The Arithmetic and Geometric Means

In this paper, we show new ways of proving the arithmetic-geometric mean AGM inequality through the first product and the second product inequalities. In addition, we prove the AGM inequality through the binomial inequalities. These methods are alternative ways of proving AGM inequalities. 2010 Mathematics Subject Classifications: 44B51 44B52

2009

In the paper, we provide an alternative and united proof of a double inequality for bounding the arithmetic-geometric mean.

IJARW, 2023

Functional inequalities are very difficult and arise all areas of mathematics, even more, science, engineering, and social sciences. They appear at all levels of mathematics. The theory of functional inequations were born very early. Many authors studied functional inequalities. In this paper, we introduce some problems about functional inequalities induced by arithmetic average of oder arbitrary to geometric average.

2018

This article is the fourth in the ‘Inequalities’ series. This time, we present a novel proof of the general AM-GM inequality, based on iteration. Following this, we present some applications of the inequality. The (generally referred to as the AM-GM inequality; said to be part of the daily diet for aspiring mathletes, arithmetic mean-geometric mean inequality and routinely used in many branches of mathematics) is well-known. New proofs come up once in a while. The following iterative proof is highly unusual and will be of interest to some reader

Applied Mathematical Sciences

Some new refinements of the arithmetic, geometric and harmonic mean inequalities are presented which improve on the inequalities of P. R. Mercer given in [5]. In addition, we present a new method to obtain inequalities. We discuss a few applications to probability theory and obtain bounds for certain central moments of positive random variables in terms of these means.

In this note, we derive non trivial sharp bounds related to the weighted harmonic-geometric-arithmetic means inequalities, when two out of the three terms are known. As application, we give an explicit bound for the trace of the inverse of a symmetric positive definite matrix and an inequality related to the coefficients of polynomials with positive roots.

We present some techniques used to prove a variety of algebraic and geometric inequalities. 1 Main Theorem Let ∆ (x, y, z) = 2xy + 2yz + 2zx − x 2 − y 2 − z 2. Lemma 1. Let α, β, γ be positive real numbers such that ∆ (α, β, γ) > 0. Then αvw + βuw + γuv ≤ 0 (A) for all real numbers u, v, w with u + v + w = 0. In addition, αvw + βuw + γuv = 0 if and only if u = v = w = 0. Proof. Indeed, αvw − βwu − γuv = −γuv + (u + v) (αv + βu) = − (α + β − γ) uv + αv 2 + βu 2 = α v + (α + β − γ) u 2α 2 + u 2 2αβ + 2βγ + 2γα − α 2 − β 2 − γ 2 4α ≥ 0. It easily follows that equality occurs if and only if u = v = w = 0. Corollary. Let α, β, γ be positive real numbers such that α + β + γ = 1 and ∆ (α, β, γ) > 0. If x, y, z are real numbers with x + y + z = 1, then 2 cyc xβγ − cyc αyz ≥ 3αβγ. (B) Proof. Let u = x − α, v = y − β, w = z − γ. Then x = u + α , y = v + β, z = w + γ, u + v + w = 0, and 2 cyc xβγ − cyclic αyz = 2 cyclic (u + α) βγ − cyclic α (v + β) (w + γ) = 6αβγ + 2 cyc uβγ − 3αβγ − cyclic αvw − cyclic (αγv + αβw) = 3αβγ − cyclic αvw. Mathematical Reflections 2 (2009)

Elemente Der Mathematik, 2003

In this paper we prove Rado and Popoviciu type inequalities for pseudo arithmetic and geometric means a n and g n

Proceedings of the American Mathematical Society, 1972

Fan has proven an inequality relating the arithmetic and geometric means of (x¡, • • • , x") and (1-*,, • • •, 1-x"), where 0<x,^|, i=l, ■ ■ ■ ,n. Levinson has generalized Fan's inequality; his result involves functions with positive third derivatives on (0, 1). In this paper, the above condition that requires 0<JC¡ísi has been replaced by a condition which only weights the x¡ to the left side of (0, 1) in pairs, and Levinson's differentiability requirement has been replaced by the analogous condition on third differences.