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)

TRAN VAN TAN A,∗ AND NGUYEN VAN THIN B

Abstract 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−r12 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

ϕ(r)(1 − r) → ∞ as r → 1−, (1)

and

Ra(z) := ϕ(|a + z/ϕ(|a|)|)

ϕ(|a|) → 1, as |a| → 1

−

(2)

uniformly on compact subsets of C

In 2011, Aulaskari and R¨atty¨a [1] introduced the concept of ϕ -normal mero-morphic 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

||f ||Nϕ := supz∈D

f#(z) ϕ(|z|) < ∞, (3)

where f#(z) := (1+|f (z)||f0(z)2 ) is the spherical derivative of f

Denote by Nϕ 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 Nϕ∗ = Nϕ 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

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

(10) ϕ(r)(1 − r) ≥ 1 for all r ∈ [0, 1)

If we take ϕ0 = 1−r1 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 (10) and (2))

The first main suppose of this paper is to establish the types of Lohwater-Pommerenke-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;

(ii) a sequence {zn} ⊂ D with zn→ z∗ ∈ D and wn:= an+ zn

ϕ(|an|) ∈ D;

(iii) a sequence {ρn}, ρn→ 0+

such that gn(ξ) := ρβnf (wn+ ρn

ϕ(|an|)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

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;

(iii) points zn, |zn| < r satisfying wn:= an+ zn

ϕ(|an|) ∈ D;

(iv) a sequence {ρn}, ρn→ 0+

such that gn(ξ) =

f (wn+ ρn

ϕ(|an|)ξ)

ρα → g(ξ) locally uniformly with respect to the

spherical metric, where g is a nonconstant meromorphic function on C, all of whose zeros have multiplicity at least k, g#(ξ) ≤ g#(0) = 3

2kA + 1.

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 positive integers satisfying 1

m+

1

n < 1 Let α1, α2 be two distinct nonzero complex numbers and the set S = {α1, α2}, and let M be positive number If |f0(z)| ≤

M ϕ(|z|) whenever f (z) ∈ S, then f is a ϕ− normal function

Take ϕ := 1−|z|1 , and M := max{|α1|, |α2|}, we get the following corollary of Theorem 3

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 positive integers satisfying 1

m+

1

n < 1 Let α1, α2 be two distinct nonzero complex numbers and the set S = {α1, α2} If f0(z) and f (z) share S − IM (in the sense

of that {z : f (z) ∈ S} = {z : f0(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 |f0(z)| ≤ M ϕ(|z|) whenever f (z) = α, then f is a ϕ− normal function Take ϕ := 1−|z|1 , 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 f0(z) and f (z) share α − IM (in the sense of that f (z) = α iff f0(z) = α), then f is a normal function

Next, we study the solution of a differential algebraic equation.

We consider the following differential algebraic equation on the unit disc D

dkw

dzk

n

=

m

X

j=1

Pj(z, w)Dj[w]

(4)

Here, k, n are positive integers; Pj(z, w) :=Pm j

i=0aij(z)wi, where aij(z) are holo-morphic functions; and Dj[w] is a differential monomial in w of the form

Dj[w] = (dw

dz)

j 1(d

2w

dz2)j2 (d

lw

dzl)jl, j1, , jl∈ N ∪ {0}

We define the weight of Dj by νj := j1+2j2+· · ·+ljland the weight of P [w](z) :=

Pm

j=1Pj(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

|z|→1

 1 ϕ(|z|)

nk−ν(P )

max

1≤j≤m

m j

X

i=0

|aij(z)| < +∞, (5)

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

j=1, ,mΓDj,

a0j ≡ 0 for all j ∈ {1, , m} and

lim

|z|→1

 1 ϕ(|z|)

nk−max j=1, ,m ΓDj

max

1≤j≤m

m j

X

i=1

|aij(z)| < +∞

(6)

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

Theorem 7 Let f be a ϕ− normal meromorphic function Then, the order ρ(f )

of f satisfies

ρ(f ) := lim sup

r→1 −

log T0(r, f )

− log(1 − r) ≤ 2χ, where χ is the order of ϕ, defined by

χ = lim sup

r→1 −

log ϕ(r)

− log(1 − r), and T0(r, f ) is the Ahlfors-Shimizu characteristic function of f

Acknowledgements: This research is funded by Vietnam National Foun-dation 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 func-tions 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) = ραnfn(zn + ρnξ) → 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 ∈ Nϕ if and only if the family {fa(z) := f (a + z

ϕ(|a|))}a∈D is normal in D.

Proof It is clear that

|a + z ϕ(|a|)| ≤ |a| +

|z|

ϕ(|a|) ≤ |a| + |z|(1 − |a|) < 1,

for all z ∈ D Therefore, functions fa(z) (a ∈ D) are well defined on D.

• Assume that f ∈ Nϕ

We have

(fa)#(z) =

f#(a + z

ϕ(|a|)) ϕ(|a|) =

f#(a + z

ϕ(|a|)) ϕ(|a + z

ϕ(|a|)|)

ϕ(|a + z

ϕ(|a|)|) ϕ(|a|) . (2.1)

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 6∈ Nϕ, then there exists a sequence {zn} ⊂ D such that limn→∞|zn| = 1 and

f#(zn) ϕ(|zn|) → ∞ as n → ∞.

(2.2)

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 constant M > 0 such that f

#(zn) ϕ(|zn|) = (fzn)#(0) ≤ M This contradicts with (2.2) Then, f ∈ Nϕ

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+

such that gn(ξ) = f (zn+ ρnξ)

ρα → g(ξ) locally uniformly with respect to the spher-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∈Dis not normal (see the definition of functions {fa} in Lemma

2) Hence, there exists the sequence am ∈ D such that

Fam =

n

fa m : fa m(z) = f (am+ z

ϕ(|am|))

o

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) = ρβnfatn(zn+ ρnξ) 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

We have fatn(zn+ ρnξ) = f (at n+zn+ ρnξ

ϕ(|at n|)) Take wn= atn+ zn

ϕ(|at n|), we get

Proof of Theorem 2 Suppose that f is not ϕ− normal Then by Lemma 2, we have that the family {f (a + z

ϕ(|a|)) : a ∈ D} is not normal in D Thus, there exists the sequence am→ 1− such that

Fam =nfam : fam(z) = f (am+ z

ϕ(|am|))

o

is not normal in D Then, without loss of the generality, we may assume that

|am| → 1−

Since limm→∞

ϕk(|am+ z

ϕ(|am|)|)

ϕk(|am|) = 1, uniformly on compact subsets of C, there exists a positive integer M such that

ϕk(|am+ z

ϕ(|am|)|)

ϕk(|am|) ≤

3 2 (3.1)

for all z ∈ D, and m ≥ M

For any m ≥ M and for any z∗ ∈ D satisfying fam(z∗) = 0, we have also

f (am+ z∗

ϕ(|am|)) = 0 Therefore, by the assumption, we have

|f(k)(am+ z∗

ϕ(|am|))| ≤ Aϕ

k(|am+ z∗

ϕ(|am|)|).

Then, by (3.1), we have

|fa(k)

m(z∗)| = 1

ϕk(|am|)|f

(k)(am+ z∗

ϕ(|am|))| ≤ A

ϕk(|am+ z∗

ϕ(|am|)|)

ϕk(|am|) ≤

3

2A. Then, by Lemma 3 (for the family {fa }m≥M), for all 0 ≤ α ≤ k, there exist:

(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+

such that gn(ξ) = fan(zn+ ρnξ)

ρα → g(ξ) locally uniformly with respect to the spherical metric, where g is a nonconstant meromorphic function on C such that

g#(ξ) ≤ g#(0) = 3

2kA + 1.

Since fa n(zn+ρnξ) = f (an+zn+ ρnξ

ϕ(|an|) ), by taking wn= an+

zn

ϕ(|an|), we complete

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;

(ii) a sequence {zn} ⊂ D: zn→ z∗ ∈ D, and wn= an+ zn

ϕ(|an|); (iii) the sequence ρn→ 0+

such that gn(ξ) = f (wn + ρn

ϕ(|an|)ξ) → g(ξ) locally uniformly with respect to 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

ϕ(|an|)ξn) = gn(ξn) = αi. Then, by the assumption, we have

f0(wn+ ρn

ϕ(|an|)ξn)

≤ M ϕ(|wn+ ρn

ϕ(|an|)ξn|).

Hence,

|gn0(ξn)| = ρn

ϕ(|an|)

f0(wn+ ρn

ϕ(|an|)ξn)

(3.2)

≤ M ρn

ϕ(|an+zn+ ρnξn

ϕ(|an|) |) ϕ(|an|) . From (3.2) and (2), we get

g0(ξ0) = lim

n→∞g0n(ξn) = 0

On the other hand, since g is non-constant and all zeros of g have multiplicity at least m(≥ 2), we get that g0(ξ) 6≡ 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 character-istic 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 2Tg(r) ≤ N (r, g) + N (r,1

g) + N (r,

1

g − α1

) + N (r, 1

g − α2

) + o(Tg(r))

≤ 1

nN (r, g) +

1

mN (r,

1

g) +

1

2N (r,

1

g − α1

) +1

2N (r,

1

f − α2

) + o(Tg(r))

≤ (1

m +

1

n+ 1)Tg(r) + o(Tg(r)).

(3.3)

This contradicts to the condition 1

m +

1

n < 1.

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:

Tg(r) ≤ N (r,1

g) + N (r,

1

g − α) + o(Tg(r))

≤ 1

3N (r,

1

g) +

1

2N (r,

1

g − α) + o(Tg(r))

≤ 5

6Tg(r) + o(Tg(r)).

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;

(ii) a sequence {zv} ⊂ D, zv → z∗ ∈ D, and wv = av+ zv

ϕ(|av|); (iii) a sequence ρv → 0+

such that gv(ξ) = f (wv+ ρv

ϕ(|av|)ξ) → g(ξ) locally uniformly with respect to the spherical metric, where g is a nonconstant meromorphic function on C

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) 6= 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

ϕ(|av|)

−`

g(`)v (ξ) = f(`)(wv+ ρv

ϕ(|av|)ξ), for all ` ∈ N∗ Therefore, since f is a solution of equation (4), we get

 ρv

ϕ(|av|)

−k

gv(k)(ξ0)

n

≤

m

X

j=1

Pj(wv+ ρv

ϕ(|av|)ξ0, gv(ξ0))

|Dj[gv](ξ0)|

 ρv

ϕ(|av|)

−νj

This implies that

g(k)v (ξ0)

n

≤

m

X

j=1

Pj(wv+ ρv

ϕ(|av|)ξ0, gv(ξ0))

|Dj[gv](ξ0)|

 ρv

ϕ(|av|)

nk−ν j

≤

m

X

j=1

Pj(wv+ ρv

ϕ(|av|)ξ0, gv(ξ0))

|Dj[gv](ξ0)| ρv

ϕ(|av|)

nk−ν(P )

,

(the last inequality holds for all v >> 0)

Thus, there exist a constant M > 0 such that

g(k)v (ξ0)

n

≤ M ρnk−ν(P )v  1

ϕ(|av|)

nk−ν(P )

· max

1≤j≤m

m j

X

i=0

aij(wv+ ρv

ϕ(|av|)ξ0)

, (3.5)

(note that gv(ξ0) → g(ξ0) and Dj[gv](ξ0) → Dj[g](ξ0), hence, they have an upper

bound)

This implies

gv(k)(ξ0)

n

≤ M ρnk−ν(P )

v

ϕ



av+ zv+ ρvξ0

ϕ(|av|)

 ϕ(|av|)

nk−ν(P )

·

max

1≤j≤m

m j

P

i=0

aij(wv+ ρv

ϕ(|av|)ξ0)

ϕ(

wv+ ρv ϕ(|av|)ξ0

)nk−ν(P )

(3.6)

On the other hand, by the definition of function ϕ (condition (2)), we have

lim

v→∞

ϕ

av+zv+ ρvξ0

ϕ(|av|)

 ϕ(|av|) = 1, (3.7)

(note that zv+ ρvξ0 ∈ D for v >> 0)

we get Claim

For each meromorphic function u, denote by Tu(r) the Nevanlinna character-istic function... ρv

ϕ(|av|)ξ) → g(ξ) locally uniformly with respect to the spherical metric, where g is a nonconstant meromorphic function on C