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There are many relations involving the geometric means $G_{n} (x)$ and power means ${[A_{n} (x^{γ})]}^{1 / γ}$ for positive $n$ -vectors $x$ . Some of them assume the form of inequalities involving parameters. There then is the question of sharpness, which is quite difficult in general. In this paper we are concerned with inequalities of the form $(1 - λ) G_{n}^{γ} (x) + λ A_{n}^{γ} (x) \geq A_{n} (x^{γ})$ and $(1 - λ) G_{n}^{γ} (x) + λ A_{n}^{γ} (x) \leq A_{n} (x^{γ})$ with parameters $λ \in ℝ$ and $γ \in (0, 1) .$ We obtain a necessary and sufficient condition for the former inequality, and a sharp condition for the latter. Several applications of our results are also demonstrated....

Optimal weighted harmonic interpolations between Seiffert means

Alfred Witkowski (2013)

Colloquium Mathematicae

We provide a set of optimal estimates of the form (1-μ)/𝓐(x,y) + μ/ℳ (x,y) ≤ 1/ℬ(x,y) ≤ (1-ν)/𝓐(x,y) + ν/ℳ (x,y) where 𝓐 < ℬ are two of the Seiffert means L,P,M,T, while ℳ is another mean greater than the two.

We present comparison theorems for the weighted quasi-arithmetic means and for weighted Bajraktarević means without supposing in advance that the weights are the same.

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On the homogeneous functions with two parameters and its monotonicity.

On the Schwab-Borchardt mean.

On the Schwab-Borchardt mean. II.

Optimal inequalities among various means of two arguments.

Optimal inequalities for generalized logarithmic, arithmetic, and geometric means.

Optimal power mean bounds for the weighted geometric mean of classical means.

Optimal sublinear inequalities involving geometric and power means

Optimal weighted harmonic interpolations between Seiffert means

Positive semidefinite matrices, exponential convexity for majorization, and related Cauchy means.

Proof for A conjecture on general means.

Proof of one optimal inequality for generalized logarithmic, arithmetic, and geometric means.

Quotient mean series.

Refinements of Carleman's inequality.

Refinements of results about weighted mixed symmetric means and related Cauchy means.

Refinements of the Hermite-Hadamard inequality for convex functions.

Remarks on the comparison of weighted quasi-arithmetic means

Schur geometric convexity for differences of means.

Schur-convexity of averages of convex functions.

Schur-convexity of two types of one-parameter mean values in $n$ variables.

Sharp power mean bounds for the combination of Seiffert and geometric means.

Currently displaying 101 – 120 of 144