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Search: All articles in the CMB digital archive with keyword inequalities

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1. CMB Online first

Lehec, Joseph
Short Probabilistic Proof of the Brascamp-Lieb and Barthe Theorems
We give a short proof of the Brascamp-Lieb theorem, which asserts that a certain general form of Young's convolution inequality is saturated by Gaussian functions. The argument is inspired by Borell's stochastic proof of the Prékopa-Leindler inequality and applies also to the reversed Brascamp-Lieb inequality, due to Barthe.

Keywords:functional inequalities, Brownian motion
Categories:39B62, 60J65

2. CMB 2012 (vol 57 pp. 3)

Adamczak, Radosław; Latała, Rafał; Litvak, Alexander E.; Oleszkiewicz, Krzysztof; Pajor, Alain; Tomczak-Jaegermann, Nicole
A Short Proof of Paouris' Inequality
We give a short proof of a result of G.~Paouris on the tail behaviour of the Euclidean norm $|X|$ of an isotropic log-concave random vector $X\in\mathbb{R}^n,$ stating that for every $t\geq 1$, \[\mathbb{P} \big( |X|\geq ct\sqrt n\big)\leq \exp(-t\sqrt n).\] More precisely we show that for any log-concave random vector $X$ and any $p\geq 1$, \[(\mathbb{E}|X|^p)^{1/p}\sim \mathbb{E} |X|+\sup_{z\in S^{n-1}}(\mathbb{E} |\langle z,X\rangle|^p)^{1/p}.\]

Keywords:log-concave random vectors, deviation inequalities
Categories:46B06, 46B09, 52A23

3. CMB 2012 (vol 57 pp. 25)

Bourin, Jean-Christophe; Harada, Tetsuo; Lee, Eun-Young
Subadditivity Inequalities for Compact Operators
Some subadditivity inequalities for matrices and concave functions also hold for Hilbert space operators, but (unfortunately!) with an additional $\varepsilon$ term. It seems not possible to erase this residual term. However, in case of compact operators we show that the $\varepsilon$ term is unnecessary. Further, these inequalities are strict in a certain sense when some natural assumptions are satisfied. The discussion also stresses on matrices and their compressions and several open questions or conjectures are considered, both in the matrix and operator settings.

Keywords:concave or convex function, Hilbert space, unitary orbits, compact operators, compressions, matrix inequalities
Categories:47A63, 15A45

4. CMB 2011 (vol 55 pp. 355)

Nhan, Nguyen Du Vi; Duc, Dinh Thanh
Convolution Inequalities in $l_p$ Weighted Spaces
Various weighted $l_p$-norm inequalities in convolutions are derived by a simple and general principle whose $l_2$ version was obtained by using the theory of reproducing kernels. Applications to the Riemann zeta function and a difference equation are also considered.

Keywords:inequalities for sums, convolution
Categories:26D15, 44A35

5. CMB 2010 (vol 54 pp. 159)

Sababheh, Mohammad
Hardy Inequalities on the Real Line
We prove that some inequalities, which are considered to be generalizations of Hardy's inequality on the circle, can be modified and proved to be true for functions integrable on the real line. In fact we would like to show that some constructions that were used to prove the Littlewood conjecture can be used similarly to produce real Hardy-type inequalities. This discussion will lead to many questions concerning the relationship between Hardy-type inequalities on the circle and those on the real line.

Keywords:Hardy's inequality, inequalities including the Fourier transform and Hardy spaces
Categories:42A05, 42A99

6. CMB 2010 (vol 53 pp. 327)

Luor, Dah-Chin
Multidimensional Exponential Inequalities with Weights
We establish sufficient conditions on the weight functions $u$ and $v$ for the validity of the multidimensional weighted inequality $$ \Bigl(\int_E \Phi(T_k f(x))^q u(x)\,dx\Bigr)^{1/q} \le C \Bigl (\int_E \Phi(f(x))^p v(x)\,dx\Bigr )^{1/p}, $$ where 0<$p$, $q$<$\infty$, $\Phi$ is a logarithmically convex function, and $T_k$ is an integral operator over star-shaped regions. The condition is also necessary for the exponential integral inequality. Moreover, the estimation of $C$ is given and we apply the obtained results to generalize some multidimensional Levin--Cochran-Lee type inequalities.

Keywords:multidimensional inequalities, geometric mean operators, exponential inequalities, star-shaped regions
Categories:26D15, 26D10

7. CMB 1999 (vol 42 pp. 478)

Pruss, Alexander R.
A Remark On the Moser-Aubin Inequality For Axially Symmetric Functions On the Sphere
Let $\scr S_r$ be the collection of all axially symmetric functions $f$ in the Sobolev space $H^1(\Sph^2)$ such that $\int_{\Sph^2} x_ie^{2f(\mathbf{x})} \, d\omega(\mathbf{x})$ vanishes for $i=1,2,3$. We prove that $$ \inf_{f\in \scr S_r} \frac12 \int_{\Sph^2} |\nabla f|^2 \, d\omega + 2\int_{\Sph^2} f \, d\omega- \log \int_{\Sph^2} e^{2f} \, d\omega > -\oo, $$ and that this infimum is attained. This complements recent work of Feldman, Froese, Ghoussoub and Gui on a conjecture of Chang and Yang concerning the Moser-Aubin inequality.

Keywords:Moser inequality, borderline Sobolev inequalities, axially symmetric functions
Categories:26D15, 58G30

8. CMB 1999 (vol 42 pp. 321)

Kikuchi, Masato
Averaging Operators and Martingale Inequalities in Rearrangement Invariant Function Spaces
We shall study some connection between averaging operators and martingale inequalities in rearrangement invariant function spaces. In Section~2 the equivalence between Shimogaki's theorem and some martingale inequalities will be established, and in Section~3 the equivalence between Boyd's theorem and martingale inequalities with change of probability measure will be established.

Keywords:martingale inequalities, rearrangement invariant function spaces
Categories:60G44, 60G46, 46E30

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