Approximations for the complete elliptic integral of the second $\hbox {Kind}$

Abstract

In this paper, we present the best possible parameters $\alpha _{1}$, $\alpha _{2}$, $\alpha _{3}$, $\alpha _{4}$, $\beta _{1}$, $\beta _{2}$, $\beta _{3}$, $\beta _{4} \in {\mathbb {R}}$ such that

$$\begin{aligned} \frac{\alpha _{1}}{H(x, y)}+\frac{1-\alpha _{1}}{L(x, y)}&<\frac{1}{V(x, y)}<\frac{\beta _{1}}{H(x, y)}+\frac{1-\beta _{1}}{L(x, y)},\\ \frac{\alpha _{2}}{H(x, y)}+\frac{1-\alpha _{2}}{P(x, y)}&<\frac{1}{V(x, y)}<\frac{\beta _{2}}{H(x, y)}+\frac{1-\beta _{2}}{P(x, y)},\\ \frac{\alpha _{3}}{H(x, y)}+\frac{1-\alpha _{3}}{N S(x, y)}&<\frac{1}{V(x, y)}<\frac{\beta _{3}}{H(x, y)}+\frac{1-\beta _{3}}{N S(x, y)},\\ \frac{\alpha _{4}}{H(x, y)}+\frac{1-\alpha _{4}}{T(x, y)}&<\frac{1}{V(x, y)}<\frac{\beta _{4}}{H(x, y)}+\frac{1-\beta _{4}}{T(x, y)} \end{aligned}$$

hold for all $x, y>0$ with $x \ne y$, where H(x, y), G(x, y), L(x, y), A(x, y), NS(x, y), P(x, y) and T(x, y) are respectively the harmonic, geometric, logarithmic, arithmetic, Neuman-Sándor, and first and second Seiffert means of two distinct positive numbers x and y, and

$$\begin{aligned} V(x,y)=\pi G^{2}(x, y) /\left[ 2\int _{0}^{\pi /2}\sqrt{A^{2}(x,y) \cos ^{2}\varphi +G^{2}(x,y)\sin ^{2}\varphi }d\varphi \right] \end{aligned}$$

is a new Seiffert-like mean. As applications, some new inequalities for the complete elliptic integral of the second kind are given.