NORMAL CRITERIA FOR MEROMORPHIC FUNCTIONS

In 2011, Aulaskari and R.. atty .. a (J. Michigan. Math. Vol. 60) introduced the concept of ϕ normal meromorphic functions on the unit disc D, where the function ϕ(r) : 0, 1) → (0, ∞) admits a sufficient regularity near 1 and exceeds 1 1−r2 in growth. They examined the class of meromorphic functions f on D satisfying f (z) = O(ϕ(|z|), as |z| → 1 −. In this paper, we establish some ϕ normal criteria for meromorphic functions. Moreover, as a corollary of our results, for a special case of ϕ, we obtain some normal criteria for meromorphic functions under a condition where functions and their derivative share a set (rather than just sharing a value as in known results)

ϕ- NORMAL CRITERIA FOR MEROMORPHIC FUNCTIONS TRAN VAN TANA,∗ AND NGUYEN VAN THINB .. .. Abstract. In 2011, Aulaskari and Rattya (J. Michigan. Math. Vol. 60) introduced the concept of ϕ- normal meromorphic functions on the unit disc D, where the function ϕ(r) : [0, 1) → (0, ∞) admits a sufficient regularity 1 near 1 and exceeds 1−r 2 in growth. They examined the class of meromorphic functions f on D satisfying f # (z) = O(ϕ(|z|), as |z| → 1− . In this paper, we establish some ϕ- normal criteria for meromorphic functions. Moreover, as a corollary of our results, for a special case of ϕ, we obtain some normal criteria for meromorphic functions under a condition where functions and their derivative share a set (rather than just sharing a value as in known results). 1. Introduction An increasing function ϕ : [0, 1) → (0, ∞) is called smoothly increasing if (1) ϕ(r)(1 − r) → ∞ as r → 1− , and ϕ(|a + z/ϕ(|a|)|) → 1, as |a| → 1− ϕ(|a|) uniformly on compact subsets of C. In 2011, Aulaskari and R¨atty¨a [1] introduced the concept of ϕ -normal meromorphic functions as follows: For a smoothly increasing function ϕ, we say that the meromorphic function f in the unit disc D is ϕ-normal if (2) (3) Ra (z) := ||f ||N ϕ := supz∈D f # (z) < ∞, ϕ(|z|) |f (z) where f # (z) := (1+|f is the spherical derivative of f. (z)|2 ) ϕ Denote by N the set of all ϕ-normal meromorphic functions on D. For each smooth increasing function ϕ, set ϕ∗ (r) := ϕ(r) + (1 − r)−1 , then ϕ∗ is also ∗ smoothly increasing and N ϕ = N ϕ . Furthermore, the function ϕ∗ satisfies the condition: ϕ∗ (r)(1 − r) ≥ 1 for all r ∈ [0, 1). Therefore, in [1], the authors ∗ Corresponding author. Email addresses: tranvantanhn@yahoo.com (T.V.Tan); nguyenvanthintn@gmail.com (N.V.Thin). 2000 Mathematics Subject Classification. Primary 30D45, 30D35. Key words: ϕ− normal function, differential algebraic equation, Nevanlinna theory. 1 2 considered that the smoothly increasing ϕ satisfies the added condition ϕ(r)(1 − r) ≥ 1 for all r ∈ [0, 1). We refer readers to [1] for comments on the concept of ϕ-normal meromorphic functions. In this paper, we consider a larger family of functions ϕ : in the above definition of smoothly increasing functions mentioned above, condition (1) is replaced by the following condition (1 ) ϕ(r)(1 − r) ≥ 1 for all r ∈ [0, 1). 1 If we take ϕ0 = 1−r then the concept ϕ0 -normal functions coincides with the concept of normal functions (note that ϕ0 does not satisfy condition (1), but it satisfies conditions (1 ) and (2)). The first main suppose of this paper is to establish the types of LohwaterPommerenke-Zalcman’s criterion (see [4], [5]) for ϕ− normal functions. Theorem 1. Let ϕ : [0, 1) → (0, ∞) be a smoothly increasing function, and let f be a meromorphic function on the unit disc D. Assume that all zeros and all poles of f have multiplicity at least p, q, respectively. Let β be a real number satisfying −p < β < q. If f is not ϕ- normal then, there exist (i) a sequence {an } ⊂ D, |an | → 1; zn ∈ D; (ii) a sequence {zn } ⊂ D with zn → z∗ ∈ D and wn := an + ϕ(|an |) (iii) a sequence {ρn }, ρn → 0+ ρn such that gn (ξ) := ρβn f (wn + ξ) → g(ξ) locally uniformly with respect to ϕ(|an |) the spherical metric, where g is a nonconstant meromorphic function on C, all of whose zeros and poles have multiplicity at least p, q respectively. We note that in the case where β = 0, the above theorem is similar to Theorem 6 in [1]. Theorem 2. Let ϕ : [0, 1) → (0, ∞) be a smoothly increasing function, and let f be a meromorphic function on the unit disc D which has all zeros with multiplicity at least k. Assume that there exists a constant A ≥ 1 such that |f (k) (z)| ≤ Aϕk (|z|) whenever f (z) = 0. If f is not ϕ− normal, then for each 0 ≤ α ≤ k, there exist, (i) a sequence {an } ⊂ D, |an | → 1; (ii) a number 0 < r < 1; zn (iii) points zn , |zn | < r satisfying wn := an + ∈ D; ϕ(|an |) (iv) a sequence {ρn }, ρn → 0+ ρn f (wn + ξ) ϕ(|an |) such that gn (ξ) = → g(ξ) locally uniformly with respect to the ραn 3 spherical metric, where g is a nonconstant meromorphic function on C, all of 3 whose zeros have multiplicity at least k, g # (ξ) ≤ g # (0) = kA + 1. 2 Remark 1. Since ϕ(|z|) ≥ 1, for all z ∈ D, the above theorem remains valid if the condition |f (k) (z)| ≤ Aϕk (|z|) whenever f (z) = 0 is replaced by the condition |f (k) (z)| ≤ A whenever f (z) = 0. By using the above types of Lohwater-Pommerenke-Zalcman’s lemma, now we establish some ϕ− normal criteria. Theorem 3. Let f be a meromorphic function on the unit disc D, all of whose zeros and poles are multiplicity at least m and n respectively, where m, n are 1 1 + < 1. Let α1 , α2 be two distinct nonzero complex positive integers satisfying m n numbers and the set S = {α1 , α2 }, and let M be positive number. If |f (z)| ≤ M ϕ(|z|) whenever f (z) ∈ S, then f is a ϕ− normal function. Take ϕ := Theorem 3. 1 1−|z| , and M := max{|α1 |, |α2 |}, we get the following corollary of Corollary 1. Let f be meromorphic function on the unit disc D, all of whose zeros and poles are multiplicity at least m and n respectively, where m, n are 1 1 + < 1. Let α1 , α2 be two distinct nonzero complex positive integers satisfying m n numbers and the set S = {α1 , α2 }. If f (z) and f (z) share S − IM (in the sense of that {z : f (z) ∈ S} = {z : f (z) ∈ S}), then f is a normal function. We would like to emphasize here that so far, there are many normal criteria for meromorphic functions under a condition where functions and their derivative share values (or share functions). Corollary 1 is the first criterion for the case where the functions and their derivative share a set. Theorem 4. Let f be an entire function in a domain D, all of whose zeros are multiplicity at least 3. let α be a nonzero complex number and M be positive number. If |f (z)| ≤ M ϕ(|z|) whenever f (z) = α, then f is a ϕ− normal function. Take ϕ := 1 1−|z| , and M := |α|, we get the following corollary of Theorem 4. Corollary 2. Let f be an entire function on the unit disc D, all of whose zeros have multiplicity at least 3, let α be a nonzero complex number. If f (z) and f (z) share α − IM (in the sense of that f (z) = α iff f (z) = α), then f is a normal function. 4 Next, we study the solution of a differential algebraic equation. We consider the following differential algebraic equation on the unit disc D. dk w dz k (4) m n = Pj (z, w)Dj [w]. j=1 m j Here, k, n are positive integers; Pj (z, w) := i=0 aij (z)wi , where aij (z) are holomorphic functions; and Dj [w] is a differential monomial in w of the form Dj [w] = ( dl w dw j1 d2 w j2 ) ( 2 ) . . . ( l )jl , dz dz dz j1 , . . . , jl ∈ N ∪ {0}. We define the weight of Dj by νj := j1 +2j2 +· · ·+ljl and the weight of P [w](z) := m j=1 Pj (z, w)Dj [w] by ν(P ) := maxj=1,...,m {νj }. Theorem 5. Asumme that nk > ν(P ) and f is a meromorphic solution of (4) such that all zeros of f have multiplicity at least k. If the coefficients {aij } satisfy the condition lim (5) |z|→1 1 ϕ(|z|) mj nk−ν(P ) |aij (z)| < +∞, max 1≤j≤m i=0 then f is ϕ− normal. We note that for the case where k = 1 the above theorem is similar to the one obtained by Aulaskari-Wulan [3] in 2001 on the strong normal criteria. In the following theorem, we examine equation (4) in the case where kn can be smaller than ν(P ). Set ΓDj = j1 + · · · + jl , ΥDj = j1 + 2j2 + · · · + ljl . Theorem 6. Assume that ΥDj ≥ 1, (ΓDj + 1)k > ΥDj , nk > max ΓDj , j=1,...,m a0j ≡ 0 for all j ∈ {1, . . . , m} and (6) lim |z|→1 1 ϕ(|z|) nk−maxj=1,...,m ΓDj mj |aij (z)| < +∞. max 1≤j≤m i=1 Assume that f is a meromorphic solution of (4) such that of zeros of f have multiplicity at least k. Then f is a ϕ− normal. Finally, we give the following estimate for the order of ϕ− normal functions. 5 Theorem 7. Let f be a ϕ− normal meromorphic function. Then, the order ρ(f ) of f satisfies log T0 (r, f ) ρ(f ) := lim sup ≤ 2χ, r→1− − log(1 − r) where χ is the order of ϕ, defined by log ϕ(r) χ = lim sup , r→1− − log(1 − r) and T0 (r, f ) is the Ahlfors-Shimizu characteristic function of f. Acknowledgements: This research is funded by Vietnam National Foundation for Science and Technology Development (NAFOSTED). The first named author was partially supported by Vietnam Institute for Advanced Study in Math´ ematics, the Institut des Hautes Etudes Scientifiques (France), and a travel grant from Simons Foundation. 2. Some Lemmas In order to prove our theorems, we need the following lemmas. Lemma 1 (Zalcman’s Lemma, see [5]). Let F be a family of meromorphic functions on the unit disc D whose all zeros and poles have multiplicity at least p, q, respectively. Let α be a real number satisfying −p < α < q. Then, F is not normal at z0 if and only if there exist (i) a number r, 0 < r < 1; (ii) points zn with |zn | < r, zn → z0 ; (iii) functions fn ∈ F; (iv) positive numbers ρn → 0+ ; such that gn (ξ) = ραn fn (zn + ρn ξ) → g(ξ) locally uniformly with respect to the spherical metric, where g is a nonconstant meromorphic function on C, all of whose zeros and poles have multiplicity at least p, q respectively. Moreover, g has order at most 2. Similarly to Theorem 3 in [1], we give the following lemma: Lemma 2. Let ϕ : [0, 1) → (0, ∞) be a smoothly increasing function, and let f is a meromorphic function on D. Then f ∈ N ϕ if and only if the family z )}a∈D is normal in D. {fa (z) := f (a + ϕ(|a|) Proof. It is clear that |a + z |z| | ≤ |a| + ≤ |a| + |z|(1 − |a|) < 1, ϕ(|a|) ϕ(|a|) 6 for all z ∈ D. Therefore, functions fa (z) (a ∈ D) are well defined on D. • Assume that f ∈ N ϕ . We have z z z f # (a + ) f # (a + ) ϕ(|a + |) ϕ(|a|) ϕ(|a|) ϕ(|a|) # (fa ) (z) = (2.1) = . . z ϕ(|a|) ϕ(|a|) ϕ(|a + |) ϕ(|a|) Therefore, by the definition of the function ϕ and since f is ϕ-normal, we have, for any compact subset K ⊂ D, Supz∈K (fa )# (z) < ∞. Thus, the family {fa (z) : a ∈ D} is normal in D. • Assume that the family {fa (z) : a ∈ D} is normal in D. If f ∈ N ϕ , then there exists a sequence {zn } ⊂ D such that limn→∞ |zn | = 1 and f # (zn ) (2.2) → ∞ as n → ∞. ϕ(|zn |) On the other hand, the family {fa (z) : a ∈ D} is normal in D. Hence, {fzn (z)}∞ n=1 is also normal in D. In particular, it is normal at z = 0. Thus, there exists a f # (zn ) constant M > 0 such that = (fzn )# (0) ≤ M. This contradicts with (2.2). ϕ(|zn |) Then, f ∈ N ϕ . We have completed the proof of Lemma 2. Lemma 3 (see [6]). Let F be a family of meromorphic functions on the unit disc, all of whose zeros have multiplicity at least k, and suppose that there exists A ≥ 1 such that |f (k) (z)| ≤ A whenever f (z) = 0. If F is not normal, then there exist, for each 0 ≤ α ≤ k, (i) a number r, 0 < r < 1, (ii) points zn , |zn | < r, (ii) functions fn ∈ F, and (iv) the sequence {ρn } → 0+ f (zn + ρn ξ) such that gn (ξ) = → g(ξ) locally uniformly with respect to the spherραn ical metric, where g is a nonconstant meromorphic function on C such that g # (ξ) ≤ g # (0) = kA + 1. 3. Proof of Theorems Proof of Theorem 1. Suppose that f is not ϕ- normal. Then by Lemma 2, the family F = {fa }a∈D is not normal (see the definition of functions {fa } in Lemma 7 2). Hence, there exists the sequence am ∈ D such that z Fam = fam : fam (z) = f (am + ) ϕ(|am |) is not normal at some point z∗ . It is clear that there exists a subsequence, denote again {aj } such that |aj | → 1− . By Lemma 1, for all −p < β < q, there exist (i) points {zn } ⊂ D with, zn → z∗ ; (ii) functions fatn ∈ Fam ; (iii) positive numbers ρn → 0+ ; such that gn (ξ) = ρβn fatn (zn + ρn ξ) locally uniformly with respect to the spherical metric, where g is a nonconstant meromorphic function on C, all of whose zeros and poles have multiplicity at least p, q respectively. zn + ρn ξ zn ). Take wn = atn + , we get We have fatn (zn + ρn ξ) = f (atn + ϕ(|atn |) ϕ(|atn |) the conclusion of Theorem 1. Proof of Theorem 2. Suppose that f is not ϕ− normal. Then by Lemma 2, we z ) : a ∈ D} is not normal in D. Thus, there have that the family {f (a + ϕ(|a|) exists the sequence am → 1− such that z ) Fam = fam : fam (z) = f (am + ϕ(|am |) is not normal in D. Then, without loss of the generality, we may assume that |am | → 1− . z ϕk (|am + |) ϕ(|am |) Since limm→∞ = 1, uniformly on compact subsets of C, ϕk (|am |) there exists a positive integer M such that z ϕk (|am + |) 3 ϕ(|am |) (3.1) ≤ k 2 ϕ (|am |) for all z ∈ D, and m ≥ M. For any m ≥ M and for any z∗ ∈ D satisfying fam (z∗ ) = 0, we have also z∗ f (am + ) = 0. Therefore, by the assumption, we have ϕ(|am |) z∗ z∗ |f (k) (am + )| ≤ Aϕk (|am + |). ϕ(|am |) ϕ(|am |) Then, by (3.1), we have |fa(k) (z∗ )| = m 1 ϕk (|am |) |f (k) (am + z∗ )| ≤ A ϕ(|am |) z∗ |) 3 ϕ(|am |) ≤ A. k 2 ϕ (|am |) ϕk (|am + Then, by Lemma 3 (for the family {fam }m≥M ), for all 0 ≤ α ≤ k, there exist: 8 (i) a number 0 < r < 1, (ii) points zn , |zn | < r, (iii) a sub-sequence, denote again by {fan }, and (iv) a sequence ρn → 0+ fa (zn + ρn ξ) such that gn (ξ) = n → g(ξ) locally uniformly with respect to the ραn spherical metric, where g is a nonconstant meromorphic function on C such that 3 g # (ξ) ≤ g # (0) = kA + 1. 2 zn + ρn ξ zn Since fan (zn +ρn ξ) = f (an + ), by taking wn = an + , we complete ϕ(|an |) ϕ(|an |) the proof of Theorem 2. Proof of Theorem 3. Suppose that f is not ϕ− normal. Then, by Theorem 1 (with β = 0), there exist: (i) a sequence {an } ⊂ D, |an | → 1; zn (ii) a sequence {zn } ⊂ D: zn → z∗ ∈ D, and wn = an + ; ϕ(|an |) (iii) the sequence ρn → 0+ ρn ξ) → g(ξ) locally uniformly with respect to such that gn (ξ) = f (wn + ϕ(|an |) the spherical metric, where g is a nonconstant meromorphic function on C, all of zeros and poles of g have multiplicity at least m and n, respectively. Claim 1: All zeros of g − αi (i = 1, 2) are multiple. Indeed, for any zero point ξ0 of g − αi (for some i ∈ {1, 2}), by Hurwitz’s theorem, there exists a sequence {ξn } → ξ0 such that f (wn + ρn ξn ) = gn (ξn ) = αi . ϕ(|an |) Then, by the assumption, we have f (wn + ρn ρn ξn ) ≤ M ϕ(|wn + ξn |). ϕ(|an |) ϕ(|an |) Hence, (3.2) ρn ρn f (wn + ξn ) ϕ(|an |) ϕ(|an |) zn + ρn ξn ϕ(|an + |) ϕ(|an |) ≤ M ρn . ϕ(|an |) |gn (ξn )| = From (3.2) and (2), we get g (ξ0 ) = lim gn (ξn ) = 0. n→∞ 9 On the other hand, since g is non-constant and all zeros of g have multiplicity at least m(≥ 2), we get that g (ξ) ≡ 0. Hence, ξ0 is a multiple zero of g − αi . Then, we get Claim 1. For each meromorphic function u, denote by Tu (r) the Nevanlinna characteristic function of u (in the disc {z : |z| < r}), and denote by N (r, u1 ) the counting funtion (counted multiplicity) of zeros (in the disc {z : |z| < r}) of u, (and by N (r, u1 )) for the case of regardless multiplicity). By the First and the Second Main Theorems (in Nevanlinna theory), we have 1 1 1 ) + N (r, ) + o(Tg (r)) 2Tg (r) ≤ N (r, g) + N (r, ) + N (r, g g − α1 g − α2 1 1 1 1 1 1 1 ) + N (r, ) + o(Tg (r)) ≤ N (r, g) + N (r, ) + N (r, n m g 2 g − α1 2 f − α2 1 1 ≤ ( + + 1)Tg (r) + o(Tg (r)). (3.3) m n 1 1 + < 1. This contradicts to the condition m n We have completed the proof of Theorem 3. Proof of Theorem 4. We can get the proof of Theorem 4 by an argument similar to the proof of Theorem 3 with the following remark: • Claim 1 is replaced by the claim: all zeros of g − α are multiple. • Inequality (3.3) is replaced by the following estimate: 1 1 Tg (r) ≤ N (r, ) + N (r, ) + o(Tg (r)) g g−α 1 1 1 1 ≤ N (r, ) + N (r, ) + o(Tg (r)) 3 g 2 g−α 5 ≤ Tg (r) + o(Tg (r)). 6 This is a contradiction. Proof of Theorem 5. Assume that f be a solution of equation (4) all of whose zeros have multiplicity at least k, but f is not ϕ- normal. By Theorem 1 (with β = 0), there exist: (i) a sequence {av } ⊂ D, |av | → 1; zv (ii) a sequence {zv } ⊂ D, zv → z∗ ∈ D, and wv = av + ; ϕ(|av |) (iii) a sequence ρv → 0+ ρv such that gv (ξ) = f (wv + ξ) → g(ξ) locally uniformly with respect to the ϕ(|av |) spherical metric, where g is a nonconstant meromorphic function on C. 10 Since all zeros of f have multiplicity at least k, and by Hurwitz’s theorem, all zeros of g also have multiplicity at least k. Hence, there exist ξ0 ∈ C such that g (k) (ξ0 ) = 0. (3.4) Indeed, otherwise if g (k) (ξ) ≡ 0, then g is a (non-constant) polynomial with degree at most k − 1. This contradicts to the fact that all zeros of g have multiplicity at least k. We have − ρv ρv gv( ) (ξ) = f ( ) (wv + ξ), for all ∈ N∗ . ϕ(|av |) ϕ(|av |) Therefore, since f is a solution of equation (4), we get ρv ϕ(|av |) −k gv(k) (ξ0 ) n m ≤ Pj (wv + j=1 ρv ρv ξ0 , gv (ξ0 )) |Dj [gv ](ξ0 )| ϕ(|av |) ϕ(|av |) −νj . This implies that gv(k) (ξ0 ) n m ≤ Pj (wv + ρv ρv ξ0 , gv (ξ0 )) |Dj [gv ](ξ0 )| ϕ(|av |) ϕ(|av |) nk−νj ρv ρv ξ0 , gv (ξ0 )) |Dj [gv ](ξ0 )| ϕ(|av |) ϕ(|av |) nk−ν(P ) Pj (wv + j=1 m ≤ j=1 , (the last inequality holds for all v >> 0). Thus, there exist a constant M > 0 such that (3.5) gv(k) (ξ0 ) n ) ≤ M ρnk−ν(P v 1 ϕ(|av |) nk−ν(P ) mj · max 1≤j≤m aij (wv + i=0 ρv ξ0 ) , ϕ(|av |) (note that gv (ξ0 ) → g(ξ0 ) and Dj [gv ](ξ0 ) → Dj [g](ξ0 ), hence, they have an upper bound). This implies (3.6) mj ρv zv + ρv ξ0 max aij (wv + ξ0 ) ϕ a + v n nk−ν(P ) 1≤j≤m i=0 ϕ(|av |) ϕ(|av |) (k) nk−ν(P ) gv (ξ0 ) ≤ M ρv · . nk−ν(P ) ρv ϕ(|av |) ϕ( wv + ξ0 ) ϕ(|av |) On the other hand, by the definition of function ϕ (condition (2)), we have zv + ρv ξ0 ϕ av + ϕ(|av |) (3.7) lim = 1, v→∞ ϕ(|av |) (note that zv + ρv ξ0 ∈ D for v >> 0). 11 Therefore, by (5) and by taking limits both sides in (3.6), we get that g (k) (ξ0 ) = (k) limv→∞ gv (ξ0 ) = 0 (note that nk > ν(P )). This contradicts to (3.4). We have completed the proof of Theorem 5. Proof of Theorem 6. It is clear that m j=1 Pj (z, f )Dj [f ] = 0 on {z : f (z) = 0} (note that a0j ≡ 0 for j = 1, . . . , m). Thus for a solution f of (4), we have |f (k) (z)| = 0 < 1 for all z such that f (z) = 0. Hence, if f is not ϕ-normal. Then, by Theorem 2 (for α = k) and Remark 1, there exist (i) a sequence {av } ⊂ D, |av | → 1; (ii) a number 0 < r < 1; zv ; (iii) points zv , |zv | < r, and wv = av + ϕ(|av |) (iv) a sequence ρv → 0+ ρv f (wv + ξ) ϕ(|av |) → g(ξ) locally uniformly with respect to such that gv (ξ) = ρkv the spherical metric, where g is a nonconstant meromorphic function on C, all of whose zeros have multiplicity at least k. By an argument as in the proof of Theorem 5, there exists ξ0 ∈ C such that g (k) (ξ0 ) = 0. (3.8) We have j (j) (j) ρk−j (wv + v ϕ (|av |)gv (ξ) = f ρv ξ), ϕ(|av |) for all j ∈ N∗ . Therefore, since f is a solution of (4), we have m |ϕ(|av |) nk (gv(k) )n (ξ0 )| ≤ Pj (wv + j=1 ΓD k−ΥDj ρv Γ ξ0 , gv (ξ0 )) |Dj [gv ](ξ0 )|ϕ(|av |) Dj ρv j . ϕ(|av |) Thus, there exist a constant M > 0 such that (3.9) min j=1,...,m |(gv(k) )n (ξ0 )| ≤ M ρv {(ΓDj +1)k−ΥDj } 1 ϕ(|av |) nk− max mj · max 1≤j≤m aij (wv + i=0 j=1,...,m ΓDj ρv ξ0 ) . ϕ(|av |) 12 This implies (3.10) min j=1,...,m |(gv(k) )n (ξ0 )| ≤ M ρv mj i=1 max · {(ΓDj +1)k−ΥDj } 1≤j≤m ϕ( wn + aij (wn + ρn ξ0 ) ϕ(|an |) zn + ρn ξ0 ϕ(|an |) ϕ(|an |) ϕ an + ρn ξ0 ) ϕ(|an |) nk− max j=1,...,m nk− max j=1,...,m ΓDj . ΓDj On the other hand, by the definition of ϕ (and note that zv + ρv ξ0 ∈ D for v >> 0), we have zv + ρv ξ0 ϕ(|av |) ϕ(|av |) ϕ av + lim v→∞ = 1. (k) Hence, by taking limits both sides in (3.10), we get g (k) (ξ0 ) = limv→∞ gv (ξ0 ) = 0, which contradicts to (3.8). We have completed the proof of Theorem 6. Proof of Theorem 7. Since f is ϕ− normal, there exist a constant M > 0 such that f # (z) ≤ M ϕ(|z|), for all z ∈ D Then, we have S(t) = 1 π |f # (z)|2 dxdy ≤ |z|≤t M2 π ϕ2 (|z|)dxdy. |z|≤t On the other hand, ϕ is a increasing function. Hence, we have ϕ2 (t)dxdy = M 2 ϕ2 (t)t2 . S(t) ≤ |z|≤t By the definition of the Ahlfors-Shimizu characteristic function, we have r T0 (r, f ) = 0 S(t) dt ≤ M 2 t r ϕ2 (t)tdt 0 2 2 = O r ϕ (r) . Hence, ρ(f ) = lim sup r→1− log T0 (r, f ) log ϕ2 (r) ≤ lim sup = 2χ. − log(1 − r) r→1− − log(1 − r) 13 References .. .. [1] R. Aulaskari and J. Rattya, Properties of meromorphic ϕ -normal function, J. Michigan. Math, 60(2011), 93-111. .. .. [2] R. Aulaskari, S. Makhmutov and J. Rattya, Results on meromorphic φ-normal functions, Complex Variable and Elliptic Equations , 54(2009), 855-863. [3] R. Aulaskari and H. Wulan, A Version of the Lohwater-Pommerenke Theorem for Strongly Normal Functions, Computational Methods and Function Theory, 1(2001), 99-105. [4] A.J. Lohwater and Ch. Pommerenke, On meromorphic functions, Ann. Acad. Sci. Fenn. Ser. A I Math. 550(1973),1-12. [5] L. Zalcman, Normal families: New perspective, Bull. Amer. Mat. Soc, 35(1998), 215-230. [6] X. Pang and L. Zalcman, Normal families and shared values, Bull. London. Math. Soc, 32(2000), 325-331. A Department of Mathematics, Hanoi National University of Education, 136 Xuan Thuy Street, Cau Giay, Hanoi, Viet Nam. B Department of Mathematics, Thai Nguyen University of Education, Luong Ngoc Quyen Street, Thai Nguyen City, Vietnam [...]... Results on meromorphic φ -normal functions, Complex Variable and Elliptic Equations , 54(2009), 855-863 [3] R Aulaskari and H Wulan, A Version of the Lohwater-Pommerenke Theorem for Strongly Normal Functions, Computational Methods and Function Theory, 1(2001), 99-105 [4] A.J Lohwater and Ch Pommerenke, On meromorphic functions, Ann Acad Sci Fenn Ser A I Math 550(1973),1-12 [5] L Zalcman, Normal families: New... f (z) = 0 Hence, if f is not ϕ -normal Then, by Theorem 2 (for α = k) and Remark 1, there exist (i) a sequence {av } ⊂ D, |av | → 1; (ii) a number 0 < r < 1; zv ; (iii) points zv , |zv | < r, and wv = av + ϕ(|av |) (iv) a sequence ρv → 0+ ρv f (wv + ξ) ϕ(|av |) → g(ξ) locally uniformly with respect to such that gv (ξ) = ρkv the spherical metric, where g is a nonconstant meromorphic function on C, all... the Ahlfors-Shimizu characteristic function, we have r T0 (r, f ) = 0 S(t) dt ≤ M 2 t r ϕ2 (t)tdt 0 2 2 = O r ϕ (r) Hence, ρ(f ) = lim sup r→1− log T0 (r, f ) log ϕ2 (r) ≤ lim sup = 2χ − log(1 − r) r→1− − log(1 − r) 13 References [1] R Aulaskari and J Rattya, Properties of meromorphic ϕ -normal function, J Michigan Math, 60(2011), 93-111 [2] R Aulaskari, S Makhmutov and J Rattya, Results on meromorphic. ..11 Therefore, by (5) and by taking limits both sides in (3.6), we get that g (k) (ξ0 ) = (k) limv→∞ gv (ξ0 ) = 0 (note that nk > ν(P )) This contradicts to (3.4) We have completed the proof of Theorem 5 Proof of Theorem 6 It is clear that m j=1 Pj (z, f )Dj [f ] = 0 on {z : f (z) = 0} (note that a0j ≡ 0 for j = 1, , m) Thus for a solution f of (4), we have |f (k) (z)| = 0 < 1 for all z such... definition of ϕ (and note that zv + ρv ξ0 ∈ D for v >> 0), we have zv + ρv ξ0 ϕ(|av |) ϕ(|av |) ϕ av + lim v→∞ = 1 (k) Hence, by taking limits both sides in (3.10), we get g (k) (ξ0 ) = limv→∞ gv (ξ0 ) = 0, which contradicts to (3.8) We have completed the proof of Theorem 6 Proof of Theorem 7 Since f is ϕ− normal, there exist a constant M > 0 such that f # (z) ≤ M ϕ(|z|), for all z ∈ D Then, we have S(t) = 1... have multiplicity at least k By an argument as in the proof of Theorem 5, there exists ξ0 ∈ C such that g (k) (ξ0 ) = 0 (3.8) We have j (j) (j) ρk−j (wv + v ϕ (|av |)gv (ξ) = f ρv ξ), ϕ(|av |) for all j ∈ N∗ Therefore, since f is a solution of (4), we have m |ϕ(|av |) nk (gv(k) )n (ξ0 )| ≤ Pj (wv + j=1 ΓD k−ΥDj ρv Γ ξ0 , gv (ξ0 )) |Dj [gv ](ξ0 )|ϕ(|av |) Dj ρv j ϕ(|av |) Thus, there exist a constant... and Ch Pommerenke, On meromorphic functions, Ann Acad Sci Fenn Ser A I Math 550(1973),1-12 [5] L Zalcman, Normal families: New perspective, Bull Amer Mat Soc, 35(1998), 215-230 [6] X Pang and L Zalcman, Normal families and shared values, Bull London Math Soc, 32(2000), 325-331 A Department of Mathematics, Hanoi National University of Education, 136 Xuan Thuy Street, Cau Giay, Hanoi, Viet Nam B Department ... f (z) ∈ S}), then f is a normal function We would like to emphasize here that so far, there are many normal criteria for meromorphic functions under a condition where functions and their derivative... condition ϕ(r)(1 − r) ≥ for all r ∈ [0, 1) We refer readers to [1] for comments on the concept of ϕ -normal meromorphic functions In this paper, we consider a larger family of functions ϕ : in the... multiplicity at least k Then f is a ϕ− normal Finally, we give the following estimate for the order of ϕ− normal functions 5 Theorem Let f be a ϕ− normal meromorphic function Then, the order ρ(f