A Criterion for Quasinormality in Cn

Proc. Indian Acad. Sci. (Math. Sci.) (2019) 129:11 https://doi.org/10.1007/s12044-018-0449-5

A criterion for quasinormality in Cn

GOPAL DATT1,2 and SANJAY KUMAR3,∗

1School of Mathematics, Harish-Chandra Research Institute (HBNI), Chhatnag Road, Jhunsi, Allahabad 211 019, India 2Department of Mathematics, Indian Institute of Science, Bangalore 560 012, India 3Department of Mathematics, Deen Dayal Upadhyaya College, University of Delhi, Delhi 110 078, India *Corresponding author. E-mail: [email protected]; [email protected]; [email protected]; [email protected]

MS received 18 April 2017; revised 8 December 2017; accepted 25 January 2018; published online 18 December 2018

Abstract. In this article, we give a Zalcman type renormalization result for the quasinormality of a family of holomorphic functions on a domain in Cn that takes values in a complete complex Hermitian manifold.

Keywords. Analytic set; Holomorphic mapping; Normal family; Quasi-normal family.

2010 Mathematics Subject Classiﬁcation. 32A19.

1. Introduction The convergence of a family of functions always has far reaching consequences. In his path-breaking paper of 1907, Montel [10] gave a result of the convergence of the family of holomorphic functions which states that a sequence of uniformly bounded holomorphic functions has a subsequence that is locally uniformly convergent. Later in 1912, Montel [11] introduced the term normal family for a family satisfying this convergence property. In a subsequent paper [12], he introduced the notion of quasinormality of a family of functions in one complex variable. All these ideas are well documented in his influential book [13]. The normality of a family of functions is one of the most fundamental concepts in function theory of one and several complex variables. It has been extensively used in the study of dynamical properties of functions of one or more complex variables. Normality plays a deep role in certain results in complex dynamics. It can be seen that normality is the central part of the definition of the Fatou set of a holomorphic function f , which is defined as the domain of normality of the family of iterations of f . The Fatou set and the Julia set, simply by definition (as the complement of the Fatou set), set up a dichotomy. In a different direction, Beardon and Minda [3] discussed normal families in terms of maps that satisfy certain types of uniform Lipschitz conditions with respect to various conformal metrics and more background materials can be found in [4,13,18]. While all these provide sufficient conditions for normality, Zalcman in [22] proved a striking result that studies

consequence of non-normality. Roughly speaking, it says that in an inﬁnitesimal scaling, the family gives a non-constant entire function under the compact-open topology. We state this renormalization result which is known as Zalcman’s Lemma:

Zalcman’s Lemma. A family F of functions meromorphic (analytic) on the unit disc is not normal if and only if there exist (a) a number r, 0 < r < 1; (b) points z j , |z j | < r; (c) functions { f j }⊆F; + (d) numbers ρ j → 0 ; such that

f j (z j + ρ j ζ) → g(ζ )

spherically uniformly (uniformly) on compact subsets of C, where g is a non-constant meromorphic (entire) function on C.

This lemma leads to a heuristic principle in function theory. The principle says that any property which forces an entire function to be constant will also force a family of holomorphic functions to be normal. The source is Marty’s inequality which gives a necessary and sufficient condition for the normality of a family of holomorphic or meromorphic functions on a domain ⊂ C. It is very natural to explore the extension of Zalcman’s Lemma in several complex variables. In [2], Aladro and Krantz gave an analogue of Zalcman’s Lemma for families of n holomorphic mappings f j from a hyperbolic domain of C into a complete complex Her- mitian manifold M (also see, Lemma 5.1 [7]). Their analysis was completed by Thai, Trang and Huong in [20], which addresses the possibility of compact divergence of the renor- malized mappings g j (ζ ) = f j (z j + ρ j ζ). In the same paper, Thai et al. [20] also defined the concept of Zalcman space. Loosely speaking, a complex space X is a Zalcman space if for each non-normal family of holomorphic mappings of the unit disc {z ∈ C :|z| < 1} into X, we get a non-constant holomorphic mapping g : C → X under the compact-open topology after an infinitesimal scaling. This work is further studied in [19,21]. In this paper, our goal is to prove an analogue of Zalcman’s lemma for quasi-normal families in several complex variables (c.f. Theorem 1.1). We have illustrated our results with examples. The theory of quasinormality is well studied in one complex variable. Chuang, in his text [4], introduced the notion of Qm-normality (m ≥ 0) as an extension of quasinormality in the complex plane, Q0 and Q1-normality are the usual normality and quasinormality respectively. Loosely speaking, a Qm-normal family on a domain D is normal outside a subset of D whose m-th order derived set is empty. He introduced the notion of μm- point and established some characterizations of Qm-normality. Roughly speaking, a point z0 ∈ D is a μ1-point of a family F if the family violates the Marty’s criterion on z0 and μ2- point is the accumulation point of μ1-points. Inductively, a μm-point is an accumulation point of μm−1-points. In this paper, we extend the notions of μ1 and μ2-points in higher dimensions whereas we could not generalize the notion of μm-points for m ≥ 3 in several variables due to the nature of zeros of holomorphic mappings in higher dimensions. It seems that the ‘order of quasinormality’ as given in one variable is not plausible in higher dimension. It is interesting to note that using the notion of μm-points, Nevo [14] proved a Zalcman type renormalization result for Qm-normal families on planar domains . Proc. Indian Acad. Sci. (Math. Sci.) (2019) 129:11 Page 3 of 13 11

In several complex variables, the theory of quasinormality has its origin in the work of Rutishauser [17] and Fujimoto [6]. Fujimoto [6], extending the work of Rutishauser, introduced the notion of meromorphic convergence. In a recent article, Ivashkovich and Neji [7] discussed several notions of convergence, namely, strong convergence, weak convergence and gamma convergence. It can be seen easily from the deﬁnitions that weakly-normal implies quasi-normal. In this paper, we have also given a renormalization result for weakly- normal family of holomorphic mappings. It is instructive to note here a survey article [5] by Dujardin, where he gives a sufﬁcient condition for quasinormality of a family of holomorphic mappings from a complex manifold to a compact Kähler manifold in terms of a suitable sequence of bidegree (1,1) currents. The main result of this paper provides an analogue of the Zalcman’s lemma for the quasi-normal families. Our main result is as follows:

Theorem 1.1. Let ⊆ Cn be a hyperbolic domain. Let M be a complete complex Her- mitian manifold of dimension k. Let F ={fα}α∈A ⊆ Hol(, M). The family F is not quasi-normal if and only if there exist a subset E ⊂ which is either a non-analytic subset or the closure E¯ has non-empty interior and corresponding to each p ∈ E, there exist ( ) {w }∞ ⊂ w → a a sequence of points j,p j=1 such that j,p p, (b) a sequence of functions { f j }⊂F, (c) a sequence of positive real numbers ρ j,p → 0 such that

n g j (ζ ) := f j (w j,p + ρ j,pξ), ξ ∈ C (p ∈ E)

satisﬁes one of the following two assertions: n (i) The sequence {g j } is compactly divergent on C . n (ii) The sequence {g j } converges uniformly on compact subsets of C to a non-constant n holomorphic map gp : C → M.

Here we give an example to elucidate Theorem 1.1

{ ( , ) = n} C2 Example 1.2. Consider the family of holomorphic mappings fn z1 z2 z1 from into C . Clearly, fn is not normal in E ={(z1, z2) :|z1|=1}. Therefore, { fn} is not quasi- 2 normal in C . To see this, ﬁx 0 ≤ θ<2π and consider the sequence {z j }={(z1 j , z2 j )}, iθ/j where z1 j = e and z2 j ∈ C. For this sequence {z j } and ζ = (ζ1,ζ2), the sequence ζ +iθ f j (z j + ρ j ζ)converges to a non-constant holomorphic function e 1 .

2. Preliminary definitions and main results Let ⊂ Cn be an open domain and be the unit disc in C.Ifz ∈ and ξ ∈ Cn, then by the work of Royden [16], the infinitesimal form of the Kobayashi pseudo-metric for at z in the direction ξ is defined as ξ F(z,ξ)= : f : → f ( ) = z, K inf ( ) is holomorphic, 0 f f 0

and f (0) is a constant multiple of ξ , 11 Page 4 of 13 Proc. Indian Acad. Sci. (Math. Sci.) (2019) 129:11 where . represents the Euclidean length. The Kobayashi pseudo-distance between z and w in is defined as 1 K(z,w)= inf F (γ (t), γ (t))dt, γ K 0 where the infimum is taken over all C1-curves γ :[0, 1]→ such that γ(0) = z and γ(1) = w. In this work, we shall use the following definition of (Kobayashi) hyperbolicity which, as shown by Royden [16], is equivalent to the original definition.

DEFINITION 2.1 [2,16]

A domain ⊆ Cn is called hyperbolic at a point z ∈ if there is a neighborhood V of z in and a positive constant c such that

( ,ξ)≥ ξ ∈ ξ ∈ Cn. FK y c for all y V and all

We say that is hyperbolic if it is hyperbolic at each point.

Let M be a complete complex Hermitian manifold of dimension k and let Tp(M) denotes the complexiﬁed tangent space to M at p. We denote the metric for M at p in the n direction of the vector ξ ∈ Tp(M) by EM (p; ξ). Let ⊆ C be a hyperbolic domain. We denote the set of all holomorphic mappings from into M by Hol(, M).

DEFINITION 2.2

Let F be a family of holomorphic mappings of a domain in Cn into a complete complex manifold M. F is said to be a normal family on if F is relatively compact in Hol(, M) in the compact open topology.

DEFINITION 2.3

Let X, Y be complex spaces and F ⊂ Hol(X, Y ). A sequence { f j }⊂F is compactly divergent if for every compact K ⊂ X and for every compact L ⊂ Y there is a number J = J(K, L) such that f j (K )∩ L =∅for all j ≥ J. If F contains no compactly divergent sequences, then F is called not compactly divergent.

Let ⊆ Cn be a domain. A subset S of is called a complex analytic subset if for any z ∈ there exist a neighborhood U of z and holomorphic functions f1,..., fl on U such that S ∩ U ={z ∈ U : f1(z) = ... = fl (z) = 0}. Notice that analytic subsets are closed and nowhere dense in .

DEFINITION 2.4

n A sequence { f j } of holomorphic mappings from a domain ⊂ C into a complete complex Hermitian manifold M is said to be weakly-regular on if any z ∈ has a connected neighborhood U with the property that { f j } converges uniformly on compact subsets of U \ E or compactly diverges on U \ E, where E ⊂ U is an analytic subset of codimension at least 2. Proc. Indian Acad. Sci. (Math. Sci.) (2019) 129:11 Page 5 of 13 11

DEFINITION 2.5

Let F be a family of holomorphic mappings from a domain in Cn into a complete complex Hermitian manifold M. F is said to be a weakly-normal family on if any sequence in F has a weakly-regular subsequence on .

DEFINITION 2.6

n A sequence { f j } of holomorphic mappings from a domain ⊂ C into a complete complex Hermitian manifold M is said to be quasi-regular on if any z ∈ has a connected neighborhood U with the property that { f j } converges uniformly on compact subsets of U \ E or compactly diverges on U \ E, where E ⊂ U is a proper complex analytic subset of U.

DEFINITION 2.7

Let F be a family of holomorphic mappings from a domain in Cn into a complete complex Hermitian manifold M. F is said to be a quasi-normal family on if any sequence in F has a quasi-regular subsequence on .

Theorem A [1,2]. Let ⊆ Cn be a hyperbolic domain. Let M be a complete complex Hermitian manifold of dimension k with metric EM . Let F ={fα}α∈A ⊆ Hol(, M). If the family F ={fα}α∈A is a normal family, then for each compact set L (i.e., Lis n relatively compact in ), there is a constant CL such that for all z ∈ L and all ξ ∈ C , it holds that

| ( ( ); ( ) ( ).ξ)|≤ ( ,ξ). sup EM fα z fα ∗ z CL FK z (2.1) α∈A

Conversely, if (2.1) holds and if for some p ∈ ,all fα(p) are in some compact set Q of M, then F ={fα}α∈A is a normal family. Aladro and Krantz [2] gave an extension of the Zalcman’s lemma to the higher- dimensional setting. A case missing from the analysis in [2] was provided by Thai et al. [20]. Their result is as follows:

Theorem B [20]. Let be a domain in Cn. Let M be a complete complex Hermitian space. Let F ⊂ Hol(, M). Then the family F is not normal if and only if there exist sequences {p j }⊂ with p j → p0 ∈ , { f j }⊂F, {ρ j }⊂R with ρ j > 0 and ρ j → 0 such that

n g j (ξ) := f j (p j + ρ j ξ), ξ ∈ C

satisﬁes one of the following two assertions:

n (i) The sequence {g j } j≥1 is compactly divergent on C . n (ii) The sequence {g j } j≥1 converges uniformly on compact subsets of C to a non-constant holomorphic map g : Cn → M. 11 Page 6 of 13 Proc. Indian Acad. Sci. (Math. Sci.) (2019) 129:11

3. Proof of the main result Before giving the proof of our main result, we give some deﬁnitions and lemmas, whose one-dimensional analogue can be found in [4,14]. Throughout section 3, ⊆ Cn is a hyperbolic domain and M denotes a complete complex Hermitian manifold. Here we extend the notions μ1-point and μ2-point of a sequence in one dimension to a sequence { f j }⊂Hol (, M) in higher dimension.

DEFINITION 3.1

Let ⊆ Cn be a hyperbolic domain. Let M be a complete complex Hermitian manifold of dimension k. Consider a sequence { f j }⊂Hol(, M). A point p0 ∈ is said to be a μ1-point of { f j }, if for each subset K containing p0,

lim sup |EM ( f j (p); ( f j )∗(p) · ξ)|=∞. →∞ j p∈K, ξ =1

(1) A point p0 is called a μ2-point of { f j } if there exists an analytic set K ⊂ of codimension at most 1, containing p0, such that each point of K is a μ1-point of { f j }. (2) We say p0 is a q-point of { f j } if there exists a subset K ⊂ , containing p0, such that ¯ closure K = and each point of K is a μ1-point of { f j }. (3) We say p0 is a λ-point of { f j } if there exists a non-analytic subset K ⊂ containing p0 such that each point of K is a μ1-point of { f j }.

2 Example 3.2. Let { fn} be a family of holomorphic mappings from C onto itself such that fn(z) = nz, where z = (z1, z2). Then z = (0, 0) is a μ1-point of { fn}.

Example 3.3. Let { fn} be a family of holomorphic mappings deﬁned on the polydisc D ={(z1, z2) :|z1| < 1 and |z2| < 1} such that fn(z1, z2) = nz1z2. Then each point of E ={(z1, z2) : z1z2 = 0} is a μ2-point of { fn}.

2 Example 3.4. Let { fn} be a family of holomorphic mappings deﬁned on C such that nz fn(z1, z2) = e 1 . Then each point of E ={(z1, z2) : Re z1 = 0} is a λ-point of { fn}.

Lemma 3.5. Let ⊆ Cn be a hyperbolic domain. Let M be a complete complex Hermitian manifold of dimension k. A family F ⊂ Hol(, M) is normal in if and only if each sequence { f j } of F has no μ1-point in .

Proof. Suppose F is normal. Then by Theorem A there is no μ1-point for any sequence { f j } of F. Let F ⊂ Hol(, M) such that every sequence { fn} of functions of F has no μ1-point in . Then (2.1) holds and { fn} is not compactly divergent. Otherwise, there is a point p0 ∈ such that we can not ﬁnd a ball ={p : p − p0 < r}, and a number N > 0 such that for j ≥ 1, we have

¯ |EM ( f j (p); ( f j )∗(p).ξ)|≤N in . Proc. Indian Acad. Sci. (Math. Sci.) (2019) 129:11 Page 7 of 13 11

Take two sequences of positive real numbers {rk}→0 and {Nk}→∞such that the ball ¯ k ={p : p − p0 ≤rk} is contained in . Then there is an integer j1 ≥ 1 such that

| ( ( ); ( ) ( ) · ξ)| > . sup EM f j1 p f j1 ∗ p N1 ¯ p∈ 1, ξ =1

Next there is an integer j2 > j1 such that

| ( ( ); ( ) ( ) · ξ)| > . sup EM f j2 p f j2 ∗ p N2 ¯ p∈ 2, ξ =1

Continuing in this manner, we get a sequence of integers { jk},(k = 1, 2,...)such that for k ≥ 1, we have

| ( ( ); ( ) ( ) · ξ)| > . sup EM f jk p f jk ∗ p Nk ¯ p∈ k , ξ =1 ¯ Now consider a ball : p − p0 < r such that . Let k0 ≥ 1 be an integer such that rk < r for k ≥ k0. Then for k ≥ k0,wehave

< | ( ( ); ( ) ( ) · ξ)| Nk sup EM f jk p f jk ∗ p ¯ p∈ k , ξ =1 ≤ | ( ( ); ( ) ( ) · ξ)|. sup EM f jk p f jk ∗ p p∈ ,¯ ξ =1

Hence

lim sup |EM ( f j (p); ( f j )∗(p) · ξ)|=∞. →∞ k k k p∈ ,¯ ξ =1

This implies p0 is a μ1-point of the sequence { f j } which is a contradiction.

Lemma 3.6. Let ⊆ Cn be a hyperbolic domain. Let M be a complete complex Hermitian manifold of dimension k. A family F ⊂ Hol(, M) is weakly-normal in if and only if each sequence { f j } of F has neither a μ2-point nor a λ-point in .

Proof. Suppose that F is weakly-normal in .Let{ f j } be a sequence of functions of F { } { } . Then we can extract a weakly-regular subsequence f jk from f j . On the contrary, we assume that { f j } has a μ2-point p0 in . Since F is weakly-normal, therefore we { } can ﬁnd a neighborhood U0 of p0 in such that f jk converges uniformly on compact subsets of U0 \ E, or diverges compactly on U0 \ E, where E is an analytic subset of U0 of ∈ \ { } codimension at least 2. For each p U0 E, f jk converges or diverges compactly and { } \ { } μ \ . hence f jk is normal in U0 E.SobyLemma3.5, f jk has no 1-point in U0 E But by deﬁnition of μ2-point, there exist an analytic set K ⊂ of codimension at most 1, such μ { } { } that each point of K is a 1-point of f jk and hence of f j , which is a contradiction. A similar argument can be given if p0 is a λ-point. Conversely, suppose that F has neither a μ2-point nor a λ-point and F is not weakly- normal on . Consider a set K .Let{ f j } be a sequence of functions of F. Then we can not extract a subsequence which is weakly-regular in K . It follows from Lemma 3.5 11 Page 8 of 13 Proc. Indian Acad. Sci. (Math. Sci.) (2019) 129:11 that { f j } must have μ1-points in K . Also, the set V of all μ1-points contains either a non-empty analytic subset V1 ⊂ of codimension at most 1 or a non-analytic set V2 ⊂ , otherwise { f j } constitutes a weakly-normal family. Since V1 is a set of codimension at most 1, then for p ∈ V1 there exists a neighborhood N1 of p and each point of N1 ∩ V1 is a μ1-point of { f j }. Thus p is a μ2-point of { f j }.Also,V2 is a non-analytic set. Then for p ∈ V2, there exists a neighborhood N2 of p and each point of N2 ∩ V2 is a μ1-point of { f j }. Thus p is a λ-point of { f j }. In either case, we get a contradiction.

We can prove the following result on similar lines.

Lemma 3.7. Let ⊆ Cn be a hyperbolic domain. Let M be a complete complex Hermitian manifold of dimension k. A family F ⊂ Hol(, M) is quasi-normal in if and only if each sequence { f j } of F has neither a q-point nor a λ-point in .

We now give a local version of Zalcman’s lemma for normal families. Lemma 3.8 is a local version of Theorem B. In what follows, the term normality (or quasi-normality) at a point z0 will be normality (or quasi-normality) on an open neighborhood of the point z0.

Lemma 3.8. Let ⊆ Cn be a hyperbolic domain. Let M be a complete complex Hermitian manifold of dimension k. Let F ={fα}α∈A ⊆ Hol(, M). The family F is not normal at p0 ∈ if and only if there exist (a) a sequence {p j }⊂ such that p j → p0; (b) a sequence of functions { f j }⊂F; (c) a sequence of positive real numbers ρ j → 0 such that

n g j (ξ) := f j (p j + ρ j ξ), ξ ∈ C satisﬁes one of the following two assertions: n (i) The sequence {g j } is compactly divergent on C . n (ii) The sequence {g j } converges uniformly on compact subsets of C to a non-constant holomorphic map g : Cn → M.

Proof. Assume that F is not normal at p0. Then by Theorem A, there exists a compact set K0 ⊂{p : p − p0 ≤ρ}=K1 for some ρ>0 and a sequence f j ⊂ F, {q j }⊂K0 and n {ξ j }⊂C , such that

| ( ( ); ( ) ( ) · ξ )|≥ ( ,ξ ). EM f j q j f j ∗ q j j jFK q j j (3.1)

1 Let k0 ∈ N be such that √ <ρ. Then for k ≥ k0, there is fk ∈ F with k0 |E ( f (q ); ( f )∗(q ) · ξ )|≥kF (q ,ξ ), for all k M k k k k k K k k 1 ≥ k0 and qi ∈ p : p − p0 ≤ √ . (3.2) 2 k

Now deﬁne for k ≥ k0, p gk(p) = fk p0 + √ . k Proc. Indian Acad. Sci. (Math. Sci.) (2019) 129:11 Page 9 of 13 11

Each gk is defined on B(0, 1) ={p : p < 1} and satisfies qk 1 qk |EM (gk (qk ); ( fk )∗(qk ) · ξk |=EM fk p0 + √ ; √ ( fk )∗ p0 + √ · ξk √ k k k ≥ ( ,ξ), kFK qk (3.3) therefore {gk} is not normal in B(0, 1). Now by Theorem B, there exist (1) a compact set K B(0, 1), ( ) { ∗}⊂ 2 a sequence p j K , ( ) { }⊂{ } 3 a sequence gk j gk , ( ) ρ∗ → 4 a sequence of positive real numbers j 0 (ξ) = ∗ + ρ∗ξ ,ξ∈ Cn Cn such that hk j gk j p j j either compactly diverges on or converges uniformly on compact subsets of Cn to a non-constant holomorphic map g : Cn → p∗ ρ∗ + √ j + j ξ ,ξ∈ Cn Cn M.Thisissameas fk j p0 either compactly diverges on k j k j or converges uniformly on compact subsets of Cn to a non-constant holomorphic map g : Cn → M.Nowset ∗ ρ∗ p j j p j = + p0 and ρ j = . k j k j { ∗}⊂ Since the sequence p j K is bounded, its limit points will be of finite modulus (in fact, of modulus less than one), and the subsequence {k j } of natural numbers converges to ∞ ∗ p j as j goes to ∞. Consequently, converges to zero and we get that p j converges to p0. k j Also, ρ j converges to 0. This proves the necessity part of the lemma. Conversely, assume that the conditions of the lemma are satisfied and suppose, on the contrary, that F is normal at p0. Then by Theorem A, for compact subsets K0 and K1 with p0 ∈ K0 K1 , there exists a number N > 0 such that

sup |EM ( f (p); ( f )∗(p) · ζ)|≤N, for each f ∈ F. (3.4) p∈K1, ζ =1

n Now, suppose g j (ξ) = f j (p j + ρ j ξ) converges uniformly on compact subsets of C to a non-constant holomorphic map g : Cn → M.Wehave

| ( (ξ); (ξ) · ζ)|=| ( ( + ρ ξ); ρ ( ) ( + ρ ξ)· ζ)| EM g j g j EM f j p j j j f j ∗ p j j

≤ ρ j N. (3.5)

Taking the limit, we get

| ( (ξ); (ξ).ζ )|=| ( (ξ); (ξ) · ζ)|= . lim EM g j g j EM g g 0 j→∞

Then g (ξ) = 0 for every ξ ∈ Cn, therefore g is a constant function which is a contradiction. 11 Page 10 of 13 Proc. Indian Acad. Sci. (Math. Sci.) (2019) 129:11

Next, suppose that g j (ξ) = f j (p j + ρ j ξ) is compactly divergent. Since the family F is normal, without any loss of generality, we may assume that the sequence { f j }→ f. Andwegetg j (ξ) → f (p0), which is not possible as {g j } is compactly divergent. This completes the proof.

2 Example 3.9. Let D ={(z1, z2) :|z1| < 1 and|z2| < 1} be the polydisc in C .We nz z consider a family of holomorphic mappings { fn} from D into C, where fn(z1, z2) = e 1 2 for all n ∈ N. Since { fn} has no subsequence which is convergent at any point in the set E ={{(Re(z1), 0) × (0, Im(z2))}∪{(0, Im(z1)) × (Re(z2), 0)}} ∩ D so { fn} is not normal in D.As { fn} is not normal at (0, 0), we get a sequence {pn} in D such that z0 z0 p = √1 , √2 , where (z0, z0) is a ﬁxed point in D. Notice that {p }→(0, 0). Also, n n n 1 2 n we have a sequence of positive real numbers {ρ }→0, where ρ = √1 such that for all n n n 2 ξ = (z1, z2) ∈ C ,wehave

0 0 (z +z1)(z +z2) gn(pn + ρnξ) = fn(pn + ρnξ) → e 1 2 .

Now we are ready to prove Theorem 1.1.

Proof of Theorem 1.1. Suppose all the conditions of the theorem are satisﬁed. Since E is ¯ either a non-analytic subset or the closure E has non-empty interior, then for each p0 ∈ E, we can get a sequence p j in E such that p j → p0. By Lemma 3.8, F is not normal at p0. Since p0 is an arbitrary point of E and E is either a dense subset or a non-analytic subset of , F is not quasi-normal in . Conversely, suppose F is not a quasi-normal family in . Then by Lemma 3.7, there exists a sequence S ={h j } of F which has either a q-point or a λ−point p0 ∈ .This implies that there exists a subset V which is either dense in or a non-analytic subset containing p0 so that each point of V is a μ1-point of S . Since V is either dense or non-analytic, we can choose a sequence of positive real numbers {ri } such that {ri }→0 and for each open ball B(p0, ri ) ={p ∈ : p − p0 < ri },thesetV ∩ B(p0, ri ) has at least one μ1-point. Now we proceed inductively to get the conditions of the theorem. Step 1. There exists

(A1) a μ1-point p1 ∈ such that p1 ∈ V ∩ B(p0, r1). So S is not normal at p1. Therefore, by Lemma 3.8 there exist (B1) a sequence {w j,1}⊂ such that {w j,1}→p1;

(C1) a subsequence S1 ={h j,1} of S ; (D1) a sequence of positive real numbers {ρ j,1}→0 such that n n h j,1(w j,1 + ρ j,1ξ), ξ ∈ C either compactly diverges on C or converges uniformly on n n compact subsets of C to a non-constant holomorphic map g1 : C → M.

Step 2. Since p0 is also a q-point or a λ-point of S1, there exists

(A1) a μ1-point p2 ∈ , p2 = p1 (by Deﬁnition 3.1) such that p2 ∈ V ∩ B(p0, r2), 0 < r2 < r1. So S1 is not normal at p2. Therefore, by Lemma 3.8, there exist (B1) a sequence {w j,2}⊂ such that {w j,2}→p2; (C1) a subsequence S2 ={h j,2} of S1; (D1) a sequence of positive real numbers {ρ j,2}→0 such that Proc. Indian Acad. Sci. (Math. Sci.) (2019) 129:11 Page 11 of 13 11

n n h j,2(w j,2 + ρ j,2ξ), ξ ∈ C , either compactly diverges on C or converges uniformly on n n compact subsets of C to a non-constant holomorphic map g2 : C → M. Continuing in this manner, we get sequences {p j }→p0, {wi, j }, {ρi, j }, {g j } and {hi, j }. = = ; w = Now we use the Cantor’s diagonal method and, choose E V ; fi hi,i i,po w ; ρ = ρ . ≥ { }∞ (w + i,i i,p0 i,i Then for each j 1, fi i= j is a subsequence of S j and fi i,p0 ρ ξ),ξ ∈ Cn, Cn i,p0 either compactly divergent on or converges uniformly on compact Cn : Cn → subsets of to a non-constant holomorphic map gp0 M. This completes the proof of the theorem.

For weakly-normal family, we propose the following theorem.

Theorem 3.10. Let ⊆ Cn be a hyperbolic domain. Let M be a complete complex Hermitian manifold of dimension k. Let F ={fα}α∈A ⊆ Hol(, M). The family F is not weakly-normal if and only if there exist a subset E ⊂ which is not locally contained in an analytic subset of codimension 2, and corresponding to each p ∈ E, there exist {w }∞ ⊂ w → (a) a sequence of points j,p j=1 such that j,p p, (b) a sequence of functions { f j }⊂F, (c) a sequence of positive real numbers ρ j,p → 0 such that

n g j (ζ ) := f j (w j,p + ρ j,pξ), ξ ∈ C (p ∈ E) satisﬁes one of the following two assertions: n (i) The sequence {g j } is compactly divergent on C . n (ii) The sequence {g j } converges uniformly on compact subsets of C to a non-constant n holomorphic map gp : C → M.

The proof of Theorem 3.10 is merely a formality. It can be proven on similar lines as in the proof of Theorem 1.1 using Lemma 3.6 instead of Lemma 3.7. The following examples elucidate Theorem 3.10.

Example 3.11. Let { fn} be a family of holomorphic mappings deﬁned on the polydisc D ={(z1, z2) :|z1| < 1 and|z2| < 1} such that fn(z1, z2) = nz1z2. Then { fn} is not weakly-normal on D,as{ fn} converges compactly in D \ E, where E ={(z1, z2) : z1z2 = } ( 0, 0) ∈ 0 is an analytic subset of codimension 1 of D.Let z1 z2 E be any arbitrary point. 0 = . { }→( , 0) Without loss of generality, we take z 1 0 Then we get a sequence pn 0 z2 of points in E, where p = 0, z0 + √1 . Also, we have a sequence of positive real numbers n 2 n {ρ }→ , ρ = 1 ξ = ( , ) ∈ C2 n 0 where n n such that for all z1 z2 , we get

( + ρ ξ) = ( + ρ ξ ) → 0 . gn pn n fn pn n n z2z1

Example 3.12. Let { fn} be a family of holomorphic mappings deﬁned on the polydisc D ={(z1, z2) :|z1| < 1 and |z2| < 1} such that fn(z1, z2) = cos(nz1z2). Then { fn} is not weakly-normal on D as { fn} is not compactly convergent in any open subset of D ={( , ) : = }, ( 0, 0) containing E z1 z2 z1z2 0 which is of codimension 1. Let z1 z2 be any 0 = . arbitrary point of E. Without loss of generality, we take z2 0 Then we get a sequence 11 Page 12 of 13 Proc. Indian Acad. Sci. (Math. Sci.) (2019) 129:11 {p }→(z0, 0) of points in E, where p = z0 + √1 , 0 . Also, we have a sequence of n 1 n 1 n {ρ }→ , ρ = 1 ξ = ( , ) ∈ C2 positive real numbers n 0 where n n such that for all z1 z2 ,we obtain

( + ρ ξ) = ( + ρ ξ) → ( 0 ). gn pn n fn pn n cos z1z2

Remark 3.13. One can observe that the notions of normality and quasinormality are the local phenomena, and therefore our results holds true even if hyperbolicity is dropped.

Acknowledgements The authors would like to thank Gautam Bharali and Kaushal Verma, IISc, Bangalore for stimulating discussions about this work as well as for critical comments. They are also grateful to the referee for his diligence and helpful suggestions to enhance the quality of the paper. The ﬁrst author is thankful to the organizers of HAYAMA Symposium on Complex Analysis of Several Variables XVIII, held in Japan (July 2016) for giving him an opportunity to present this work. The research work of the ﬁrst author is supported by the postdoctoral fellowship of Harish-Chandra Research Institute, Allahabad and the National Postdoctoral Fellowship SERB (DST) India. The research of the second author is supported by a Minor Research Project grant of UGC (India).

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Communicating Editor: Kaushal Verma