Monotonicity rule for the quotient of two functions and its application.

In the article, we provide a monotonicity rule for the function [Formula: see text], where [Formula: see text] is a positive differentiable and decreasing function defined on [Formula: see text] ([Formula: see text]), and [Formula: see text] and [Formula: see text] are two real power series converging on [Formula: see text] such that the sequence [Formula: see text] is increasing (decreasing) with [Formula: see text] and [Formula: see text] for all [Formula: see text]. As applications, we present new bounds for the complete elliptic integral [Formula: see text] ([Formula: see text]) of the second kind.

Introduction

The most commonly used monotonicity rule in elementary calculus is that f is increasing (decreasing) on if is continuous on and has a positive (negative) derivative on , and it can be proved easily by the Lagrange mean value theorem. The functions whose monotonicity we prove in this way are usually polynomials, rational functions, or other elementary functions. But we often find that the derivative of a quotient of two functions is quite messy and the process is tedious. Therefore, the improvements, generalizations and refinements of the method for proving monotonicity of quotients have attracted the attention of many researchers.

In 1955, Biernacki and Krzyż [1] (see also [2], Lemma 2.1, [3]) found an important criterion for the monotonicity of the quotient of power series as follows.

Theorem 1.1

[1]

Let and be two real power series converging on () with for all k. If the non-constant sequence is increasing (decreasing) for all k, then the function is strictly increasing (decreasing) on .

In [4], Cheeger et al. presented the monotonicity rule for the quotient of two functions.

Theorem 1.2

[4]

If f and g are positive integrable functions on such that is decreasing, then the function is decreasing.

Unaware of Theorem 1.2, Anderson et al. [5], Lemma 2.2 (see also [6], Theorem 1.25) established l’Hôpital’s monotone rule that can be applied to a wide class of quotients of functions.

Theorem 1.3

[5]

Let , be continuous on and differentiable on , and on . If is increasing (decreasing) on , then so are the functions If is strictly monotone, then the monotonicity in the conclusion is also strict.

Pinelis [7] provided the following monotonicity theorem.

Theorem 1.4

[7]

Let f and g be differentiable and never vanish on an open interval . Then the following statements are true:

If on , and is increasing on , then on .

If on , and is decreasing on , then on .

If on , and is increasing on , then on .

If on , and is increasing on , then on .

Recently, Yang et al. [8], Theorem 1.2, established a more general monotonicity rule for the ratio of two power series.

Theorem 1.5

[8]

Let and be two real power series converging on and for all k, and . Suppose that for certain , the non-constant sequence is increasing (decreasing) for and decreasing (increasing) for . Then the function is strictly increasing (decreasing) on if and only if . Moreover, if , then there exists such that the function is strictly increasing (decreasing) on and strictly decreasing (increasing) on .

The foregoing monotonicity rules have been used very effectively in the study of special functions [9-23], differential geometry [4, 24], probability [25] and approximation theory [26]. The main purpose of the article is to present the monotonicity rule for the function and to provide new bounds for the complete elliptic integral of the second kind. Some complicated computations are carried out using Mathematica computer algebra system.

Monotonicity rule

Theorem 2.1

Let be a positive differentiable and decreasing function defined on (), let and be two real power series converging on . If , for all and the non-constant sequence is increasing (decreasing), then the function is strictly increasing (decreasing) on .

Proof

Let , and Then differentiating gives Note that can be rewritten as If , for all and the non-constant sequence is increasing (decreasing), then we clearly see that for all and for all .

It follows from is a positive differentiable and decreasing function on that for all .

Therefore, (<) 0 for all follows easily from (2.1)-(2.6), and the proof of Theorem 2.1 is completed. □

Bounds for the complete elliptic integral of the second kind

For , Legendre’s complete elliptic integral [27] of the second kind is given by

It is well known that , , and is the particular case of the Gaussian hypergeometric function where and () is the gamma function. Indeed, we have

Recently, the bounds for the complete elliptic integral of the second kind have been the subject of intensive research. In particular, many remarkable inequalities for can be found in the literature [28-41]. Vuorinen [42] conjectured that the inequality holds for all , where, and in what follows, . Inequality (3.2) was proved by Barnard et al. in [43].

Very recently, the accurate bounds for in terms of the Stolarsky mean were given in [44, 45]: where .

In this section, we shall use Theorem 2.1 to present new bounds for the complete elliptic integral of the second kind. In order to prove our main result, we need three lemmas, which we present in this section.

Lemma 3.1

see [46], Lemma 7

Let and with , for all , and Then there exists such that , for and for .

Lemma 3.2

see [47, 48]

The double inequality holds for all and .

Lemma 3.3

Let , , , and be defined by respectively. Then for all .

Let , , , and be defined by respectively.

Then from (3.5)-(3.13) and elaborated computations we get

From Lemma 3.1 and (3.19) together with the facts that and for , we clearly see that there exists such that the sequence is increasing for and decreasing for , which implies that

It follows from Lemma 3.2 that for all .

From (3.10), (3.11), (3.17), (3.18), (3.20) and (3.21) we get for all .

Therefore, Lemma 3.3 follows easily from (3.14)-(3.16) and (3.22) together with the facts that , and for . □

Theorem 3.4

The double inequality holds for all , where

Let , , , , and be defined by respectively.

Then from (3.1) and (3.24)-(3.28) we have where where and is defined by (3.9).

It follows from (3.9), (3.29), (3.30) and elaborated computations that where is defined by (3.8).

It is not difficult to verify that for all .

From Lemma 3.3, (3.30), (3.33) and (3.34) we know that for all , and the sequence is decreasing.

Equation (3.25) implies that for , and is decreasing on .

It follows from Theorem 2.1, (3.31) and (3.32) together with the monotonicity of the sequence and the function on that the function is strictly decreasing on and for all .

Note that (3.24), (3.28) and (3.31) lead to the conclusion that

Therefore, Theorem 3.4 follows from (3.28), (3.37) and (3.38). □

Remark 3.5

Let where is defined by (3.24). Then simple computations lead to

From (3.3), (3.4), (3.23) and (3.39)-(3.45) we clearly see that there exists small enough such that the lower bound given in (3.23) for is better than the lower bound given in (3.3) for , the lower bound given in (3.23) for is better than the lower bound given in (3.4) for , the upper bound given in (3.23) for is better than the upper bound given in (3.3) for , and the upper bound given in (3.23) for is better than the upper bound given in (3.4) for .

Corollary 3.6

Let be defined by (3.24). Then the double inequality holds for all .

Let be defined by (3.28) and

Then we clearly see that

From (3.49) and the proof of Theorem 3.4 we know that is strictly decreasing on and is strictly increasing on . Therefore, inequality (3.46) follows from (3.47) and (3.48) together with the monotonicity of on the interval . □

Corollary 3.7

Let be defined by (3.24). Then the double inequality holds for all .

Let be defined by (3.47) and

Then we clearly see that

From (3.53) and the proof of Corollary 3.6 we know that both and are strictly increasing on . Therefore, inequality (3.50) follows from (3.51) and (3.52) together with the monotonicity of on the interval . □

Remark 3.8

From Corollaries 3.6 and 3.7 we have for all , which implies that both the absolute and relative errors using to approximate are less than 0.14%.

Conclusions

In this paper, we find a monotonicity rule for the function . As applications, we present new bounds for the complete elliptic integral () of the second kind, and we show that our bounds are sharper than the previously known bounds for some .