Journal of Function Spaces
Volume 2018 (2018), Article ID 7028597, 6 pages
https://doi.org/10.1155/2018/7028597

Research Article

On the Maximum Term and Central Index of Entire Functions and Their Derivatives

Jianren Long1,2 and Zhigao Qin¹

¹School of Mathematical Sciences, Guizhou Normal University, Guiyang 550001, China
²School of Computer Sciences and School of Sciences, Beijing University of Posts and Telecommunications, Beijing 100876, China

Correspondence should be addressed to Jianren Long; longjianren2004@163.com

Received 4 January 2018; Revised 16 March 2018; Accepted 16 April 2018; Published 7 June 2018

Academic Editor: Raúl E. Curto

Copyright © 2018 Jianren Long and Zhigao Qin. 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 shall establish some criteria on entire series with finite logarithmic order in terms of maximum term and central index.

1. Introduction

A function is called meromorphic, if it is nonconstant and analytic in the complex plane except at possible isolated poles. If no poles occur, then reduces to an entire function. In what follows, we assume that the reader is familiar with the standard notation and fundamental results in Nevanlinna theory of meromorphic functions; see [1, 2] or [3] for more details. We often use the order of growth and the lower order of growth to measure the growth of a meromorphic function. For a meromorphic function in , the order of growth and the lower order of growth of are defined by

and

respectively. If is entire function in , the order and lower order of are defined also by ; i.e., is replaced with in above equalities, where . By the following inequalities which can be found in [3, p. 10], then the order and lower order are same by definition of and :

which hold for all .

The theory of meromorphic functions of finite positive order is fairly complete as compared to the theory of functions of order zero. Techniques that work well for functions of finite positive order often do not work for functions of order zero. In order to make some progress with functions of order zero, Chern introduced the concept of logarithmic order in [4]. For an entire function of zero order, the logarithmic order of is defined by

For an nonconstant entire function , we must have , by the usual proof of Liouville’s theorem. It is easily seen that if has logarithmic order then so has the function for . Furthermore, the function is again of logarithmic order , while has logarithmic order . It is clear that for a polynomial of degree the logarithmic order is 1. There exists also transcendental entire series such that its logarithmic order is of one; for each positive number , put , and set

by a calculation [5, p. 6] we have ; the example can also be found in [6]. On the other hand, there exists transcendental entire series such that its logarithmic order is bigger than one. For each positive number (>1), put and , and set

and by a direct calculation [5, p. 6] we have ; the example can also be found in [6]. Another case is of infinite logarithmic order; let and suppose that ; then

is of order zero, but its logarithmic order is infinite [7]. More results regarding logarithmic order can be found in [8–11].

Wiman-Valiron theory is one of the important concepts in entire function theory; in the present paper, we study the properties of entire functions by Wiman-Valiron theory. To this end, we also need the following notations. Let be a transcendental entire series in . Then the maximum term and central index of are denoted as and .

In [6], Chern and Kim consider some criteria conditions of logarithmic order with terms of maximum term and central index and proved the following consequence.

Theorem A. Let be a transcendental entire series with finite logarithmic order; then the following statements are equivalent.

(1) has logarithmic order .(2) has logarithmic order .(3) has logarithmic order .(4) has logarithmic order .

Although, for any given entire series of positive finite order, and both have the same order, the proof can be found in [12] or [2], but the situation is different for function of finite logarithmic order; from the Theorem A, we have

In [7], Berg and Pedersen described the logarithmic order by using Taylor coefficient of entire function and obtained the following result.

Theorem B. Let be a transcendental entire series with finite logarithmic order; then its logarithmic order satisfies

Now the purpose of the paper is that the logarithmic order is described by using other forms in terms of maximum term and central index. To the end, we also need the following notations. Let and , where , , denotes the th-derivative of . We reckon , , from the first term of the series of . For the uniformity in the notation we write and . We denote the th-derivative of by at the point of its existence in . It is easily seen that the functions and , , are positive, nondecreasing, and unbounded functions of , having only ordinary discontinuities and . For the entire function , in the present paper, we find a precise measure of the rates of growth of , , and as in terms of the parameters defined in (4). These results will be shown in Section 2.

This paper is organized as follows. In Section 2, we will state main results and prove them. In Section 3, we will discuss some further results.

2. Main Results and Proofs

In the proof of our theorems, the growth relationship between meromorphic function and its th-derivative is needed. We prove the following result by using similar way in [[8], Theorem 3] and then omit the proof of details.

Lemma 1. Let be a transcendental meromorphic function in with zero order; then and , , have the same logarithmic order.

The following two consequences are due to G. Valiron [13], which can also be found in [12] or [2].

Lemma 2. Let be an entire function in . Then, for any ,

Lemma 3. Let be an entire function in . Then, for all ,

The first result is stated as follows.

Theorem 4. Let be a transcendental entire series with finite logarithmic order . Then

Proof. By Lemma 1, we know that the logarithmic order of a function and its derivative are the same. In view of (8), we have

Now, for , let , where , and let and ; then

It follows that

On the other hand, for , we have

This implies

By using (15) and (17), we get

By (18) and a simple calculation, we have

This implies

From (20), we get

Combining the inequality above and (13), we obtain

The proof of Theorem 4 is completed.

Theorem 5. Let be a transcendental entire series with finite logarithmic order . Let , . Then, for , one has

Proof. Let , then, for , we have

and

The functions and are constants in intervals and have at most an enumerable number of discontinuities and so their derivatives vanish almost everywhere except possibly at a set of measure zero. Taking logarithmic order of both the sides of (25), differentiating with respect to , and denoting the derivative of by at the point of its existence, we have for almost all values of

By (26), for sufficiently large , we get

By using Lemma 2, for , we have

Combining (27) and (28), for sufficiently large , we have

i.e.,

Thus, for ,

In view of Theorem 4, (31) and (12) yield (23). This completes the proof of Theorem 5.

Theorem 6. Let be a transcendental entire series with finite logarithmic order . Then, for and for almost all values of satisfying ,

and

where is set of with zero measure which does not exist.

Proof. Since is step function with , so is differential everywhere except at an enumerable set of points of discontinuities of and . Hence, we have, at the points of existence of ,

for the derivatives of and vanish almost everywhere.
This implies

By differentiating (35) at the points of existence of and , we get

Combining (35) and (36), we have

On repeating the differentiation times, we have

This proves (32).
Now, using (38), for , and then multiplying the -inequalities thus obtained give

Thus,

Combining (40) and (13), we get (33), on proceeding to limits as and . This completes the proof of Theorem 6.

By the proof of Theorem 6, we can get the following.

Corollary 7. Let be a transcendental entire series. Then, for ,

and

as and .

3. Further Discussion

In [4], Chern introduced the definition of logarithmic order of meromorphic function; however, there are not discussions of the lower logarithmic order. Hence, by using similar definition of lower order, we can define the lower logarithmic order and discuss some properties of entire functions in terms of lower logarithmic order.

Definition 8. Let be an entire function with zero order. Then its lower logarithmic order is defined by

where .

In Theorem A, Chern obtained the growth relationship of logarithmic order by using maximum modulus, maximum term, and central index. In this section, we will try to find the growth relationship of lower logarithmic order by using maximum modulus, maximum term, and central index. To this end, let be a transcendental entire series and set

and

Then we have the following consequences.

Theorem 9. Let be a transcendental entire series with finite logarithmic order. Then .

Proof. By using Cauchy inequality, we have for all . Hence, . Thus we just only prove . By Lemma 2, there exists , such that, for all ,

By Lemma 3 for , we get

Combining (46) and (47), we have

where is positive constant. By (43) and (48), we have .

Theorem 10. Let be a transcendental entire series with finite logarithmic order. Then .

Proof. By using similar way of Theorem 9, we get (46); hence is obvious.
On the other hand, set ; then, we get . Hence,

where is positive constant number. This implies that . So, we have .

Under Theorems 9 and 10, we will consider similar results of Section 2 with respect to lower logarithmic order. To this end, let denote the class of functions of transcendental entire function in , satisfying

By using similar way to Section 2, we can also prove the following consequences.

Theorem 11. Suppose that belongs to . Then, for ,

Theorem 12. Suppose that belong to . Then, for and ,

Theorem 13. Suppose that belongs to . Then, for ,

where is set of with zero measure where does not exist.

Data Availability

The data used to support the findings of this study are included within the article.

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

Authors’ Contributions

The first author wrote the main part of the manuscript, and the second author pointed out many valuable ideas to modify the present manuscript. All authors read and approved the final manuscript.

Acknowledgments

This research work is supported by the National Natural Science Foundation of China (Grant no. 11501142), the Foundation of Science and Technology of Guizhou Province of China (Grant no. 2112), the Foundation of Doctoral Research Program of Guizhou Normal University 2016, and the Foundation of Qian Ceng Ci Innovative Talents of Guizhou Province 2016.

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