#### 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 . 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  studied the -trigonometric functions depending on two parameters. For , these functions reduce to the so-called -trigonometric functions introduced by Lindqvist . The following ()-eigenvalue problem with Dirichlét boundary condition was considered by Drábek and Manásevich . 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 , whereFor , reduces to ; see . In order to give the definition of the function , first of all we define its inverse function and then the function itself. For , setThe function is an increasing homeomorphism, andis defined on the interval . The function can be extended to byBy oddness, the further extension can be made to . Finally, the function is extended to whole by -periodicity; see . 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  with two parameters. For all and , the complete -elliptic integrals of the first and second kinds  are defined byandrespectively. Here, and . The complete -elliptic integrals can be expressed in terms of hypergeometric functions as follows:andIf , we can obtain the classically complete elliptic integrals as follows: for ,If , we can obtain the generalized complete elliptic integrals with single parametersIt is worth noting that Bhayo and Yin also gave new -complete elliptic integrals in . For more on this topic, the readers can see related references . 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.  discovered that the complete elliptic integral of the first kind can be approximated by the inverse hyperbolic tangent function:For , they provedLater, Qi and Huang  obtained the following inequality by using Chebyshev inequality:In 2004, Alzer and Qiu  proved that, for , we havewith 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 byConsidering Lemma 1, we only need to discuss the monotonicity of the sequence defined by Sincewe 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. DefineAnd is defined by Simple computation yieldsThis 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 , 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 : 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 haveby 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) iffor all . Recently, generalized convexity/concavity with respect to general mean values has been studied by Anderson et al. in . In , Baricz studied that if the function is differentiable, then it is -convex (concave) on if and only if is increasing (decreasing). In , 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 ). Let be integrable functions, both increasing or both decreasing. Furthermore, let be a positive, integrable function. ThenIf one of the functions or is nonincreasing and the other is nondecreasing, then the inequality in (39) is reversed.

Lemma 9 (see ). For , thenif the function is convex on .

Theorem 10. Let and . Then

Proof. Let us suppose without loss of generality. Define Simple computation yieldsThis 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 inSince the functions and are strictly increasing, we may obtainby taking in Lemma 8. This is equivalent to Noting that log-convexity implies the convexity and Lemma 9, we havewhere we apply the known inequality .

Theorem 11. Let and . Then

Proof. This proof is similar to Theorem 10. We still suppose . Direct calculation yieldsSince the functions and are strictly increasing, we haveby 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 haveand whenever ; then find conditions on and for which the inequalityholds 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 yieldsThis 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).