Monotonicity properties and bounds for the complete p-elliptic integrals.

We generalize several monotonicity and convexity properties as well as sharp inequalities for the complete elliptic integrals to the complete p-elliptic integrals.

Introduction

Let and . Then the function and the number are defined by and respectively, where B is the classical beta function. The inverse function of defined on is said to be the generalized sine function and denoted by . From (1.1) and (1.2) we clearly see that and . The generalized sine function and appeared in the eigenvalue problem of one-dimensional p-Laplacian

Indeed, the eigenvalues are given by and the corresponding eigenfunction to is for each  . In the same way one can define the generalized cosine and tangent functions and their inverse functions [1-3].

Let , and a, b and c be the real numbers with  . Then the Gaussian hypergeometric function [4-11] is defined by where denotes the shifted factorial function ,  , and for . The well-known complete elliptic integrals and [12-15] of the first and second kinds are respectively defined by and

Let and . Then the complete p-elliptic integrals and [16, 17] of the first and second kinds are respectively defined by and

From (1.4) and (1.5) we clearly see that the complete p-elliptic integrals and respectively reduce to the complete elliptic integrals and if . Recently, the complete p-elliptic integrals and and their special cases and have attracted the attention of many mathematicians [18-30].

Takeuchi [31] generalized several well-known theorems for the complete elliptic integrals and , such as Legendre’s formula, Gaussian’s approximation formulas for π, differential equations, and other similar results of the theory of complete elliptic integrals to the complete p-elliptic integrals and , and proved that

Anderson, Qiu, and Vamanamurthy [32] discussed the monotonicity and convexity properties of the function and proved that the double inequality holds for all . Both inequalities given in (1.8) are sharp as , while the second inequality is also sharp as . Here and in what follows, we denote , , and .

Alzer and Richards [33] proved that the function is strictly increasing and convex from onto , and the double inequality holds for all with the best constant and .

Inequalities (1.8) and (1.9) have been generalized to the generalized elliptic integrals by Huang et al. in [34].

The main purpose of the article is to generalize inequalities (1.8) and (1.9) to the complete p-elliptic integrals. We discuss the monotonicity and convexity properties of the functions and present their corresponding sharp inequalities.

Lemmas

In order to prove our main results, we need several formulas and lemmas, which we present in this section.

The following formulas for the hypergeometric function and complete p-elliptic integrals can be found in the literature [5, 1.20(10), (1.16), 1.19(4), (1.48)]], [18], and [35, Equation (26)]: where is the classical gamma function.

Lemma 2.1

Let . Then the function is strictly increasing and convex from onto .

Proof

It follows from (1.3), (1.6), and (1.7) that where . Therefore, and is strictly increasing and convex on due to .

From (1.5), (1.6), (2.4), (2.7), and (2.8) we clearly see that  □

Lemma 2.2

(see [18, Lemma 2.3])

Let be an interval and be two positive real-valued functions. Then the product fg is convex on I if both f and g are convex and increasing (decreasing) on I.

Lemma 2.3

Let . Then the function is strictly increasing from onto .

Let Then it follows from (1.3), (1.6), (1.7), (2.9), and (2.10) that Equations (2.11) and (2.12) lead to It is easy to verify that for and .

Therefore, the monotonicity of on the interval follows easily from (2.13) and (2.14).

From (2.3), (2.4), (2.10), (2.11), and (2.13) we clearly see that and  □

Lemma 2.4

(see [5, Theorem 1.25])

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

Main results

Theorem 3.1

Let and be defined by (1.10). Then is strictly increasing and convex from onto , and the double inequality holds for all if and only if and . Moreover, both inequalities in (3.1) are sharp as , while the second inequality is sharp as .

Let be defined by (2.7). Then can be rewritten as

From Lemma 2.1 we know that both the functions and are positive and strictly increasing on , hence is also strictly increasing on .

Next, we prove that is convex on . It follows from (2.6) and (2.7) that where where is defined by (2.9).

From (1.3), (1.6), (1.7), and Lemma 2.1 we clearly see that

Equations (3.3)–(3.5) and Lemmas 2.3 and 2.4 lead to the conclusion that the function is strictly decreasing on , which implies that the function is strictly increasing on and is convex on .

Therefore, is convex on follows from Lemmas 2.1 and 2.2 together with (3.2) and the convexity of .

The limit values follow easily from Lemma 2.1 and (3.2).

It follows from (2.8) and (3.2) that

Therefore, inequality (3.1) holds for all if and only if , and follows easily from (3.6) and (3.7) together with the monotonicity and convexity of on . From (3.6) we clearly see that both inequalities in (3.1) are sharp as and the second inequality is sharp as . □

Theorem 3.2

Let and be defined in (1.11). Then is strictly increasing and convex from onto , and the double inequality holds for all if and only if and . Moreover, both inequalities in (3.8) are sharp as , while the second inequality is sharp as .

Let Then from (1.3), (1.6), (1.7), and (1.11) we get It follows from (2.1), (2.2), (2.5), and (3.10) that Equations (1.3) and (3.12)–(3.14) lead to for .

Therefore, the monotonicity and convexity for on the interval follow from (3.11) and (3.15).

It follows from (1.2), (1.3), (2.3), (3.9), and (3.10) that

Therefore, the desired results in Theorem 3.2 follow easily from (3.11) and (3.16) together with the monotonicity and convexity of on the interval . □

Remark 3.3

Let . Then we clearly see that inequalities (3.1) and (3.8) reduce to inequalities (1.8) and (1.9), respectively.

Corollary 3.4

Let , be defined by (1.11) and Then the double inequality holds for all .

It follows from (3.17) and the proof of Theorem 3.2 that for all .

From (3.17)–(3.19) we get

Therefore, Corollary 3.4 follows easily from Theorem 3.2 and (3.20). □

Methods

The main purpose of the article is to generalize inequalities (1.8) and (1.9) for the complete elliptic integrals to the complete p-elliptic integrals. To achieve this goal we discuss the monotonicity and convexity properties for the functions given by (1.10) and (1.11) by use of the analytical properties of the Gaussian hypergeometric function and the well-known monotone form of l’Hôpital’s rule given in [5, Theorem 1.25].

Results and discussion

In the article, we present the monotonicity and convexity properties and provide the sharp bounds for the functions and on the interval .

The obtained results are the generalization of the well-known results on the classical complete elliptic integrals given in [32, 33].

Conclusion

In this paper, we generalize the monotonicity, convexity, and bounds for the functions involving the complete elliptic integrals to the complete p-elliptic integrals. The given idea may stimulate further research in the theory of generalized elliptic integrals.