Journal of Mathematics

Volume 2013 (2013), Article ID 429083, 9 pages

http://dx.doi.org/10.1155/2013/429083

Research Article

On the Iterated Exponent of Convergence of Solutions of Linear Differential Equations with Entire and Meromorphic Coefficients

Rabab Bouabdelli and Benharrat Belaïdi

Laboratory of Pure and Applied Mathematics, Department of Mathematics, University of Mostaganem (UMAB), BP 227, 27000 Mostaganem, Algeria

Received 27 September 2012; Accepted 17 November 2012

Academic Editor: Jen-Chih Yao

Copyright © 2013 Rabab Bouabdelli and Benharrat Belaïdi. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

We investigate the zeros of the difference of the derivative of solutions of the higher-order linear differential equations and small functions, where , are entire or meromorphic functions of finite iterated order.

1. Introduction and Main Results

In this paper, a meromorphic function will mean meromorphic in the whole complex plane. We shall use the standard notations in Nevanlinna value distribution of meromorphic functions [1, 2] such as , , . For the definition of the iterated order of a meromorphic function, we use the same definition as in [3], [4, p. 317], [2, p. 129]. For all , we define and , . We also define for all sufficiently large and . Moreover, we denote by , , and .

Definition 1 (see [2, 3]). Let be a meromorphic function. Then the iterated order of is defined by

For , this notation is called order and for hyperorder. If is an entire function, then the iterated order of is defined by

where .

Definition 2 (see [3]). The finiteness degree of the order of a meromorphic function is defined by

Definition 3 (see [3]). The iterated convergence exponent of the sequence of zeros of a meromorphic function is defined by

where is the counting function of zeros of in . Similarly, the iterated convergence exponent of the sequence of distinct zeros of is defined by

where is the counting function of distinct zeros of in .

Definition 4 (see [3]). The finiteness degree of the iterated convergence exponent of the sequence of zeros of a meromorphic function is defined by

Remark 5. Similarly, we can define the finiteness degree of .

Definition 6 (see [5]). Let be an entire function. Then the iterated type of , with iterated order is defined by

where .

We define the linear measure of a set by and the logarithmic measure of a set by , where is the characteristic function of a set .

Recently, Xu, Tu, and Zheng have investigated the relationship between the small functions and the derivative of solutions of higher-order linear differential equation:

where are entire or meromorphic functions and have obtained the following results.

Theorem A (see [6]). Let    be entire functions with finite order and satisfy one of the following conditions:

(i). (ii) and .

Then for every solution of (8) and for any entire function satisfying , we have

Theorem B (see [6]). Let    be polynomials and be a transcendental entire function. Then for every solution of (8) and for any entire function of finite order, we have

(i), (ii), .

When the coefficients are meromorphic functions, they have proved the following theorem.

Theorem C (see [6]). Let    be meromorphic functions satisfying and . Then for every meromorphic solution of (8) and for any meromorphic function satisfying , we have

In this paper, we improve and extend the above results by considering the iterated order, and we obtain the following theorems. For some closely related to applications of this paper, see the papers of Gupta et al. [7, 8].

Theorem 7. Let    be entire functions of finite iterated order with and satisfy one of the following conditions:

(i). (ii) and .

Then for every solution of (8) and for any entire function satisfying , we have

Theorem 8. Let    be polynomials and be a transcendental entire function with . Then for every solution of (8) and for any entire function of finite iterated order , we have

(i). (ii).

Theorem 9. Let    be meromorphic functions of finite iterated order with satisfying and . Then for every meromorphic solution of (8) whose poles are of uniformly bounded multiplicity and for any meromorphic function satisfying , we have

Corollary 10. Under the assumptions of Theorem 7, if , then, for every solution of (8), we have

Corollary 11. Under the assumptions of Theorem 8, if , then, for every solution of (8), we have

(i). (ii), , .

Corollary 12. Under the assumptions of Theorem 9, if , then for every meromorphic solution of (8) whose poles are of uniformly bounded multiplicity, we have

2. Auxiliary Lemmas

In order to prove our theorems, we need the following lemmas.

Lemma 13 (see [6]). Assume that is a solution of (8), set , and then satisfies the equation

where

Lemma 14 (see [3]). Let be a meromorphic function for which and , and let be an integer. Then for any ,

outside of a possible exceptional set of finite linear measure.

Lemma 15 (see [1]). Let be a transcendental meromorphic function and be an integer. Then

outside of a possible exceptional set of finite linear measure, and if is of finite order of growth, then

Lemma 16 (see [9]). Let and be monotone nondecreasing functions such that for all , where is a set of finite logarithmic measure. Let be a given constant. Then there exists an such that for all .

Lemma 17 (see [10]). Let be a meromorphic solution of (8), assuming that not all coefficients are constants. Given a real constant , and denoting , we have

outside of an exceptional set with .

Remark 18. We note that in the above lemma, corresponds to Euclidean measure and to logarithmic measure.

Lemma 19 (see [11]). Let be a meromorphic function of finite iterated order satisfying . Then there exists a set with infinite logarithmic measure such that for all , we have

Lemma 20. Let be entire functions of finite iterated order with and satisfy and set

where ;  ,   and ,  . Then there exists a set with infinite logarithmic measure such that for all

Proof. We prove this lemma by using mathematical induction. First, when , we have ,   and .

When , that is, . Then we have

From , we have

When , from the definitions of , we have

. Since are entire functions with , then by Lemma 19, there exists a set with infinite logarithmic measure such that for all

By this, (26), and Lemma 14, we get for all

for some constant outside of a possible exceptional set of finite linear measure. From (24), (25), and (28) we get for all

Now, suppose that (23) holds, for ; that is, there is a set with infinite logarithmic measure such that for all

Next, we prove that (23) holds for . Since , then we have

where ;  . When , we have . Then we obtain that

Since , then we have

When , from the definitions of ,   and , we have

From (30)–(34), there exists a set with infinite logarithmic measure such that for all

Thus, the proof of Lemma 20 is completed.

Lemma 21. Let be meromorphic functions of finite iterated order. If

and there exists a set with infinite logarithmic measure such that

holds for all , then every meromorphic solution of (8) satisfies and .

Proof. Suppose that is a meromorphic solution of (8). If , then from (8) we have

By using Lemma 14, we obtain

outside of a possible exceptional set of finite linear measure. By the hypothesis of Lemma 21, there exists a set having infinite logarithmic measure such that for all , we have for any

and so by Lemma 16, we obtain

and from this we obtain a contradiction. Hence .

Now, assume that is a meromorphic solution of (8) with . By (8) we have

By Lemma 15 and (42), we have

where is a set having finite linear measure. By the hypothesis of Lemma 21, there exists a set having infinite logarithmic measure such that, for all , we have by using (43)

where . By (44) and Lemma 16, we have

Lemma 22 (see [12, Theorem 3]). Let be a transcendental meromorphic function, and let be a given constant. Then there exists a set with finite logarithmic measure and a constant that depends only on and , such that for all satisfying , we have

From the above lemma, we obtain the following result.

Lemma 23. Let be a transcendental meromorphic function with finite iterated order ,   be a finite set of distinct pairs of integers satisfying   , and let be a given constant. Then, there exists a set with finite logarithmic measure, such that, for all satisfying and for all , we have

Proof. The definition

implies that for any given there exists such that for all we have

Combining (49) with Lemma 22, for , there exists a set that has finite logarithmic measure and a constant , such that if , we obtain

Hence, there exists a constant , such that if , we have

Lemma 24 (see [5]). Let be an entire function of iterated order and iterated type . Then, for every given , there exists a set that has infinite logarithmic measure, such that, for all , we have

Lemma 25. Let be entire functions with finite iterated order and satisfy , and , and let ,   be stated as in Lemma 20. Then, for any given , there exists a set with infinite logarithmic measure such that

where and .

Proof. We prove this lemma by using mathematical induction.

(i)We first prove that    satisfy (53) when . From the definition of and , we have

From Lemmas 23, 24, and (54), for any , there exists a set with infinite logarithmic measure such that

where is a constant, not necessarily the same at each occurrence.(ii)Next, we show that    satisfy (53) when . From

we have

By the conclusions of (i) and Lemma 23, (55)–(57), we obtain for all

(iii)Now, suppose that (53) holds for ,  ; that is, for any given , there exists a set with infinite logarithmic measure such that

From and   ,,  , we have

Then, from Lemma 23 and (59)-(60), for all ,

for , and

Thus, the proof of Lemma 25 is complete.

Lemma 26. Let be meromorphic functions of finite iterated order with such that and . Then every meromorphic solution of (8) satisfies .

Proof. Assume that is a meromorphic solution of (8). By (8) we have

By Lemma 15, we obtain

where is a set having finite linear measure. By Lemma 19, there exists a set having infinite logarithmic measure such that, for all , we have

Since , then for any given and for all , by (65), we have

From (64) and (66), we have for any given and for all

By (67) and Lemma 16, we obtain

Lemma 27. Let be meromorphic functions of finite iterated order. If there exist positive constants and a set with infinite logarithmic measure such that

hold for all , then every meromorphic solution of (8) satisfies .

Proof. Assume that is a meromorphic solution of (8). By (8), we obtain

By Lemma 22, there exists a set having a finite logarithmic measure such that, for all , we have

where is a constant. Substituting (69) and (71) into (70), we obtain for all

Since , then from (72), Definitions 1, 2, and Lemma 16, we can deduce that and

Lemma 28 (see [13]). Let be an integer, and let be a meromorphic solution of the differential equation

where and are meromorphic functions:

(i)if , then , (ii)if ,  then .

Lemma 29 (see [3]). Let , be entire functions such that . If either : or , then every solution of (8) satisfies and .

Lemma 30 (see [5]). Let be entire functions of finite iterated order, and let . Assume that and . Then every solution of (8) satisfies and .

3. Proof of the Theorems

Proof of Theorem 7. We consider two cases.

Case  1. Suppose that .

(i) First, we prove that . Assume that is a solution of (8). From Lemma 29, we have . Set . Since , then and ,. By substituting into (8), we get that satisfies the following equation:

Set . If , then, from Lemma 29, we have , which is a contradiction. Hence . From the assumptions of Theorem 7, we get

From Lemma 28  (ii), we have .

(ii) Second, we prove that . Set , then . By Lemma 13, we get that satisfies the (15). Set , where    are stated as in Lemma 13. If , then, from Lemmas 20 and 21, we have , a contradiction with . Hence . By Lemma 28 (ii), we have

(iii) Now, we prove that . Set , then . By Lemma 13, we get that satisfies (15). Set , where    are stated as in Lemma 13. If , then, from Lemmas 20 and 21, we have , a contradiction with . Hence . By Lemma 28  , we have

(iv) Set , and then . From Lemmas 13, 20, 21, and 28  (ii), using the same argument as in Case 1  iii, we can get .

(v) We prove that ,  ,  . Set , and then . By Lemma 13, we get that satisfies (15). Set , where    are stated as in Lemma 13. If , then from Lemmas 20 and 21, we have , a contradiction with . Hence . By Lemma 28  , we have   .

Case . Suppose that and .

(i) First, we prove that . Assume that is a solution of (8). From Lemma 30, we have . Set . Since is an entire function satisfying , then and ,  . By substituting into (8), we get that satisfies (75). Set . If , then, from Lemma 30, we have , which is a contradiction. Hence . From the assumptions of Theorem 7, we get

From Lemma 28 (ii), we have .

(ii) Now, we prove that . Set . Since is an entire function satisfying , then . By Lemma 13, we get that satisfies the (15). If , then, from Lemmas 25 and 27, we have , a contradiction with . Hence . By Lemma 28 (ii), we have . Similar to the arguments as in Case 1 (iii)–(v) and by using Lemmas 13, 25, and 27, we can get ,  , . Thus, the proof of Theorem 7 is completed.

Proof of Theorem 8. Since    are polynomials and is a transcendental entire function with ; that is,    satisfy the conditions of Theorem 7 (i). By using the same argument as in Theorem 7 and Lemma 28 (i), we can get the conclusions of Theorem 8 easily.

Proof of Theorem 9. Assume that is a meromorphic solution of (8). By (8) we get that the poles of can only occur at the poles of . Note that the multiplicities of poles of are uniformly bounded, and thus we have

where and are some suitable positive constants. This gives

Applying now (81) with Lemma 17, we obtain

According to the conditions of Theorem 9, we can easily get the conclusions of Theorem 9 by using the similar argument as in Theorem 7, the estimation (82), and Lemmas 26 and 28 (ii).

Acknowledgments

This research is supported by ATRST (Agence Thématique de Recherche en Sciences et Technologies) and University of Mostaganem (UMAB) (PNR Project Code 8/u27/3144).

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