Normality Criteria of Lahiri's Type and Their Applications

Let denote the complex plane, and let be a nonconstant meromorphic function in . It is assumed that the reader is familiar with the standard notion used in the Nevanlinna value distribution theory such as the characteristic function , the proximity function , the counting function (see, e.g., [1–4]), and denotes any quantity that satisfies the condition as outside of a possible exceptional set of finite linear measure. A meromorphic function is called a small function with respect to , provided that .

Let and be two nonconstant meromorphic functions. Let and be small functions of and . means and have the same zeros (counting multiplicity) and means that and have the same poles (counting multiplicity). If whenever , we write . If and , we write . If , then we say that and share .

Set

(1.1)

where , are nonnegative integers. is called the differential monomial of and is called the degree of .

Let be a family of meromorphic functions defined in a domain . is said to be normal in , in the sense of Montel, if for any sequence , there exists a subsequence such that converges spherically locally uniformly in , to a meromorphic function or .

According to Bloch's principle, every condition which reduces a meromorphic function in to a constant makes a family of meromorphic functions in a domain normal. Although the principle is false in general, many authors proved normality criteria for families of meromorphic functions starting from Picard type theorems, for instance.

Theorem A (see [5]).

Let be an integer, and . If, for a meromorphic function , for all , then must be a constant.

Theorem B (see [6, 7]).

Let be an integer, , , and let , be a family of meromorphic functions in a domain . If for all , then is a normal family.

In 2005, Lahiri [8] got a normality criterion as follows.

Theorem C.

Let be a family of meromorphic functions in a complex domain . Let such that . Define

(1.2)

If there exists a positive constant such that for all whenever , then is a normal family.

In 2009, Charak and Rieppo [9] generalized Theorem C and obtained two normality criteria of Lahiri's type.

Theorem D.

Let be a family of meromorphic functions in a complex domain . Let , such that . Let , , , be positive integers such that , , , and put

(1.3)

If there exists a positive constant such that for all whenever , then is a normal family.

Theorem E.

Let be a family of meromorphic functions in a complex domain . Let , such that . Let , , , be nonnegative integers such that , and put

(1.4)

If there exists a positive constant such that for all whenever , then is a normal family.

Very recently, Xu and Cao [10] further extended Theorems D and E by replacing with ; they got

Theorem F.

Let be a family of meromorphic functions in a complex domain , all of whose zeros have multiplicity at least . Let such that . Let , , , be nonnegative integers such that , , , (if , ), and put

(1.5)

If there exists a positive constant such that for all whenever , then is a normal family.

Theorem G.

Let be a family of meromorphic functions in a complex domain , all of whose zeros have multiplicity at least . Let such that . Let , , , be positive integers such that , and put

(1.6)

If there exists a positive constant such that for all whenever , then is a normal family.

To prove Theorems D–G, the authors used a key lemma (Lemma 2.4 in this paper) besides Zalcman-Pang's Lemma. It's natural to ask whether such normality criteria of Lahiri's type still hold for the general differential monomial . We study this problem and obtain the following theorem

Theorem 1.1.

Let be a family of meromorphic functions in a complex domain , for every , all zeros of have multiplicity at least . Let , such that , let , , , , be nonnegative integers such that

(1.7)

Put

(1.8)

If there exists a positive constant such that for all whenever , then is a normal family.

Theorem 1.2.

Let be a family of meromorphic functions in a complex domain , for every , all zeros of have multiplicity at least . Let , such that , let , , , , be nonnegative integers such that , ( when or ), for all positive integers and , . Put

(1.9)

If there exists a positive constant such that for all whenever , then is a normal family.

As an application of Theorem 1.1, we obtain the following theorem.

Theorem 1.3.

Let be a family of holomorphic functions in a domain , for every , all zeros of have multiplicity at least . Let be two finite values and be nonnegative integers with , , . For every , all zeros of have multiplicity at least , if , then is normal in .

Example 1.4.

Let and . If , let . For each function , and share 0 in . However, it can be easily verified that is not normal in . Example 1.4 shows that the condition in Theorem 1.3 is sharp.

Example 1.5.

Let and . If , let , where is the root of . For each function , , in . However, it can be easily verified that is not normal in . Example 1.5 shows that the multiplicity restriction on zeros of in Theorem 1.3 is sharp (at least for ).

Lemma 2.1 (see [11]).

Let be a family of meromorphic functions on the unit disc , all of whose zeros have the multiplicity at least , then if is not normal, there exist, for each

(a)a number , ,

(b)points , ,

(c)functions , and

(d)positive numbers

such that locally uniformly with respect to the spherical metric, where is a nonconstant meromorphic function on , all of whose zeros have multiplicity at least , such that . Here, as usual, is the spherical derivative.

Lemma 2.2 (see [1, page 158]).

Let be a family of meromorphic functions in a domain . Then is normal in if and only if the spherical derivatives of functions are uniformly bounded on each compact subset of .

Lemma 2.3 (see [12]).

Let be an entire function and a positive integer. If for all , then has the order at most one.

Lemma 2.4 (see [13]).

Take nonnegative integers with , and define . Let be a transcendental meromorphic function with the deficiency . Then for any nonzero value , the function has infinitely many zeros. Moreover, if , the deficient condition can be omitted.

The following two lemmas can be seen as supplements of Lemma 2.4.

Lemma 2.5.

Take nonnegative integers with , and define . Let be a transcendental meromorphic function whose zeros have multiplicity at least . Then for any nonzero value , the function has infinitely many zeros, provided that and when . Specially, if is transcendental entire, the function has infinitely many zeros.

Proof.

If , then , this case has been considered (see [5, 12–20]).

If and if , we immediately get the conclusion from Lemma 2.4. Next we consider the case .

Let . Using the proof of Lemma 2.4 (see [13, page 161–163] ), we obtain

(2.1)

Suppose that is a zero of of multiplicity , then is a zero of of multiplicity , and thus is a pole of of multiplicity . Thereby, from (2.1) we get

(2.2)

Note that , we deduce from (2.2) that

(2.3)

If , then ; this case has been proved as mentioned above (see [13–16]).

If , then we have ; the conclusion is evident.

If , note that and we deduce that , thus the conclusion holds.

If is a transcendental entire function, we only need to consider the case . Note that (see Hu et al. [21, page 67])

(2.4)

With similar discussion as above, we obtain

(2.5)

In view of and , we get , thus we immediately obtain the conclusion. This completes the proof of Lemma 2.5.

Lemma 2.6.

Take nonnegative integers with , , and define . Let be a nonconstant rational function whose zeros have multiplicity at least . Then for any nonzero value , the function has at least one finite zero.

Proof.

Since the case has been proved by Charak and Rieppo [9], we only need to consider .

Suppose that has no zero.

Case 1.

If is a nonconstant polynomial, since the zeros of have multiplicity at least , we know that is also a nonconstant polynomial, so has at least one zero, which contradicts our assumption.

Case 2.

If is a nonconstant rational function but not a polynomial. Set

(2.6)

where is a nonzero constant and ,   .

Then by mathematical induction, we get

(2.7)

where , are constants and

(2.8)

(2.9)

It is easily obtained that

(2.10)

Combining (2.6) and (2.7) yields

(2.11)

where with .

Moreover, is not a constant, or else, we get is a constant for . The leading coefficient of is .

If is a constant, then we get

(2.12)

If is a constant, then we get

(2.13)

which implies , a contradiction with the assumption .

Then from (2.11), we obtain

(2.14)

where is a polynomial with .

Since , we obtain from (2.11) that

(2.15)

where is a nonzero constant. Then

(2.16)

where is a polynomial with .

From (2.14) and (2.16), we deduce that

(2.17)

in view of , together with (2.8), we have

(2.18)

namely

(2.19)

which is a contradiction since .

Hence has at least one finite zero.

This proves Lemma 2.6.

Remark 2.7.

Lemma 2.6 is a generalization of Lemma 2.2 in [10]. The proof of Lemma 2.6 is quite different from its proof. Actually, the proof of Lemma 2.2 in [10] is incorrect. The main problem appears at (2.2) in [10]. Since has only zero with multiplicity at least , then each zero of is of multiplicity at least , but this does not mean that each zero of is of multiplicity at least because the zeros of may not be the zeros of , and thus their multiplicity may less than . Therefore, the inequality of (2.2) in [10] is not valid, which implies that the proof of Lemma 2.2 in [10] is not correct.

Lemma 2.8.

Let , such that . Let , , , , be nonnegative integers such that , ( when or ), for all positive integers and , . Let be a meromorphic function in ; all zeros of have multiplicity at least . Define

(2.20)

Then has a finite zero.

Proof.

The algebraic complex equation

(2.21)

has always a nonzero solution, say . By Lemmas 2.5 and 2.6, the differential monomial cannot avoid it and thus there exists such that .

Under the assumptions, for all positive integers , , , , we have

(2.22)

Thus

(2.23)

This proves Lemma 2.8.

Lemma 2.9 (see [2, page 51]).

If is an entire function of order , then

(2.24)

where denotes the central-index of .

Lemma 2.10 (see [22, page 187–199] or [2, page 51]).

If is a transcendental entire function, let and be such that and that holds. Then there exists a set of finite logarithmic measure, that is, such that

(2.25)

holds for all and all .

Without loss of generality, we may assume . Suppose that is not normal at . By Lemma 2.1, for , there exist , such that , and such that locally uniformly with respect to the spherical metric, where is a nonconstant meromorphic function on , all of whose zeros have multiplicity at least . For simplicity, we denote by . By Lemmas 2.4 and 2.6, there exists such that

(3.1)

Obviously, , so in some neighborhood of , converges uniformly to . We have

(3.2)

Let , and under the assumption , we obtain

(3.3)

is the uniform limit of

(3.4)

in some neighborhood of .

By (3.1) and Hurwitz's theorem, there exists a sequence such that for all large values of and ,

(3.5)

Hence for all large values of , , it follows from the condition that

(3.6)

Since is not a pole of , there exists a positive number such that in some neighborhood of we get .

Since converges uniformly to in some neighborhood of , we get for all large values of and for all in that neighborhood of  

(3.7)

By (3.7), we get

(3.8)

which is a contradiction since as .

This completes the proof of Theorem 1.1.

Without loss of generality, we may assume . Suppose that is not normal in . By Lemma 2.1, for , there exist , , and such that locally uniformly with respect to the spherical metric, where is a nonconstant meromorphic function on , all of whose zeros have multiplicity at least . By Lemma 2.8, we get

(4.1)

for some .

We can arrive at a contradiction by using the same argument as in the latter part of proof of Theorem 1.1.

This completes the proof of Theorem 1.2.

Proof of Theorem 1.3.

We shall divide our argument into two cases.

Case 1 ( ).

Let be a positive constant with ; under the assumptions, we have

(5.1)

and for all whenever ; by Lemmas 2.5 and 2.6, using the similar proof of Theorem 1.1, we obtain the conclusion.

Case 2 ( ).

For , if for , since , we have , which is a contradiction, hence .

If for , since , we immediately get and hence , which is still a contradiction, hence .

Suppose that is not normal in , by Lemma 2.1, there exist , , , and such that locally uniformly with respect to the spherical metric, where is a nonconstant entire function, all of whose zeros have multiplicity at least . By Hurwitz's theorem, we have

(i) or , and

(ii) or .

Since is not a constant, we have . By Lemma 2.3, has the order at most 1, so , where , are two constants. Thus

(5.2)

If , we immediately get a contradiction. Hence

(5.3)

but by Lemmas 2.5 and 2.6 we get a contradiction again.

This proves Theorem 5.1.

Further more, using Theorem 1.3, we obtain a uniqueness theorem related to R. Brück's Conjecture. Firstly, we recall this conjecture.

R. Brück's Conjecture

Let be a nonconstant entire function such that the hyper-order is not a positive integer and . If and share a finite value CM, then

(5.4)

where is a nonzero constant and the hyper-order is defined as follow:

(5.5)

Since then, many results related to this conjecture have been obtained. We refer the reader to Brück [23], Gundersen and Yang [24], Yang [25], Chen and Shon [26], Li and Gao [27], and Wang [28].

It's interesting to ask what happens if is replaced by in Brück's Conjecture. Recently, Yang and Zhang [29] considered this problem and got the following theorem.

Theorem H.

Let be a nonconstant entire function. be an integer, and let . If and share 1 CM, then , and assumes the form

(5.6)

where is a nonzero constant.

Lü et al. [30] improves Theorem H and obtained the following theorem.

Theorem I.

Let be a polynomial, and let be an intege; let be a transcendental entire function, and let . If and share CM, then

(5.7)

where is a nonzero constant.

We obtain a more general result as follows.

Theorem 5.1.

Let be nonnegative integers with , , , and , be two finite nonzero values. Let be a nonconstant entire function whose zeros have multiplicity at least . If , then

(5.8)

where is a nonzero constant. Specially, if , then , where is a nonzero constant, is the root of .

Proof of Theorem 5.1.

First we assert that . Let

(5.9)

Under the assumptions of Theorem 1.3, we get that is a normal family of holomorphic functions in . By Lemma 2.2, there exists a constant such that

(5.10)

for all . Hence by Lemma 2.3, has the order at most 1.

Since , must be a transcendental entire function and

(5.11)

From (5.11), we have , hence and is a polynomial with . Note that is transcendental, we have , as . Let , where , we deduce

(5.12)

By Lemma 2.10, there exists a subset of finite logarithmic measure, namely such that for some point satisfying and , we obtain

(5.13)

as .

Rewrite (5.11) as

(5.14)

it follows from (5.12)–(5.14) and Lemma 2.8 that

(5.15)

as . Since is a polynomial, from (5.15), we deduce that is a constant. Let , then is a nonzero constant. Thus

(5.16)

Specially, if , suppose that has a zero , by letting in (5.16), we get ; hence

(5.17)

Suppose that is a zero of with multiplicity , then is a zero of with multiplicity , and a zero of with multiplicity , which is a contradiction. So has no zero, note that is a transcendental entire function and , we have , where and are two finite nonzero values. In view of (5.16) and , we deduce that

(5.18)

hence and . , is the root of .

This completes the proof of Theorem 5.1.