Growth of Logarithmic Derivatives and Their Applications in Complex Differential Equations

Abstract

We continue the study of the behavior of the growth of logarithmic derivatives. In fact, we prove some relations between the value distribution of solutions of linear differential equations and growth of their logarithmic derivatives. We also give an estimate of the growth of the quotient of two differential polynomials generated by solutions of the equation where and are entire functions.

1. Introduction and Main Results

Throughout this paper, we assume that the reader is familiar with the fundamental results and the standard notations of Nevanlinna's value distribution theory (see [1–4]). In addition, we will denote by (resp., ) and (resp., ) the exponent of convergence of zeros (resp., distinct zeros) and poles (resp., distinct poles) of a meromorphic function , to denote the order of growth of . A meromorphic function is called a small function with respect to if as except possibly a set of of finite linear measure, where is the Nevanlinna characteristic function of .

Definition 1 (see [4]). Let be a meromorphic function. Then the hyperorder of is defined as

Definition 2 (see [1, 3]). The type of a meromorphic function of order is defined as If is an entire function, then the type of of order is defined as where is the maximum modulus function.

Remark 3. For entire function, we can have . For example, if , then we have and .

Growth of logarithmic derivative of meromorphic functions has been generously considered during the last decades, among others, by Gol'dberg and Grinšteĭn [5], Benbourenane and Korhonen [6], Hinkkanen [7], and Heittokangas et al. [8]. All these considerations have been devoted to getting more detailed estimates to the proximity function than what is the original Nevanlinna estimate that essentially can be written as . The same applies for the case of higher logarithmic derivatives as well. The estimation of logarithmic derivatives plays the key role in theory of differential equations. In his paper Gundersen [9] proved some interesting inequalities on the module of logarithmic derivatives of meromorphic functions. Recently [10], the authors have studied some properties of the behavior of growth of logarithmic derivatives of entire and meromorphic functions and have obtained some relations between the zeros of entire functions and the growth of their logarithmic derivatives. In fact, they have proved the following.

Theorem A (see [10]). Let be an entire function with finite number of zeros. Then, for any integer ,

Theorem B (see [10]). Suppose that is an integer and let be a meromorphic function. Then

In [11, pages 457–458], Bank and Langley have obtained the following result.

Theorem C (see [11]). All nontrivial solutions of where is a nonconstant polynomial, satisfy .

Theorem D (see [4] (E. Picard)). Any transcendental entire function must take every finite complex value infinitely many times, with at most one exception.

Example 4. The entire function omits the value zero and every other finite value is assumed infinitely often and zero is the exceptional value of Picard.

Question 1. Under what conditions can we ensure that has no Picard's exceptional value?

Question 2. Under what conditions can we obtain the same result as Theorem C for higher order linear differential equations with entire coefficients?

The subject of this paper is to give answers to the above questions. In fact, we prove some relations between the value distribution of solutions of linear differential equations and growth of their logarithmic derivatives. Firstly, we study the relationship between the growth of logarithmic derivatives of solutions of complex differential equation and their exponents of convergence. We will also prove the nonexistence of any Picard exceptional value of solutions of (7), and we obtain the following.

Theorem 5. Let be entire functions satisfying , , and let , if . If is a nontrivial solution of (7), such that , then assume every finite complex value infinitely often.
Furthermore, if , then

Corollary 6. Under the hypotheses of Theorem 5, let be an entire function with finite order. If is a nontrivial solution of (7), such that for some , then

Corollary 7. Under the hypotheses of Theorem 5, every nontrivial solution of (7) does not have any Picard exceptional values.

Example 8. It is clear that the function satisfies the differential equation and . Obviously, the conditions of Theorem 5 are satisfied. Then .

Secondly, we consider the differential equation where and are entire functions, and let and be two differential polynomials defined by where and are entire functions not all vanishing identically. In recent years, there is a great interest to investigate the growth and oscillation of differential polynomials generated by solutions of differential equations in the complex plane (see [12–14]). In this part, we give under some conditions an estimate on the growth of the quotient of two differential polynomials generated by solutions of (11), and we obtain the following results.

Theorem 9. Let be a nontrivial solution of (11), and let and be entire functions not all vanishing identically such that is irreducible function in . If then

Remark 10. In Theorem 9, the condition (13) is necessary. For example, is a solution of the differential equation with and let If we take , , and , then

Remark 11. In Theorem 9, the condition that is irreducible function in is necessary. If , then and so By setting in Theorem 9, we can obtain the following special case.

Corollary 12. Let and be entire functions of finite order such that and if . Let be entire functions that are not all vanishing identically such that and . If is a solution of (11), such that , then

Corollary 13. Under the hypotheses of Corollary 12, if is a nontrivial solution of (11), such that , then

2. Auxiliary Lemmas

Lemma 14 (see [1]). Let be a meromorphic function and let . Then where , possibly outside a set with a finite linear measure. If is a finite order of growth, then

Lemma 15 (see [15]). Let be entire functions of finite order such that Then every solution of (7) satisfies and .

Lemma 16 (see [16]). Let be entire functions satisfying , , and let , if . Then every solution of (7) satisfies and .

Here, we give a special case of the result given by Xu et al. in [17].

Lemma 17. Let be entire functions with finite order and let them satisfy one of the following conditions:(i); (ii) and . Then, for every solution of (7) and for any entire function satisfying , we have

Lemma 18 (see [10]). Let be a meromorphic function. If there exists an integer , such that and , then Furthermore, if is entire function, then

By the principle of mathematical induction we can obtain easily the following lemma.

Lemma 19. Suppose that , where and are entire functions. Then, for an integer , there exist entire functions and such that

Lemma 20 (see [18]). Let be a solution of (7) where the coefficients are analytic functions in the disc , . Let be the number of nonzero coefficients , and let and . If such that for some , then, for all , where is a constant satisfying

Here, we give a special case of the result given by Cao et al. in [19].

Lemma 21. Let be a meromorphic function with finite order and type . Then for any given there exists a subset of that has infinite logarithmic measure such that holds for all .

Lemma 22. Let and be entire functions such that , , and let and if . If is a solution of then and .

Proof. If , then, by Lemma 15, we obtain the result. We prove only the case when and . Since , then, by (32), we have Suppose that is of finite order; then by (33) and Lemma 14 we obtain which implies the contradiction Hence . By using inequality (30) for , we have On the other hand, since , then, by (33) and Lemma 14, where , possibly outside a set with a finite linear measure. By we choose , satisfying such that, for , we have By Lemma 21, there exists a subset of infinite logarithmic measure such that By (37)–(39) we obtain, for all , which implies By using (36) and (41), we obtain .

Remark 23. Lemma 22 was obtained by Tu and Yi in [16] for higher order linear differential equations by using the type (Lemma 16 in the present paper).

Lemma 24 (see [20]). Let and be meromorphic functions such that , and , . Then we have the following.(i)If , then we obtain (ii)If and , then we get

3. Proof of Theorems and Corollaries

Proof of Theorem 5. By the conditions of Theorem 5 and Lemma 16, every nontrivial solution of (7) is of infinite order and . We need to prove that every solution attains every complex value without exception; that is, for all , the equation has infinitely many solutions. We have two cases(i)If , then, by Lemma 17, we have and the equation has infinitely many solutions.(ii)If , then we suppose that has a finite number of zeros. By Theorem A which is a contradiction, and hence assume every finite complex value infinitely often. Now, if and since , then by using Lemma 18 we have

Proof of Corollary 6. By using Lemma 17, we have for any finite order entire function Now, if , then we put . It follows that and, since , then, by applying Lemma 18, we obtain that is, and the proof of Corollary 6 is completed.

Proof of Theorem 9. By using Lemma 19, we have which we can write as where It is clear that and, since is irreducible function in , then . So On the other hand, we have which implies By (53) and (55), we obtain .

Proof of Corollary 12. Under the conditions of Corollary 12 and Lemma 22, we have and . Set Substituting into , we obtain where . In order to prove that is irreducible function in , we need only to prove that , where Since and , if , , then, by Lemma 24, we have , so . By applying Theorem 9 we obtain .

Proof of Corollary 13. Suppose that is a nontrivial solution of (11) such that . Then, by Lemma 22, we have By setting and and using Corollary 12 we have which implies by using Lemma 18 that

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.