An Alternative and United Proof of a Double Inequality for Bounding the Arithmetic-geometric Mean

For more information on the arithmetic-geometric mean and the complete elliptic integral of the first kind, please refer to [2, pp. 132–136], [4] and related references therein. In [4, Theorem 4] and [6], it was proved that the inequality M(a, b) ≥ L(a, b) (5) holds true for positive numbers a and b and that the inequality (5) becomes equality if and only if a = b, where L(a, b) = b− a ln b− ln a (6) stands for the logarithmic mean for positive numbers a and b with a 6= b. In [16, Theorem 1.3], it was turned out that M(a, b) < I(a, b) (7)