Alemeh Sheikhhosseini, Maryam Khosravi
In this note, some refinements of Young inequality and its reverse for positive numbers are proved and using these inequalities some operator versions and Hilbert-Schmidt norm versions for matrices of these inequalities are obtained.
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On refined Young inequalities and reverse inequalities
本文证明了两个正算子的精化Young不等式。我们的结果改进了两个正算子的算术平均、几何平均和调和平均之间的序关系。另外，我们给出了两个正算子的精化Young不等式的两个不同的反向不等式。
On refined Young inequalities
本文研究了与Young不等式有关的标量不等式和算子不等式的加细。作为我们的主要结果，我们证明了两个正算子的精化Young不等式。这一结果细化了算术平均、几何平均和调和平均之间的排序关系。最后，对两个正实数的精化Young不等式给出了补充.在此基础上给出了算子不等式.
In this paper we obtain some new refinements and reverses of Young's operator inequality. Extensions for convex functions of operators are also provided.
In this paper, we give a new inequality for convex functions of real variables, and we apply this inequality to obtain considerable generalizations, refinements, and reverses of the Young and Heinz inequalities for positive scalars. Applications to unitarily invariant norm inequalities involving positive semidefinite matrices are also given.
Refined Young inequalities with Specht's ratio
在本文中，我们证明了$ν$加权算术平均值大于$ν$加权几何平均和Specht比率的乘积。作为推论，我们还表明，$ν$加权几何平均值大于产品的$ν$加权调和平均和Specht的比率。这些结果改进了经典的Young不等式，因为Specht比一般大于1.此外，利用改进的Young不等式，给出了正算子的一个算子不等式.
Two reverse inequalities for Young's inequality were shown by M. Tominaga, using Specht ratio. In this short paper, we show alternative reverse inequalities for Young's inequality without using Specht ratio.
In this paper we employ some operator techniques to establish some refinements and reverses of the Callebaut inequality involving the geometric mean and Hadamard product under some mild conditions. In particular, we show \begin{align*} K&\left(\frac{M^{2t-1}}{m^{2t-1}},2\right)^{r'} \sum_{j=1}^n(A_j\sharp_{s}B_j)\circ \sum_{j=1}^n(A_j\sharp_{1-s}B_j) \nonumber\\&\,\,+\left(\frac{t-s}{t-1/2}\right)\left(\sum_{j=1}^n(A_j\sharp_{t}B_j)\circ \sum_{j=1}^n(A_j\sharp_{1-t}B_j) -\sum_{j=1}^n(A_j\sharp B_j)\circ \sum_{j=1}^n(A_j\sharp B_j)\right)\nonumber \\&\leq \sum_{j=1}^n(A_j\sharp_{t}B_j)\circ \su…
We present some reverse Young-type inequalities for the Hilbert-Schmidt norm as well as any unitarily invariant norm. Furthermore, we give some inequalities dealing with operator means. More precisely, we show that if $A, B\in {\mathfrak B}(\mathcal{H})$ are positive operators and $r\geq 0$, $A\nabla_{-r}B+2r(A\nabla B-A\sharp B)\leq A\sharp_{-r}B$ and prove that equality holds if and only if $A=B$. We also establish several reverse Young-type inequalities involving trace, determinant and singular values. In particular, we show that if $A, B$ are positive definite matrices and $r\geq 0$, then …
In this paper, we give some reverse-types of Ando's and Hölder-McCarthy's inequalities for positive linear maps, and positive invertible operators. For our purpose, we use a recently improved Young inequality and its reverse.
Some improvements of Young inequality and its reverse for positive numbers with Kontrovich constant are given. Using these inequalities some operator versions and Hilbert-Schmidt norm versions for matrices are proved.
In this article, we study the further refinements and reverses of the Young and Heinz inequalities with the Kantorovich constant. These modified inequalities are used to establish corresponding operator inequalities on Hilbert space and Hilbert-Schmidt norm inequalities.
Optimal Young's inequality and its converse:a simple proof
本文给出了由Beckner [Be]和Brascamp-Lieb [BL]首先证明的关于卷积的Young不等式的尖锐形式的一个新证明。后者在指数小于$1$的情况下也证明了一个尖锐的反向不等式。我们的证明更简单，并给出了杨氏不等式和它的逆。
In this paper, sharp results on operator Young's inequality are obtained. We first obtain sharp multiplicative refinements and reverses for the operator Young's inequality. Secondly, we give an additive result, which improves a well-known inequality due to Tominaga. We also provide some estimates for $A{{\sharp}_{v}}B-A{{\nabla }_{v}}B$ in which $v\notin \left[ 0,1 \right]$.
Jensen's operator inequality for convexifiable functions is obtained. This result contains classical Jensen's operator inequality as a particular case. As a consequence, a new refinement and a reverse of Young's inequality is given.
We give the tight bounds of Tsallis relative operator entropy by using Hermite-Hadamard's inequality. Some reverse inequalities related to Young inequalities are also given. In addition, operator inequalities for normalized positive linear map with Tsallis relative operator entropy are given.
In this article, we prove that convex functions and log-convex functions obey certain general refinements that lead to several refinements and reverses of well known inequalities for matrices, including Young's inequality, Heinz inequality, the arithmetic-harmonic and the geometric-harmonic mean inequalities.
This paper is twofold. In the first part, we present a refinement of the Rényi Entropy Power Inequality (EPI) recently obtained in \cite{BM16}. The proof largely follows the approach in \cite{DCT91} of employing Young's convolution inequalities with sharp constants. In the second part, we study the reversibility of the Rényi EPI, and confirm a conjecture in \cite{BNT15, MMX16} in two cases. Connections with various $p$-th mean bodies in convex geometry are also explored.
We give some new refinements and reverses Young inequalities in both additive-type and multiplicative-type for two positive numbers/operators. We show our advantages by comparing with known results. A few applications are also given. Some results relevant to the Heron mean are also considered.
We give some new refinements of Heinz inequality and an improvement of the reverse Young's inequality for scalars and we use them to establish new inequalities for operators and the Hilbert-Schmidt norm of matrices. We give a uniformly and abbreviated form of the inequalities presented by Kittaneh and Mansarah, and the inequalities presented by Kai and we obtain some of their operator and matrix versions.
Improved upper bounds for the Berezin numbers of operators on reproducing kernel Hilbert spaces
研究再生核希尔伯特空间上算子的贝雷津数上界，利用对称均值和奥利奇函数的插值路径及改进的杨氏不等式得出结果，改进推广了已知发现并推导了不等式。
In this paper, we give some refinements for the second inequality in $\frac{1}{2}\|A\| \leq w(A) \leq \|A\|$, where $A\in B(H)$. In particular, if $A$ is hyponormal by refining the Young inequality with the Kantorovich constant $K(\cdot, \cdot)$, we show that $w(A)\leq \dfrac{1}{\displaystyle {2\inf_{\| x \|=1}}ζ(x)}\| |A|+|A^{*}|\|\leq \dfrac{1}{2}\| |A|+|A^*|\|$, where $ζ(x)=K(\frac{\langle |A|x,x \rangle}{\langle |A^{*}|x,x \rangle},2)^{r},~~~r=\min\{λ,1-λ\}$ and $0\leq λ\leq 1$ . We also give a reverse for the classical numerical radius power inequality $w(A^{n})\leq w^{n}(A)$ for any oper…
Let $μ$ be the Haar measure of a unimodular locally compact group $G$ and $m (G)$ as the infimum of the volumes of all open subgroups of $G$. The main result of this paper is that \begin{align*} \int_{G}^{} f \circ \left( ϕ_1 * ϕ_2 \right) \left( g \right) dg \leq \int_{\mathbb{R}}^{} f \circ \left( ϕ_1^* * ϕ_2^* \right) \left( x \right) dx \end{align*} holds for any measurable functions $ϕ_1, ϕ_2 \colon G \to \mathbb{R}_{\geq 0}$ with $μ( \mathrm{supp} \; ϕ_1 ) + μ( \mathrm{supp} \; ϕ_2 ) \leq m(G)$ and any convex function $f \colon \mathbb{R}_{\geq 0} \to \mathbb{R}$ with $f(0) = 0$. Here $ϕ…
New Improvements to Heron and Heinz Inequality Using Matrix Techniques
本文研究矩阵均值插值与比较，将参数$\vartheta$扩展至$\mathbb{R}^+$，引入结合Kantorovich常数的Heinz不等式新标量变体并强化算子版本，提出专为半正定矩阵迹、行列式和范数设计的改进型Young不等式。
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