Functional Inequalities for Generalized Complete Elliptic Integrals with Two Parameters

0 tb−1 (1 − t)c−b−1 (1 − zt)−a dt (2) Re(c) > Re(a) > 0, | arg(1 − z)| < π; 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 x, y > 0, the Euler gamma function Γ and its logarithmic derivative ψ, the so-called digamma function, are defined by (cf. [1, 3]) Γ (x) = ∫∞ 0 tx−1e−tdt, B (x, y) = Γ (x) Γ (y) Γ (x + y) , ψ (x) = Γ󸀠 (x) Γ (x) . (3)


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: Journal of Function Spaces 3 For ∈ (0, 1), they proved Later, Qi and Huang [29] obtained the following inequality by using Chebyshev inequality: In 2004, Alzer and Qiu [30] proved that, for ∈ (0, 1), we have with the best possible constants = 3/4 and = 1. Here, we shall show some new inequalities for the generalized complete elliptic integrals.
Here we mainly prove (1) for the sake of simplicity. First of all, we consider the function , : (0, 1) → (0, ∞) defined by Considering Lemma 1, we only need to discuss the monotonicity of the sequence { } ≥0 defined by Since +1 = (−1/ + ) (1 + 1/ + ) (1/ + ) (1 − 1/ + 1/ + ) ≤ 1 ⇐⇒ we get that the sequence { } ≥0 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.
Simple computation yields This implies that the sequence {Ω } ≥0 is strictly increasing. Using Lemma 1, we obtain that the function , is strictly increasing on (0, 1). The limiting value follows easily. This completes the proof.

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 (37)

Let
: → (0,∞) be continuous, where is a subinterval of (0, ∞). Let and be the means defined above; then we call that the function is 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 1− ( )/ ( ) 1− 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] If one of the functions or is nonincreasing and the other is nondecreasing, then the inequality in (39) is reversed.
Proof. Let us suppose ≤ without loss of generality. Define Simple computation yields This implies that the function ( ) is strictly log-convex on (0, 1). By using the fact that the integral preserves monotonicity and log-convexity, we get that the function , ( ) is strictly increasing and log-convex on (0, 1). The substitution = , ( ) results in where we apply the known inequality ( , ) ≤ ( , ).
The proof is complete.

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.