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

We shall establish some criteria on entire series with finite logarithmic order in terms of maximum term and central index.


Introduction
A function  is called meromorphic, if it is nonconstant and analytic in the complex plane C 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 C, the order of growth and the lower order of growth of  are defined by and respectively.If  is entire function in C, the order and lower order of  are defined also by log (, ); i.e., (, ) is replaced with log (, ) in above equalities, where (, ) = max ||= |()|.By the following inequalities which can be found in [3, p. 10], then the order and lower order are same by definition of (, ) and log (, ): (, ) ≤ log  (, ) ≤  +   −   (, ) 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  log () ≥ 1, by the usual proof of Liouville's theorem.It is easily seen that if () has logarithmic order  then so has the function ( + ) for  ̸ = 0. Furthermore, the function ()  is again of logarithmic order , while (  ) has logarithmic order .It is clear that for a polynomial of degree  ≥ 1 the logarithmic order is 1.There exists also transcendental entire series () such that its logarithmic order is of one; for each positive number  > 1, put   =    , and set by a calculation [5, p. 6] we have  log () = 1; 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  =  − 1 > 0 and   =   1/ , and set and by a direct calculation [5, p. 6] we have  log () =  > 1; the example can also be found in [6].Another case is of infinite logarithmic order; let  ∈ C and suppose that 0 < || < 1; then is of order zero, but its logarithmic order is infinite [7].More results regarding logarithmic order can be found in [8][9][10][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 () = ∑ ∞ =0     be a transcendental entire series in C. Then the maximum term (, ) and central index ](, ) of  are denoted as () ≡ (, ) = max ≥0 {|  |  } and ]() ≡ ](, ) = max{ : (, ) = |  |  }.
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.
Although, for any given entire series  of positive finite order, log (, ) 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.
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.

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 C with zero order; then  and  () ,  = 1, 2, . .., have the same logarithmic order.
The following two consequences are due to G. Valiron [13], which can also be found in [12] or [2].
The first result is stated as follows.
By the proof of Theorem 6, we can get the following.

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.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 () = ∑ ∞ =0     be a transcendental entire series and set