Journal of Function Spaces
Volume 2019 (2019), Article ID 4752856, 7 pages
https://doi.org/10.1155/2019/4752856

Research Article

Functional Inequalities for Generalized Complete Elliptic Integrals with Two Parameters

Xiangkai Dou,¹ Li Yin,¹ and Xiu-Li Lin²

¹College of Science, Binzhou University, Binzhou City, Shandong Province, 256603, China
²College of Mathematics Science, Qufu Normal University, Qufu City, Shandong Province, 273165, China

Correspondence should be addressed to Li Yin; yinli_79@163.com

Received 4 May 2019; Accepted 27 June 2019; Published 7 July 2019

Academic Editor: Maria Alessandra Ragusa

Copyright © 2019 Xiangkai Dou et al. 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.

In this paper, we establish some functional inequalities for generalized complete elliptic integrals with two parameters, such as estimation of bounds and mean inequalities. Our main results give -analogues to the early results for classical complete elliptic integrals.

1. Introduction

For the given complex numbers with , the Gaussian hypergeometric function is defined by

Here for , and for is the shifted factorial or Appell symbol The following is the integral representation formula of the Gaussian hypergeometric function:

; see [1]. For more information on the history, background, properties, and applications, please refer to [1, 2] and related references. Here, we give the following definitions of some classical functions. For real number , the Euler gamma function and its logarithmic derivative , the so-called digamma function, are defined by (cf. [1, 3])

respectively.

Recently, Takeuchi [4] studied the -trigonometric functions depending on two parameters. For , these functions reduce to the so-called -trigonometric functions introduced by Lindqvist [5]. The following ()-eigenvalue problem with Dirichlét boundary condition was considered by Drábek and Manásevich [3]. Let For and

They found the complete solution to this problem. The solution to this problem also appears in [4, Thm 2.1]. In particular, for the function is a solution to this problem with , where

For , reduces to ; see [6]. In order to give the definition of the function , first of all we define its inverse function and then the function itself. For , set

The function is an increasing homeomorphism, and

is defined on the interval . The function can be extended to by

By oddness, the further extension can be made to . Finally, the function is extended to whole by -periodicity; see [7]. Similarly, the other generalized inverse trigonometric and hyperbolic functions that appeared in the current paper are defined as follows:

For the expression of the above generalized inverse trigonometric and hyperbolic functions in terms of hypergeometric functions, we have the following formula [8, 9]:

Based on the above definitions, we simply introduce generalized elliptic integrals defined by Takeuchi [10] with two parameters. For all and , the complete -elliptic integrals of the first and second kinds [11] are defined by

and

respectively. Here, and . The complete -elliptic integrals can be expressed in terms of hypergeometric functions as follows:

and

If , we can obtain the classically complete elliptic integrals as follows: for ,

If , we can obtain the generalized complete elliptic integrals with single parameters

It is worth noting that Bhayo and Yin also gave new -complete elliptic integrals in [12]. For more on this topic, the readers can see related references [12–25]. Some recent results regarding the case of semielliptical crack in a cylindrical rod for viscoelastic medium can be found in [26, 27].

2. Lower and Upper Bounds for Generalized Complete Elliptic Integrals

In 1992, Anderson et al. [28] discovered that the complete elliptic integral of the first kind can be approximated by the inverse hyperbolic tangent function:

For , they proved

Later, Qi and Huang [29] obtained the following inequality by using Chebyshev inequality:

In 2004, Alzer and Qiu [30] proved that, for , we have

with the best possible constants and Here, we shall show some new inequalities for the generalized complete elliptic integrals.

Lemma 1 (see [31, Lemma 1]). Consider the power series and , where and for all , and suppose that both converge on If the sequence is increasing (decreasing), then the function is increasing (decreasing) on

Theorem 2. Let and let be as (11) and (12). Then we have the following:
(1) The function is strictly decreasing from onto

(2) The function is strictly decreasing from onto
(3) The function is strictly decreasing from onto
(4) The function is strictly decreasing from onto

Proof. The proofs of assertion (1)-(4) are similar to each other. Here we mainly prove (1) for the sake of simplicity. First of all, we consider the function defined by

Considering Lemma 1, we only need to discuss the monotonicity of the sequence defined by

Since

we get that the sequence is strictly decreasing. On the other hand, making use of Gauss formula

we may conclude that

The proof of assertion (1) is complete. Due to (4), we only note the known identity

Similar method can complete the proof. Here we omit the details.

As a direct result of Theorem 2, we have the following Corollary 3.

Corollary 3. For and , we have
(1)
(2)
(3)
(4)
where the constants in the above inequalities are the best possible.

Theorem 4. Let , and . The function is strictly increasing from onto As a result, we have the following inequality:

Proof. Define

And is defined by

Simple computation yields

This implies that the sequence is strictly increasing. Using Lemma 1, we obtain that the function is strictly increasing on . The limiting value follows easily. This completes the proof.

Remark 5. Corollary 3 and Theorem 4 give -analogues to the right of (18), (20), and inequality (19). Considering the left of (20), we naturally pose the following open question: for and , the following inequality holds true:

Theorem 6. Let and . Then

Proof. Using the known fact [32], for , the function is strictly increasing from onto We easily complete the proof by taking

Theorem 7. Let and . Then

Proof. We apply the following fact [32]: for , the function is strictly decreasing on if and only if By taking , we easily verify the condition based on . Due to limiting value, we have

by using the formula So, the proof is complete.

3. Mean Inequalities for Generalized Complete Elliptic Integrals

For two distinct positive real numbers and , the arithmetic mean, geometric mean, logarithmic mean, and identric mean are, respectively, defined by

Let be continuous, where is a subinterval of . Let and be the means defined above; then we call that the function is -convex (concave) if

for all . Recently, generalized convexity/concavity with respect to general mean values has been studied by Anderson et al. in [28]. In [33], Baricz studied that if the function is differentiable, then it is -convex (concave) on if and only if is increasing (decreasing). In [34], Bhayo and Vuorinen established all kinds of mean inequalities for the generalized trigonometric functions. In this section, we shall show logarithmic and identic means inequalities for the generalized complete elliptic integrals by using Chebyshev inequality.

Lemma 8 (see [29]). Let be integrable functions, both increasing or both decreasing. Furthermore, let be a positive, integrable function. Then

If one of the functions or is nonincreasing and the other is nondecreasing, then the inequality in (39) is reversed.

Lemma 9 (see [35]). For , then

if the function is convex on .

Theorem 10. Let and . Then

Proof. Let us suppose without loss of generality. Define

Simple computation yields

This implies that the function is strictly log-convex on . By using the fact that the integral preserves monotonicity and log-convexity, we get that the function is strictly increasing and log-convex on . The substitution results in

Since the functions and are strictly increasing, we may obtain

by taking in Lemma 8. This is equivalent to

Noting that log-convexity implies the convexity and Lemma 9, we have

where we apply the known inequality .

Theorem 11. Let and . Then

Proof. This proof is similar to Theorem 10. We still suppose . Direct calculation yields

Since the functions and are strictly increasing, we have

by taking in Chebyshev inequality of Lemma 8. Applying Lemma 9 and the known inequality , we easily prove

The proof is complete.

Remark 12. Our method may be an effective way to deal with Baricz’s conjecture: if are two-variable means, i.e., for and for all , we have

and whenever ; then find conditions on and for which the inequality

holds true for all , where .

4. Grünbaum Type Inequalities for Generalized Complete Elliptic Integrals

Lemma 13 (see [9, Lemma 3, p246]). Let us consider the function , where . If the function , defined by , is increasing on , then, for the function , defined by , we have the following Grünbaum type inequality:

where and . If the function is decreasing, then inequality (54) is reversed.

Lemma 14. As a function of , the function is increasing if and only if . In particular, the function is increasing on .

Proof. Simple calculation yields

This completes the proof.

Theorem 15. Let and . Then

Proof. By defining and applying Lemmas 13 and 14, the proof can easily be completed.

Remark 16. Since is increasing on (0,1) and is decreasing on (0,1), we also obtain the following Grünbaum type inequalities by applying Lemma 13:

5. Conclusion

We mainly established some functional inequalities for generalized complete elliptic integrals with two parameters. First of all, our results give -analogues to the early results for classical complete elliptic integrals. Moreover, we show logarithmic and identic means inequalities by using Chebyshev inequality. Furthermore, we show a Grünbaum type inequality.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no competing interests.

Authors’ Contributions

All authors contributed equally to the manuscript and read and approved the final manuscript.

Acknowledgments

This work was supported by National Natural Science Foundation of China (Grant No. 11701320), the Science and Technology Foundations of Shandong Province (Grants Nos. J18KB105 and J17KA161), and Science Foundation of Binzhou University (Grant No. BZXYLG1903).

References

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables, vol. 55 of Applied Mathematics Series, National Bureau of Standards, Washington, DC., USA, 1965.

G. E. Andrews, R. Askey, and R. Roy, Special Functions, Cambridge University Press, Cambridge, UK, 1999.

P. Drábek and R. Manásevich, “On the closed solution to some p-Laplacian nonhomogeneous eigenvalue problems,” Differential and Integral Equations, vol. 12, no. 6, pp. 773–788, 1999.

S. Takeuchi, “Generalized Jacobian elliptic functions and their application to bifurcation problems associated with p-Laplacian,” Journal of Mathematical Analysis and Applications, vol. 385, no. 1, pp. 24–35, 2012.

P. Lindqvist, “Some remarkable sine and cosine functions,” Ricerche di Matematica, vol. 44, no. 2, pp. 269–290, 1995.

Á. Baricz, A. Bhayo, and M. Vuorinen, “Turán type inequalities for generalized inverse trigonometric functions,” Filomat, vol. 29, no. 2, pp. 303–313, 2015.

D. E. Edmunds, P. Gurka, and J. Lang, “Properties of generalized trigonometric functions,” Journal of Approximation Theory, vol. 164, no. 1, pp. 47–56, 2012.

Á. Baricz, B. A. Bhayo, and T. K. Pogány, “Functional inequalities for generalized inverse trigonometric and hyperbolic functions,” Journal of Mathematical Analysis and Applications, vol. 417, no. 1, pp. 244–259, 2014.

Á. Baricz, B. A. Bhayo, and R. Klén, “Convexity properties of generalized trigonometric and hyperbolic functions,” Aequationes Mathematicae, vol. 89, no. 3, pp. 473–484, 2015.

S. Takeuchi, “The complete p-elliptic integrals and a computation formula of πp for p=4,” Ramanujan Journal, vol. 46, no. 2, pp. 309–321, 2018.

T. Kamiya and S. Takeuchi, “Complete (p,q)-elliptic integrals with application to a family of means,” Journal of Classical Analysis, vol. 10, no. 1, pp. 15–25, 2012.

B. A. Bhayo and L. Yin, “On generalize d(p,q)-elliptic integrals,” http://arxiv.org/abs/1507.00031.

G. D. Anderson, S.-L. Qiu, M. K. Vamanamurthy, and M. Vuorinen, “Generalized elliptic integrals and modular equation,” Pacific Journal of Mathematics, vol. 192, no. 1, pp. 1–37, 2000.

G. D. Anderson, M. K. Vamanamurthy, and M. Vuorinen, “Functional inequalities for the complete elliptic integrals and their ratios,” SIAM Journal on Mathematical Analysis, vol. 21, no. 2, pp. 536–549, 1990.

Á. Baricz, “Turán type inequalities for hypergeometric functions,” Proceedings of the American Mathematical Society, vol. 136, no. 09, pp. 3223–3229, 2008.

Á. Baricz, “Turán type inequalities for generalized complete elliptic integrals,” Mathematische Zeitschrift, vol. 256, no. 4, pp. 895–911, 2007.

B. A. Bhayo and M. Vuorinen, “On generalized complete elliptic integrals and modular functions,” Proceedings of the Edinburgh Mathematical Society, vol. 55, no. 3, pp. 591–611, 2012.

B. A. Bhayo and L. Yin, “On a function involving generalized complete (p, q)-elliptic integrals,” Arabian Journal of Mathematics, 2019, http://arxiv.org/abs/1606.03621.

V. Heikkala, M. K. Vamanamurthy, and M. Vuorinen, “Generalized elliptic integrals,” Computational Methods and Function Theory, vol. 9, no. 1, pp. 75–109, 2009.

G. Wang, X. Zhang, and Y. Chu, “Inequalities for the generalized elliptic integrals and modular functions,” Journal of Mathematical Analysis and Applications, vol. 331, no. 2, pp. 1275–1283, 2007.

S. Takeuchi, “Legendre-type relations for generalized complete elliptic integrals,” Journal of Classical Analysis, no. 1, pp. 35–42, 2012.

L. Yin and L.-G. Huang, “Inequalities for generalized trigonometric and hyperbolic functions with two parameters,” Journal of Nonlinear Sciences and Applications, vol. 8, no. 4, pp. 315–323, 2015.

L. Yin and L. G. Huang, “A survey for generalized trigonometric and hyperbolic functions,” Journal of Mathematical Inequalities, vol. 13, no. 3, 2019, http://rgmia.org/papers/v20/v20a38.pdf.

L. Yin, L. Huang, Y. Wang, and X. Lin, “An inequality for generalized complete elliptic integral,” Journal of Inequalities and Applications, vol. 2017, no. 1, article no 303, 2017.

L. Yin and L. F. Mi, “Landen type inequalities for generalized complete elliptic integrals,” Advanced Studies in Contemporary Mathematics, vol. 26, no. 4, pp. 717–722, 2012.

B. Mostefa, D. Abdelkader, A. Aid, and N. Benseddiq, “Time dependent J-integral solution for semi-elliptical surface crack in HDPE,” CMC-Computers Materials & Continua, vol. 45, no. 3, pp. 163–186, 2015.

M. Ruggieri, “Kink solutions for a class of generalized dissipative equations,” Abstract and Applied Analysis, vol. 2012, Article ID 237135, 7 pages, 2012.

G. D. Anderson, M. K. Vamanamurthy, and M. Vuorinen, “Generalized convexity and inequalities,” Journal of Mathematical Analysis and Applications, vol. 335, no. 2, pp. 1294–1308, 2007.

F. Qi and Z. Huang, “Inequalities of the complete elliptic integrals,” Tamkang Journal of Mathematics, vol. 29, no. 3, pp. 165–169, 1998.

H. Alzer and S.-L. Qiu, “Monotonicity theorems and inequalities for the complete elliptic integrals,” Journal of Computational and Applied Mathematics, vol. 172, no. 2, pp. 289–312, 2004.

Á. Baricz, “Landen inequalities for special functions,” Proceedings of the American Mathematical Society, vol. 142, no. 9, pp. 3059–3066, 2014.

G. D. Anderson, R. W. Barnard, K. C. Richards, M. K. Vamanamurthy, and M. Vuorinen, “Inequalities for zero-balanced hypergeometric functions,” Transactions of the American Mathematical Society, vol. 347, no. 5, pp. 1713–1723, 1995.

Á. Baricz, “Geometrically concave univariate distributions,” Journal of Mathematical Analysis and Applications, vol. 363, no. 1, pp. 182–196, 2010.

B. A. Bhayo and M. Vuorinen, “On generalized trigonometric functions with two parameters,” Journal of Approximation Theory, vol. 164, no. 10, pp. 1415–1426, 2012.

J. C. Kuang, Applied Inequalities, Shan Dong Science and Technology Press, Jinan, China, 2nd edition, 2002.