Optimal bounds for Seiffert-like elliptic integral mean by harmonic, geometric, and arithmetic means

In this article, we present the optimal bounds for a special elliptic integral mean in terms of the harmonic combinations of harmonic, geometric, and arithmetic means. As consequences, several new bounds for the complete elliptic integral of the second kind are discovered, which are the improvements of many previously known results.


Introduction
For r ∈ (0, 1), Legendre's complete elliptic integrals of the first kind K(r) and second kind E(r) [1][2][3][4][5][6][7][8]  It is well known that K(r) is strictly increasing from (0, 1) onto (π/2, ∞) and E(r) is strictly decreasing from (0, 1) onto (1, π/2), they satisfy the derivative formulas The Author(s) 2022. This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/. and Landen identities where and in what follows we denote r = √ 1r 2 for r ∈ (0, 1). Let a, b > 0 with a = b. Then the harmonic mean H(a, b), geometric mean G(a, b), arithmetic mean A(a, b), arithmetic-geometric mean AG(a, b) [9][10][11], and Toader mean TD(a, b) [12][13][14][15] are given by Recently, the complete elliptic integrals K(r) and E(r) of the first and second kinds have attracted the attention of many researchers [16][17][18][19][20][21][22] because they have wide applications in many branches of mathematics including the geometric function theory, differential equations, number theory, and mean value theory. For instance, the perimeter L(a, b) of an ellipse with semi-axes a, b and eccentricity e = √ 1b 2 /a 2 is given by Many remarkable inequalities and properties for the complete elliptic integrals K(r) and E(r) can be found in the literature [23][24][25][26][27][28][29][30][31]. Barnard et al. [32] and Alzer and Qiu [33] proved that λ = 3/2 and μ = log 2/ log(π/2) are the best possible constants such that the double inequality holds for all r ∈ (0, 1). Later, Wang and Chu [34] improved the lower bound of (1.3) and proved that the double inequality holds for all r ∈ (0, 1) with the best possible constants α = 4+ Very recently, Yang et al. [35] found the high accuracy asymptotic bounds for E(r) and proved that for all r ∈ (0, 1), where J(r) = 51r 2 + 20r √ r + 50r + 20 √ r + 51 16(5r + 2 √ r + 5) .
The following Seiffert-like elliptic integral mean was introduced by Witkowski in [36], in which Witkowski investigated the so-called Seiffert-like means where the function f : (0, 1) → R (called Seiffert function) satisfies the double inequality Inspired by (1.5), the main purpose of the article is to find the optimal bounds for V (a, b) in terms of the harmonic combinations of H(a, b) and G(a, b) (or H(a, b) and A(a, b)). Our main results are the following Theorems 1.1 and 1.2.

Theorem 1.2 The double inequality
holds for all a, b > 0 with a = b if and only if α 2 ≤ 2/π and β 2 ≥ 3/4.
To further improve and refine the lower bound in (1.6) and the upper bound in (1.7), we also establish the following Theorems 1.3 and 1.4.

Lemmas
In order to prove our main results, we need several lemmas which we present in this section. (a, b), then so is f /g.
Simple computations lead to

Proofs of Theorems 1.1-1.4
In this section, we assume that a > b > 0 because all the bivariate means H(a, b), G(a, b), A(a, b), and V (a, b) are symmetric and homogeneous of degree one.  where f (r) = 1 -2E/π 1r .
In order to compare the lower and upper bounds in Corollary 3.1, we provide Theorem 3.2 as follows.