We find the greatest values $alpha_{1} $ and $alpha_{2} $, and the least values $beta_{1} $ and $beta_{2} $ such that the inequalities $alpha_{1} C(a,b)+(1-alpha_{1} )H(a,b)

Given two positive real numbers x and y, let A(x, y), G(x, y), and I(x, y) denote their arithmetic mean, geometric mean, and identric mean, respectively. Also, let Kp(x, y) = p √ 2 3A p(x, y) + 13G p(x, y) for p > 0. In this note we prove that Kp(x, y) < I(x, y) for all positive real numbers x 6= y if and only if p ≤ 6/5, and that I(x, y) < Kp(x, y) for all positive real numbers x 6= y if and o...

Given the positive real numbers x and y, let A(x, y), G(x, y), and I(x, y) denote their arithmetic mean, geometric mean, and identric mean, respectively. It is proved that for p ≥ 2, the double inequality αA(x, y) + (1− α)G(x, y) < I(x, y) < βA(x, y) + (1− β)G(x, y) holds true for all positive real numbers x 6= y if and only if α ≤ ( 2 e )p and β ≥ 23 . This result complements a similar one est...

We prove certain new inequalities for special means of two arguments, including the identric, arithmetic, and geometric means. 2000 Mathematics Subject Classification. Primary 26D99, 65D32.

In the short note, the inequalities G ≤ L ≤ I ≤ A for the geometric, logarithmic, identric, and arithmetic means in n variables are proved.

and Applied Analysis 3 It is the aim of this paper to find the best possible lower power mean bound for the Neuman-Sándor mean M(a, b) and to present the sharp constants α and β such that the double inequality α < M(a, b) I (a, b) < β (17) holds for all a, b > 0 with a ̸ = b. 2. Main Results Theorem 1. p0 = (log 2)/ log [2 log(1 + √2)] = 1.224 . . . is the greatest value such that the inequality...

and Applied Analysis 3 Lemma 1. The double inequality

The stability and stabilizability concepts for means in two variables have been introduced in (Raïssouli in Appl. Math. E-Notes 11:159-174, 2011). It has been proved that the arithmetic, geometric, and harmonic means are stable, while the logarithmic and identric means are stabilizable. In the present paper, we introduce new concepts, the so-called sub-stabilizability and super-stabilizability,...