Related papers: Another Proof of a Famous Inequality
In this short paper we show that the inequality of arithmetic and geometric means is reduced to another interesting inequality, and a proof is provided.
In the paper, we provide an alternative and united proof of a double inequality for bounding the arithmetic-geometric mean.
We give a simpler proof of a result of Holland concerning a mixed arithmetic-geometric mean inequality. We also prove a result of mixed mean inequality involving the symmetric means.
We give an upper bound for the weighted geometric mean using the weighted arithmetic mean and the weighted harmonic mean. We also give a lower bound for the weighted geometric mean. These inequalities are proven for two invertible positive…
We shall give a refinement of the arithmetic-geometric mean inequality.
In the current note, we investigate the mathematical relations among the weighted arithmetic mean-geometric mean (AM-GM) inequality, the H\"{o}lder inequality and the weighted power-mean inequality. Meanwhile, the proofs of mathematical…
We obtain simple proofs of certain inequalites for bivariate means.
In this note we revisit the classical geometric-arithmetic mean inequality and find a formula for the difference of the arithmetic and the geometric means of given $n\in\mathbb N$ nonnegative numbers $x_1,x_2,\dots,x_n$. The formula yields…
In this note we present a refinement of the AM-GM inequality, and then we estimate in a special case the typical size of the improvement.
The classical AM-GM inequality has been generalized in a number of ways. Generalizations which incorporate variance appear to be the most useful in economics and finance, as well as mathematically natural. Previous work leaves unanswered…
In the current note, we present a new, short proof of the famous AM-GM-HM inequality using only induction and basic calculus.
In this note, we present a probabilistic proof of the well-known finite geometric series. The proof follows by taking the moments of the sum and the difference of two independent exponentially distributed random variables.
A mixed arithmetic-mean, geometric-mean inequality was conjectured by F. Holland and proved by K. Kedlaya. In this note, we prove a mixed arithmetic-mean, harmonic-mean inequality and a mixed geometric-mean, harmonic-mean, and a more…
We obtain a fundamental inequality for a contraction with respect to a $C^*$-algebra valued metric space. As an application of this inequality a simple proof is given for the fixed point theorem in $C^*$-algebra valued metric space.
Inequalities for norms of different versions of the geometric mean of two positive definite matrices are presented.
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…
We present a refinement, by selfimprovement, of the arithmetic geometric inequality.
In this paper we give simple proofs for the bounds (some of them sharp) of the difference of the moduli of the second and the first logarithmic coefficient for the general class of univalent functions and for the class of convex univalent…
We establish a general criterion for the validity of inequalities of the following form: A certain convex combination of the values of a convex function at n points and of its value at a weighted mean of these n points is always greater or…
In this paper I present a kind of proof for classical Euclidean geometric problems which relies on both synthetic and analytic geometry. Using the elementary tools of polynomial algebra and multivariate calculus we manage to reduce the…