiple zeros and poles which involves sharing of a non-zero value by certain non linear differential polynomials generated by the members of the family.
1. Introduction and main results Let D be a domain on C, and F be a family of meromorphic functions defined on D. The family F is said to be normal in D, if every sequence {fn } ⊂ F has a subsequence {fnj } which converges spherically uniformly on compact subsets of D, to a meromorphic function or ∞. { [6], p. 71} Let f and g be meromorphic functions in a domain D and a ∈ C. Let zeros of f − a are zeros of g − a (ignoring multiplicity), we write f = a ⇒ g = a. Hence f = a ⇐⇒ g = a means that f −a and g −a have the same zeros (ignoring multiplicity). If f −a ⇐⇒ g −a, then we say that f and g share the value z = a IM. { [10], p. 108} In 1992, W. Schwick gave a connection between normality criteria and sharing values. He proved the following theorem : Theorem 1.1. { [7], Theorem 2} Let F be a family of meromorphic functions on a domain D and a1 , a2 , a3 be distinct complex numbers. If f and f ′ share a1 , a2 , a3 for every f ∈ F , then F is normal in D. Since then many results in this area have been obtained. The following normality criteria was proved by D. W. Meng and P. C. Hu in 2011: Theorem 1.2. { [4], Theorem 1.1} Take a positive integer k and a non-zero complex number a. Let F be a family of meromorphic functions in a domain D ⊂ C such that each f ∈ F has only zeros of multiplicity at least k + 1. For each pair (f, g) ∈ F , if f f (k) and gg (k) share a IM, then F is normal in D. Let f be a meromorphic function in D ⊆ C and a ∈ C \ {0} and n ≥ 2, we define (1.1)
D(f ) := f + a(f ′ )n ,
2010 Mathematics Subject Classification. 30D45. Key words and phrases. Meromorphic functions, Holomorphic functions, Shared values, Normal families. The research work of the first author is supported by research fellowship from UGC India. 1
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a non linear differential polynomial. In this paper we investigate the situation where we replace f f (k) and gg (k) by D(f ) and D(g) respectively and prove the following results: Theorem 1.3. Let F be a family of meromorphic functions defined in a domain D such that for each f ∈ F , f has no simple pole and has no zero of multiplicity less than 3 in D. If for each pair of functions f (z), g(z) ∈ F , D(f ) and D(g) share the value b IM, where b is a non-zero complex number, then F is normal. Theorem 1.4. Let F be a family of meromorphic functions defined in a domain D such that for each f ∈ F , f has no simple pole and has no zero of multiplicity less than 3 in D. If for each function f (z) ∈ F , D(f ) − b has at most one zero in D, where b is a non-zero complex number, then F is normal. It is natural to ask whether one can replace the value b, in Theorem 1.3 and 1.4 by a holomorphic function α(z). We investigate this situation in the following theorem: Theorem 1.5. Let α(z) be a holomorphic function such that α(z) 6= 0. Let F be a family of meromorphic functions defined in a domain D such that for each f ∈ F , f has no simple pole and has no zero of multiplicity less than 3 in D. If D(f ) and D(g) share α(z) IM for each pair f (z), g(z) ∈ F , then F is normal in D. Theorem 1.6. Let α(z) be a holomorphic function such that α(z) 6= 0. Let F be a family of meromorphic functions defined in a domain D such that for each f ∈ F , f has no simple pole and has no zero of multiplicity less than 3 in D. If D(f ) − α(z0 ) has at most one zero in D, for z0 ∈ D, then F is normal in D. 2. Some Lemmas In order to prove our results, we need following results. Lemma 2.1. { [6] p. 101; [5], Lemma 2} Let F be a family of meromorphic functions on the unit disc ∆ := {z ∈ C : |z| < 1}. Then F is not normal in ∆, if and only if there exist (1) a number r with 0 < r < 1, (2) points zn satisfying |zn | < r, (3) functions fn ∈ F , (4) positive numbers ρn → 0 as n → ∞, such that fn (zn + ρn ζ) = gn (ζ) → g(ζ) as n → ∞ locally uniformly with respect to the spherical metric, where g is a nonconstant meromorphic function on C, with g # (ζ) ≤ g # (0) = 1. In particular, g has order at most 2. P Lemma 2.2. { [14], Lemma 2.5, Lemma 2.6; [15], Lemma 2.2} Let R = Q be a rational (k) function and Q be non-constant. Then (R )∞ ≤ (R)∞ − k, where k is a positive integer,
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(R)∞ = deg(P ) − deg(Q) and deg(P ) denotes the degree of P. In 2008 the following result was proved by M.L. Fang and L. Zalcman. Lemma 2.3. { [1], Theorem 2} Let f be a transcendental meromorphic function and a be a non-zero complex number. Then D(f ) := f + a(f ′ )n assumes every complex value infinitely often for each positive integer n ≥ 2. Lemma 2.4. Let f be a non-polynomial rational function having multiple zeros and poles. Then D(f ) − b has at least two distinct zeros, where b is a non-zero complex number. Proof. We consider the following cases. Case1.
Let D(f ) − b has exactly one zero at z0 .
Since f is a non-polynomial rational function with multiple zeros and poles, then we have (2.1)
f (z) = A
(z − α1 )m1 (z − α2 )m2 . . . (z − αs )ms , (z − β1 )n1 (z − β2 )n2 . . . (z − βt )nt
where A is nonzero constant, mi ≥ 2 (i = 1, 2, . . . , s) and nj ≥ 2 (j = 1, 2, . . . , t) are integers. We write (2.2)
M=
s X i=1
mi ≥ 2s and N =
t X
nj ≥ 2t.
j=1
From (2.1), we get f ′ (z) =
(2.3)
(z − α1 )m1 −1 (z − α2 )m2 −1 . . . (z − αs )ms −1 g1 (z), (z − β1 )n1 +1 (z − β2 )n2 +1 . . . (z − βt )nt +1
where g1 (z) is a polynomial. From (2.1) and(2.3), we get (f (z))∞ = M − N
and (f ′ (z))∞ = M − N − (s + t) + deg(g1 (z)).
Since by Lemma 2.2, (f ′ (z))∞ ≤ (f (z))∞ − 1, so we get deg(g1 (z)) ≤ s + t − 1.
(2.4) From (2.3), we get (2.5)
(z − α1 )n(m1 −1) (z − α2 )n(m2 −1) . . . (z − αs )n(ms −1) a(f (z)) = g(z), (z − β1 )n(n1 +1) (z − β2 )n(n2 +1) . . . (z − βt )n(nt +1) ′
n
where g(z) is a polynomial and (2.6)
deg(g(z)) ≤ n(s + t − 1).
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From (2.1) and (2.5), we get (z − α1 )m1 (z − α2 )m2 . . . (z − αs )ms (z − β1 )n1 (z − β2 )n2 . . . (z − βt )nt (z − α1 )n(m1 −1) (z − α2 )n(m2 −1) . . . (z − αs )n(ms −1) + g(z) (z − β1 )n(n1 +1) (z − β2 )n(n2 +1) . . . (z − βt )n(nt +1) (z − α1 )m1 (z − α2 )m2 . . . (z − αs )ms h(z) = (z − β1 )n(n1 +1) (z − β2 )n(n2 +1) . . . (z − βt )n(nt +1) P = , Q
D(f ) = f + a(f ′ )n = A
(2.7)
where P, Q and h(z) are polynomials and (2.8)
deg(h(z)) ≤ max {(n − 1)N + nt, (n − 1)M − ns + deg(g(z))}.
Since D(f ) has exactly one b point at z0 (say). We get from (2.7) D(f ) = f + a(f ′ )n = b + =
(2.9)
B(z − z0 )l (z − β1 )n(n1 +1) (z − β2 )n(n2 +1) . . . (z − βt )n(nt +1)
P , Q
where B is a nonzero constant and l is a positive integer. On differentiating (2.6) and (2.9), we get
(2.10)
{D(f )}′ =
(z − α1 )m1 −1 (z − α2 )m2 −1 . . . (z − αs )ms −1 h1 (z) (z − β1 )n(n1 +1)+1 (z − β2 )n(n2 +1)+1 . . . (z − βt )n(nt +1)+1
{D(f )}′ =
(z − z0 )l−1 h2 (z) . (z − β1 )n(n1 +1)+1 (z − β2 )n(n2 +1)+1 . . . (z − βt )n(nt +1)+1
and (2.11)
From (2.7) and (2.10), we have (D(f ))∞ = M + deg(h) − nN − nt and ({D(f )}′)∞ = M − s + deg(h1 ) − nN − nt − t. Since by Lemma 2.2, ({D(f )}′)∞ ≤ (D(f ))∞ − 1, so we get (2.12)
deg(h1 ) ≤ (s + t − 1) + deg(h).
Similarly from (2.9) and (2.11), we get (2.13)
deg(h2 ) ≤ t.
Now, since αi 6= z0 for i = 1, 2, . . . , s from (2.10) and (2.11), we see that (z − α1 )n(m1 −1) (z − α2 )n(m2 −1) . . . (z − αs )n(ms −1) is a factor of h2 . Therefore (2.14)
M − s ≤ deg(h2 ) ≤ t.
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From (2.1) and(2.14), we get
(2.15)
This implies
M −s ≤t M ≤s+t M N ≤ + 2 2 M < N.
Now we consider the following two cases. Case1.1
l 6= nN + nt. Then (D(f ))∞ ≥ 0, which shows nN + nt ≤ M + deg(h).
(2.16) Again we have two cases. Case 1.1.1.
deg(h) ≤ (n − 1)N + nt. Then from (2.16), we get nN + nt ≤ M + deg(h) ≤ M + (n − 1)N + nt.
(2.17)
N ≤ M,
This implies
which contradicts (2.15). Case 1.1.2.
deg(h) ≤ (n − 1)M − ns + deg(g). Then from (2.16), we have nN + nt ≤ M + deg(h) ≤ M + (n − 1)M − ns + deg(g)
which implies
≤ M + (n − 1)M − ns + n(s + t − 1) = nM + nt − n nN ≤ nM − n N + 1 ≤ M,
(2.18) which contradicts (2.15).
Case 1.2. l = nN + nt. Then from (2.10) and (2.11), we see that (z − z0 )l−1 is a factor of h1 . So we get (2.19)
l − 1 ≤ deg(h1 ) ≤ (s + t − 1) + deg(h).
Again, we consider two cases.
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Case 1.2.1.
deg(h) ≤ (n − 1)N + nt. Then from (2.19), l ≤ s + t + deg(h) this implies which gives from (2.2)
(2.20)
this implies
nN + nt ≤ s + t + (n − 1)N + nt N ≤s+t N M + N≤ 2 2 N < M,
which contradicts (2.15). Case 1.2.2. deg(h) ≤ (n − 1)M − ns + deg(h) ≤ (n − 1)M + nt − n. Then from (2.19) and (2.2), l ≤ s + t + (n − 1)M + nt − n
(2.21)
nN + nt ≤ s + t + (n − 1)M + nt − n M N nN ≤ + + (n − 1)M − n 2 2 1 1 (n − )N ≤ (n − )M − n 2 2 N < M − 1,
which contradicts (2.15). Let D(f ) − b has no zero and f is non-polynomial rational function. Now Case 2. putting l = 0 in (2.9) and proceeding as in case1.2.1, we arrive at a contradiction. This proves the lemma. Lemma 2.5. Let f be a non-constant polynomial and f has no zero of multiplicity less than 3 in D. Then D(f ) − b has at least two distinct zeros, where b is a non-zero complex number. Proof. Since f is a non-constant polynomial with zeros of multiplicity ≥ 3. So D(f ) − b will be a polynomial of degree at least 4, hence it has a zero. Let us assume D(f ) − b has exactly one zero at z0 . So, we can write (2.22)
D(f ) − b = f + a(f ′ )n − b = A(z − z0 )m
where A is non-zero constant and m ≥ 4. Now differentiating both sides of (2.22) we get (2.23)
{D(f ) − b}′ = f ′ {1 + na(f ′ )n−2 f ′′ } = Am(z − z0 )m−1 .
which shows that z0 is the only zero of {D(f ) − b}′ = f ′ {1 + na(f ′ )n−2 f ′′ }. Clearly zeros of f ′ and 1 + na(f ′ )n−2 f ′′ are zeros of {D(f ) − b}′ . Also 1 + na(f ′ )n−2 f ′′ has at least one zero as f ′′ is non-constant and as zeros of f ′ and 1 + na(f ′ )n−2 f ′′ are different we arrive at a contradiction. Example 2.6. Take f (z) = z 2 , a = − 41 , b = 1 and n = 2. It is easy to check D(f ) − b has no zeros. This example shows that condition on multiplicity can not be weakened.
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3. Proof of Theorem 1.3 Since normality is a local property, we assume that D = ∆ = {z ∈ C : |z| < 1}. Suppose that F is not normal in ∆. Then there exists at least one point z0 such that F is not normal at the point z0 in ∆. Without loss of generality we assume that z0 = 0. Then by Lemma 2.1, there exist (1) a number r with 0 < r < 1, (2) points zn satisfying |zn | < r, (3) functions fn ∈ F , (4) positive numbers ρn → 0 as n → ∞, such that (3.1)
fj (zj + ρj ζ) = gj (ζ) → g(ζ)
locally uniformly with respect to spherical metric, where g(ζ) is a nonconstant meromorphic function on C. The zeros of g(ζ) are of multiplicity at least 3 and has no simple poles. Moreover g(ζ) is of order at most 2. We see that (3.2)
D(fj (zj + ρj ζ)) − b = D(gj (ζ)) − b → D(g(ζ)) − b as j → ∞
locally uniformly with respect to spherical metric. If D(g(ζ)) ≡ b, then g(ζ) has no zeros and poles. Since g(ζ) is a non constant meromorphic function of order at most 2, then there exist constant c1 and c2 such that (c1 , c2 ) 6= (0, 0), and 2 g(ζ) = ec0 +c1 ζ+c2ζ . Clearly, this is contrary to the case D(g(ζ)) ≡ b. Hence D(g(ζ)) 6≡ b. Since g(ζ) is a non-constant meromorphic function, by Lemma 2.3, Lemma 2.4 and Lemma 2.5, D(g(ζ)) − b has at least two distinct zeros. Let ζ0 and ζ0∗ be two distinct zeros of D(g(ζ)) − b. We choose δ > 0 small enough such that D(ζ0 , δ) ∩ D(ζ0∗ , δ) = ∅ and D(g(ζ)) − b has no other zeros in D(ζ0, δ) ∪ D(ζ0∗, δ), where D(ζ0 , δ) = {ζ : |ζ − ζ0 | < δ} and D(ζ0∗, δ) = {ζ : |ζ − ζ0∗ | < δ}. By Hurwitz’s theorem, there exist two sequences {ζj } ⊂ D(ζ0 , δ), {ζj∗} ⊂ D(ζ0∗, δ) converging to ζ0 , and ζ0∗ respectively and from (3.2), for sufficiently large j, we have D(fj (zj + ρj ζj )) − b = 0 and D(fj (zj + ρj ζj∗)) − b = 0. Since, by assumption that D(fj ) and D(fm ) share b in D = ∆ for each pair fj and fm in F , it follows that D(fm (zj + ρj ζj )) − b = 0 and D(fm (zj + ρj ζj∗)) − b = 0. We fix m and letting j → ∞, and noting zj + ρj ζj → 0, zj + ρj ζj∗ → 0, we obtain D(fm (0)) − b = 0. Since the zeros of D(f (z)) − b are isolated and we have zj + ρj ζj = 0, zj + ρj ζj∗ = 0. Hence z z ζj = − ρjj , ζj∗ = − ρjj which is not possible as D(ζ0, δ) ∩ D(ζ0∗δ) = ∅. This completes the proof.
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4. Proof of Theorem 1.4 Since normality is a local property, we assume that D = ∆ = {z ∈ C : |z| < 1}. Suppose that F is not normal in ∆. Then there exists at least one point z0 such that F is not normal at the point z0 in ∆. Without loss of generality we assume that z0 = 0. Then by Lemma 2.1, there exist (1) (2) (3) (4)
a number r with 0 < r < 1, points zn satisfying |zn | < r, functions fn ∈ F , positive numbers ρn → 0 as n → ∞,
such that (4.1)
fj (zj + ρj ζ) = gj (ζ) → g(ζ)
locally uniformly with respect to spherical metric, where g(ζ) is a non-constant meromorphic function on C. The zeros of g(ζ) are of multiplicity at least 3 and has no simple poles. Moreover g(ζ) is of order at most 2. We see that (4.2)
D(fj (zj + ρj ζ)) − b = D(gj (ζ)) − b → D(g(ζ)) − b as j → ∞
locally uniformly with respect to spherical metric. Now we claim that D(g) − b has at most 1 zero IM. Suppose that D(g) − b has two distinct zeros ζ0 and ζ0∗ and choose δ > 0 small enough such that D(ζ0 , δ)∩D(ζ0∗, δ) = ∅ and D(g(ζ)) − b has no other zeros in D(ζ0, δ) ∪ D(ζ0∗, δ), where D(ζ0 , δ) = {ζ : |ζ − ζ0 | < δ} and D(ζ0∗, δ) = {ζ : |ζ − ζ0∗ | < δ}. By Hurwitz’s theorem, there exist two sequences {ζj } ⊂ D(ζ0 , δ), {ζj∗} ⊂ D(ζ0∗, δ) converging to ζ0 , and ζ0∗ respectively and from (4.2), for sufficiently large j, we have D(fj (zj + ρj ζj )) − b = 0 and D(fj (zj + ρj ζj∗ )) − b = 0 Since zj → 0 and ρ → 0, we have zj + ρj ζj ∈ D(ζ0, δ) and zj + ρj ζj∗ ∈ D(ζ0∗ , δ) for sufficiently large j, so D(fj (z)) − b has two distinct zeros, which contradicts the fact that D(fj (z)) − b has at most one zero. But Lemma 2.3, Lemma 2.4 and Lemma 2.5 confirms the non existence of such non-constant meromorphic function. This contradiction shows that F is normal in ∆ and this proves the theorem.
5. Proof of Theorem 1.5 Since normality is a local property, we assume that D = ∆ = {z ∈ C : |z| < 1}. Suppose that F is not normal in ∆. Then there exists at least one point z0 such that F is not normal at the point z0 in ∆. Without loss of generality we assume that z0 = 0. Then by Lemma 2.1, there exist (1) (2) (3) (4)
a number r with 0 < r < 1, points zn satisfying |zn | < r, functions fn ∈ F , positive numbers ρn → 0 as n → ∞,
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such that (5.1)
fj (zj + ρj ζ) = gj (ζ) → g(ζ)
locally uniformly with respect to spherical metric, where g(ζ) is a non-constant meromorphic function on C. The zeros of g(ζ) are of multiplicity at least 3 and has no simple poles. Moreover g(ζ) is of order at most 2. We see that (5.2) D(fj (zj + ρj ζ)) − α(zj + ρj ζ) = D(gj (ζ)) − α(zj + ρj ζ) → D(g(ζ)) − α(0) as j → ∞ locally uniformly with respect to spherical metric. Since g(ζ) is a non-constant meromorphic function, by Lemma 2.3, Lemma 2.4 and Lemma 2.5, D(g(ζ)) − α(0) has at least two distinct zeros, say ζ0 and ζ0∗. So we have (5.3)
D(g(ζ0) − α(0) = 0; D(g(ζ0∗) − α(0) = 0 and ζ0 6= ζ0∗ .
We choose δ > 0 small enough such that D(ζ0 , δ) ∩ D(ζ0∗ , δ) = ∅ and D(g(ζ)) − α(0) has no other zeros in D(ζ0 , δ)∪D(ζ0∗, δ), where D(ζ0 , δ) = {ζ : |ζ −ζ0| < δ} and D(ζ0∗, δ) = {ζ : |ζ − ζ0∗| < δ}. By Hurwitz’s theorem, there exist two sequences {ζj } ⊂ D(ζ0, δ), {ζj∗} ⊂ D(ζ0∗ , δ) converging to ζ0 , and ζ0∗ respectively and from (5.2), for sufficiently large j, we have D(fj (zj + ρj ζj )) − α(zj + ρj ζj ) = 0 and D(fj (zj + ρj ζj∗)) − α(zj + ρj ζj∗ ) = 0. Since, by assumption that D(fj ) and D(fm ) share α(z) IM in D = ∆ for each pair fj and fm in F , it follows that D(fm (zj + ρj ζj )) − α(zj + ρj ζj ) = 0 and D(fm (zj + ρj ζj∗ )) − α(zj + ρj ζj∗ ) = 0. We fix m and letting j → ∞, and noting zj + ρj ζj → 0, zj + ρj ζj∗ → 0, we obtain D(fm (0)) − α(0) = 0. Since the zeros are isolated and we have zj + ρj ζj = 0, zj + ρj ζj∗ = 0. Hence ζj = z z − ρjj , ζj∗ = − ρjj which is not possible as D(ζ0 , δ) ∩ D(ζ0∗δ) = ∅. This completes the proof. 6. Proof of Theorem 1.6 Since normality is a local property, we assume that D = ∆ = {z ∈ C : |z| < 1}. Suppose that F is not normal in ∆. Then there exists at least one point z0 such that F is not normal at the point z0 in ∆. Without loss of generality we assume that z0 = 0. Then by Lemma 2.1, there exist (1) a number r with 0 < r < 1, (2) points zn satisfying |zn | < r, (3) functions fn ∈ F , (4) positive numbers ρn → 0 as n → ∞, such that (6.1)
fj (zj + ρj ζ) = gj (ζ) → g(ζ)
locally uniformly with respect to spherical metric, where g(ζ) is a non-constant meromorphic function on C. The zeros of g(ζ) are of multiplicity at least 3 and has no simple poles. Moreover g(ζ) is of order at most 2. We see that (6.2) D(fj (zj + ρj ζ)) − α(zj + ρj ζ) = D(gj (ζ)) − α(zj + ρj ζ) → D(g(ζ)) − α(0) as j → ∞
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locally uniformly with respect to spherical metric. Now we claim that D(g) − α(0) has at most one zero IM. Suppose that D(g) − α(0) has two distinct zeros ζ0 and ζ0∗ and choose δ > 0 small enough such that D(ζ0 , δ)∩D(ζ0∗, δ) = ∅ and D(g(ζ))−α(0) has no other zeros in D(ζ0 , δ)∪D(ζ0∗ , δ), where D(ζ0 , δ) = {ζ : |ζ −ζ0 | < δ} and D(ζ0∗ , δ) = {ζ : |ζ − ζ0∗| < δ}. By Hurwitz’s theorem, there exist two sequences {ζj } ⊂ D(ζ0 , δ), {ζj∗} ⊂ D(ζ0∗, δ) converging to ζ0 , and ζ0∗ respectively and from (6.2), for sufficiently large j, we have D(fj (zj + ρj ζj )) − α(0) = 0 and D(fj (zj + ρj ζj∗ )) − α(0) = 0 Since zj → 0 and ρ → 0, we have zj + ρj ζj ∈ D(ζ0, δ) and zj + ρj ζj∗ ∈ D(ζ0∗ , δ) for sufficiently large j, so D(fj (z)) − α(0) has two distinct zeros, which contradicts the fact that D(fj (z)) − α(z0 ) has at most one zero. But Lemma 2.3, Lemma 2.4 and Lemma 2.5 confirms the non existence of such non-constant meromorphic function. This contradiction shows that F is normal in ∆ and this proves the theorem. Remark 6.1. For a family of non-polynomial meromorphic functions, the condition, zeros are of multiplicity ≥ 3 can be weakened by multiplicity ≥ 2 in theorem 1.3, theorem 1.4, theorem 1.5 and theorem 1.6. References [1] M. L. Fang, L. Zalcman, On the value distribution of f + a(f ′ )n , Sci. China Ser. A: Math, 51 (2008), no. 7, 1196–1202. [2] W. K. Hayman, Meromorphic Functions, Clarendon Press, Oxford, 1964. [3] Y. Li, Normal families of meromorphic functions with multiple zeros, J. Math. Anal. Appl., 381 (2011), 344–351. [4] D. W. Meng, P. C. Hu, Normality criteria of meromorphic functions sharing one value, J. Math. Anal. Appl., 381 (2011), 724–731. [5] X. C. Pang, L. Zalcman, Normal families and shared values, Bull. London Math. Soc., 32 (2000), 325–331. [6] J. Schiff, Normal Families, Springer-Verlag, Berlin, 1993. [7] W. Schwick, Sharing Values and Normality, Arch. Math., 59 (1992), 50–54. [8] Y. F. Wang, M. L. Fang, Picard values and normal families of meromorphic functions with multiple zeros, Acta Math Sinica, new series, 14 (1998), no. 1, 17–26. [9] X. Wu, Y. Xu, Normal families of meromorphic functions and shared vales, Monatsh Math., 165 (2012) no. 3–4, 569–578. [10] C. C. Yang, H. X. Yi, Uniqueness theory of meromorphic functions, Science Press/ Kluwer Academic Publishers, 2003. [11] L. Yang, Value Distribution Theory, Springer-Verlag, Berlin, 1993. [12] L. Zalcman, A heuristic principle in complex function theory, The American Mathematical Monthly, 82 (1975), 813–817. [13] C. Zeng, Normality and shared values with multiple zeros, J. Math. Anal. Appl., 394 (2012), 683– 686. [14] S. Zeng, I. Lahiri, A normality criterion for meromorphic functions, Kodai Math. J., 35 (2012), 105–114. [15] S. Zeng, I. Lahiri, A normality criterion for meromorphic functions having multiple zeros, Ann. Polon. Math.(to appear)
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Department of Mathematics, University of Delhi, Delhi–110 007, India E-mail address:
[email protected] Department of Mathematics, Deen Dayal Upadhyaya College, University of Delhi, Delhi–110 015, India E-mail address:
[email protected]
