Gerard Maze, Urs Wagner
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.
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In the paper, the authors find the best possible constants appeared in two inequalities for bounding the Seiffert mean by the linear combinations of the arithmetic, centroidal, and contra-harmonic means.
Hölder- and Minkowski-type inequalities for generalized quasi-arithmetic means
本文研究由仅严格单调的生成函数定义的广义拟算术平均，建立一般函数不等式成立的若干充要条件，并给出Hölder型、Minkowski型和Jensen型不等式的特殊情形。
In this paper, some inequalities of bounds for the Neuman-Sándor mean in terms of weighted arithmetic means of two bivariate means are established. Bounds involving weighted arithmetic means are sharp.
In this paper, we find the greatest values $α$ and $λ$, and the least values $β$ and $μ$ such that the double inequalities $$C^α(a,b)A^{1-α}(a,b)<M(a,b)<C^β(a,b)A^{1-β}(a,b)$$ and &[C(a,b)/6+5 A(a,b)/6]^λ[C^{1/6}(a,b)A^{5/6}(a,b)]^{1-λ}<M(a,b) &\qquad<[C(a,b)/6+5 A(a,b)/6]^μ[C^{1/6}(a,b)A^{5/6}(a,b)]^{1-μ} hold for all $a,b>0$ with $a\neq b$, where $M(a,b)$, $A(a,b)$ and $C(a,b)$ denote the Neuman-Sándor, arithmetic, and contra-harmonic means of $a$ and $b$, respectively.
In this paper, we extend the Hermite-Hadamard type $\dot{I}$scan inequality to the class of symmetrized harmonic convex functions. The corresponding version for harmonic h-convex functions is also investigated. Furthermore, we establish Hermite-Hadamard type inequalites for the product of a harmonic convex function with a symmetrized harmonic convex function.
We use the method of induction-on-scales to prove certain diffeomorphism invariant nonlinear Brascamp--Lieb inequalities. We provide applications to multilinear convolution inequalities and the restriction theory for the Fourier transform, extending to higher dimensions recent work of Bejenaru--Herr--Tataru and Bennett--Carbery--Wright.
Sharp inequalities for the psi function and harmonic numbers
本文分别建立了关于psi函数$ψ$与调和数$H_n$的界的两个尖锐不等式，推广了[I. Muqattash和M. Yahdi，\textit{Infinite family of approximations of the Digamma function}，Math. Comput.建模\textbf{43}（2006），1329 \nobreakdash --1336。]并给出了一些说明。
In the paper, by finding linear relations of differences between some means, the authors supply a unified proof of some double inequalities for bounding Neuman-Sándor means in terms of the arithmetic, harmonic, and contra-harmonic means and discover some new sharp inequalities involving Neuman-Sándor, contra-harmonic, root-square, and other means of two positive real numbers.
Necessary conditions for vector-valued operator inequalities in harmonic analysis
通过一个随机构造，我们建立了调和分析中某些算子族的$L^p(\ell^q)$不等式成立的必要条件.特别是，我们考虑膨胀的卷积核与compensate支持傅里叶变换，向量极大函数的指数型整函数类，和表征的索伯列夫空间的平方函数和逐点模的光滑性。
An interplay between the sum of certain series related to Harmonic numbers and certain finite trigonometric sums is investigated. This allows us to express the sum of these series in terms of the considered trigonometric sums, and permits us to find sharp inequalities bounding these trigonometric sums. In particular, this answers positively an open problem of H. Chen (2010).
We show some non-standard Poincaré type estimates in the biparametric setting with appropriate weights. We will derive these results using variants from classical estimates exploiting the interplay between maximal functions and fractional integrals. We also provide a sharper result by using extrapolation techniques.
Optimal bounds for the ratio of differences of quadratic, arithmetic, and harmonic means
研究\(n\)个非负实数二次、算术和调和平均差之比的最优界，采用基于Cauchy和Maclaurin经典优化方法变体，证明\(n\geq3\)时的双重不等式，扩展了T. Mitev仅针对\(n = 3、4、5\)的结果。
A Note On Mixed Mean Inequalities
本文给出了Holland关于混合算术几何平均不等式的一个结果的一个简单证明。本文还证明了一个涉及对称平均的混合平均不等式的结果。
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 new stronger versions of the geometric-arithmetic mean inequality. We also find a second version of a strong geometric-arithmetic mean inequality and show that all inequalities are optimal in some sense. Anther striking novelty is, that the equality in all new inequalities holds not only in the case when all $n$ numbers are equal, but also in other cases.
In this paper, we prove that the inequalities $α[1/3 Q(a,b)+2/3 A(a,b)]+(1-α)Q^{1/3}(a,b)A^{2/3}(a,b)<M(a,b) <β[1/3 Q(a,b)+2/3 A(a,b)]+(1-β)Q^{1/3}(a,b)A^{2/3}(a,b)$ and $λ[1/6 C(a,b)+5/6 A(a,b)]+(1-λ)C^{1/6}(a,b)A^{5/6}(a,b)<M(a,b)<μ[1/6 C(a,b)+5/6 A(a,b)]++(1-μ)C^{1/6}(a,b)A^{5/6}(a,b)$ hold for all $a,b>0$ with $a\neq b$ if and only if $α\leq (3-3\sqrt[6]{2}\log(1+\sqrt{2}))/[(2+\sqrt{2}-3\sqrt[6]{2})\log(1+\sqrt{2})]=0.777...$, $β\geq 4/5$, $λ\leq (6-6\sqrt[6]{2}\log(1+\sqrt{2}))/(7-6\sqrt[6]{2}\log(1+\sqrt{2}))=0.274...$, and $μ\geq 8/25$. Here, $M(a,b)$, $A(a,b)$, $C(a,b)$, and $Q(a,b)$ …
The aim of this paper is to present some new Fejer-type results for convex functions. Improvements of Young's inequality (the arithmetic-geometric mean inequality) and other applications to special means are pointed as well.
Generalized Convexity and Inequalities
设R+ =（0，无穷大），M是R+中两个数的所有平均值的族（例如算术平均、几何平均和调和平均）。给定M中的m1，m2，我们说函数f：R+ to R+是（m1，m2）-凸的，如果对R+中的所有x，y，f（m1（x，y））<或= m2（f（x），f（y））。通常的凸性是两个平均值都是算术平均时的特殊情况。研究了（m1，m2）-凸性对m1和m2的依赖性，给出了由Maclaurin级数定义的函数的（m1，m2）-凸性的充分条件.该标准涉及麦克劳林系数。我们的结果得到了一类新的不等式，它涉及到高斯超几何函数和广义Bessel函数等特殊函数。
In this paper, we present the greatest values $α$, $λ$ and $p$, and the least values $β$, $μ$ and $q$ such that the double inequalities $αD(a,b)+(1-α)H(a,b)<T(a,b)<βD(a,b)+(1-β) H(a,b)$, $λD(a,b)+(1-λ)H(a,b)<C(a,b)<μD(a,b)+(1-μ) H(a,b)$ and $p D(a,b)+(1-p)H(a,b)<Q(a,b)<q D(a,b)+(1-q)H(a,b)$ hold for all $a,b>0$ with $a\neq b$, where $H(a,b)=2ab/(a+b)$, $T(a,b)=(a-b)/[2\arctan((a-b)/(a+b))]$, $Q(a,b)=\sqrt{(a^2+b^2)/2}$, $C(a,b)=(a^2+b^2)/(a+b)$ and $D(a,b)=(a^3+b^3)/(a^2+b^2)$ are the harmonic, Seiffert, quadratic, first contraharmonic and second contraharmonic means of $a$ and $b$, respective…
Sharp quantitative integral inequalities for harmonic extensions
本文针对Poisson算子及其伴随算子，证明了Hang、Wang和Yan的精确积分不等式的量化版本，获得了最强范数和不一定等于2的最优稳定性指数。
In the paper, by establishing the monotonicity of some functions involving the sine and cosine functions, the authors provide concise proofs of some known inequalities and find some new sharp inequalities involving the Seiffert, contra-harmonic, centroidal, arithmetic, geometric, harmonic, and root-square means of two positive real numbers $a$ and $b$ with $a\ne b$.
In this paper, we give the explicit formulas for the Neuman means $N_{AH}$, $N_{HA}$, $N_{AC}$ and $N_{CA}$, and present the best possible upper and lower bounds for theses means in terms of the combinations of harmonic mean $H$, arithmetic mean $A$ and contraharmonic mean $C$.
Weighted norm inequalities, off-diagonal estimates and elliptic operators. Part III: Harmonic analysis of elliptic operators
这是第三部分的一系列四篇文章加权范数不等式，非对角估计和椭圆算子。对于一类椭圆算子的$L$，我们研究了由$L$产生的奇异“非积分”算子的加权范数$L^p$不等式;它们是有界全纯函数$ϕ$的算子$ϕ(L)$，Riesz变换$\nabla L^{-1/2}$（或$(-Δ)^{1/2}L^{-1/2}$）及其逆$L^{1/2}(-Δ)^{-1/2}$，一些Littlewood-Paley-Stein型二次泛函$g\_{L}$和$G\_{L}$以及一些向量-值不等式，如最大$L^p$ -正则性所涉及的不等式。对于每一个，我们获得尖锐或近尖锐的PDMATHAMV范围使用的一般理论的有界性的第一部分和非对角估计的第二部分。我们还得到了与BMO函数的交换子结果。
We prove a one-parameter family of sharp integral inequalities for functions on the $n$-dimensional unit ball. The inequalities are conformally invariant, and the sharp constants are attained for functions that are equivalent to a constant function under conformal transformations. As a limiting case, we obtain an inequality that generalizes Carleman's inequality for harmonic functions in the plane to poly-harmonic functions in higher dimensions.
From Ingham to Nazarov's inequality: a survey on some trigonometric inequalities
本文综述了调和与非调和三角多项式在 L^1 和 L^2 范数下的若干不等式，重点介绍 Ingham 不等式、Littlewood 猜想的证明及 Nazarov 的非调和对应结果，并补充了小区间上的定量估计与更强条件下的下界。
In this paper, we establish the sharp critical and subcritical trace Trudinger-Moser and Adams inequalities on the half spaces and prove the existence of their extremals through the method based on the Fourier rearrangement, harmonic extension and scaling invariance. These trace Trudinger-Moser and Adams inequalities can be considered as the borderline case of the Sobolev trace inequalities of first and higher orders. Furthermore, we show the existence of the least energy solutions for a class of bi-harmonic equations with nonlinear Neumann boundary condition associated with the trace Adams in…
In the paper, the authors discover the best constants $α_{1}$, $α_{2}$, $β_{1}$, and $β_{2}$ for the double inequalities $$ α_{1}\bar{C}(a,b)+(1-α_{1}) A(a,b)< T(a,b) <β_{1} \bar{C}(a,b)+(1-β_{1})A(a,b) $$ and $$ \frac{α_{2}}{A(a,b)}+\frac{1-α_{2}}{\bar{C}(a,b)}<\frac1{T(a,b)} <\frac{β_{2}}{A(a,b)}+\frac{1-β_{2}}{\bar{C}(a,b)} $$ to be valid for all $a,b>0$ with $a\ne b$, where $$ \bar{C}(a,b)=\frac{2(a^{2}+ab+b^{2})}{3(a+b)},\quad A(a,b)=\frac{a+b}2, $$ and $$ T(a,b)=\frac{2}π\int_{0}^{π/{2}}\sqrt{a^2{\cos^2θ}+b^2{\sin^2θ}}\,\tdθ$$ are respectively the centroidal, arithmetic, and Toader means…
In this paper, we will define $a$-deformed Laguerre operators $L_{a,α}$ and $a$-deformed Laguerre holomorphic semigroups on $L^2\left(\left(0,\infty\right),dμ_{a,α}\right)$. Then we give a spherical harmonic expansion, which reduces to the Bochner-type identity when taking the boundary value $z=\frac{πi}2$, of the $(k,a)$-generalized Laguerre semigroup introduced by S. Ben Saïd, T. Kobayashi and B. Ørsted. And then we prove a Hardy inequality for fractional powers of the $a$-deformed Dunkl harmonic oscillator $\triangle_{k,a}:=\left|x\right|^{2-a}\triangle_k-\left|x\right|^a$ using this expans…
We give bounds on the difference between the weighted arithmetic mean and the weighted geometric mean. These imply refined Young inequalities and the reverses of the Young inequality. We also study some properties on the difference between the weighted arithmetic mean and the weighted geometric mean. Applying the newly obtained inequalities, we show some results on the Tsallis divergence, the Rényi divergence, the Jeffreys-Tsallis divergence and the Jensen-Shannon-Tsallis divergence.
We prove variation and oscillation $L^p$-inequalities associated with fractional derivatives of certain semigroups of operators and with the family of truncations of Riesz transforms in the inverse Gaussian setting. We also study these variational $L^p$-inequalities in a Banach-valued context by considering Banach spaces with the UMD-property and whose martingale cotype is fewer than the variational exponent. We establish $L^p$-boundedness properties for weighted difference involving the semigroups under consideration.
A classical result of Khinchin says that for almost all real numbers $α$, the geometric mean of the first $n$ digits $a_i(α)$ in the continued fraction expansion of $α$ converges to a number $K = 2.6854520\ldots$ (Khinchin's constant) as $n \to \infty$. On the other hand, for almost all $α$, the arithmetic mean of the first $n$ continued fraction digits $a_i(α)$ approaches infinity as $n \to \infty$. There is a sequence of refinements of the AM-GM inequality, Maclaurin's inequalities, relating the $1/k$-th powers of the $k$-th elementary symmetric means of $n$ numbers for $1 \leq k \leq n$. On…
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