1. CMB 2013 (vol 57 pp. 585)
 Lehec, Joseph

Short Probabilistic Proof of the BrascampLieb and Barthe Theorems
We give a short proof of the BrascampLieb 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 PrÃ©kopaLeindler inequality and applies also to the
reversed BrascampLieb 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; TomczakJaegermann, 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
logconcave 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 logconcave random vector $X$
and any $p\geq 1$,
\[(\mathbb{E}X^p)^{1/p}\sim \mathbb{E} X+\sup_{z\in
S^{n1}}(\mathbb{E} \langle
z,X\rangle^p)^{1/p}.\]
Keywords:logconcave random vectors, deviation inequalities Categories:46B06, 46B09, 52A23 

3. CMB 2012 (vol 57 pp. 25)
 Bourin, JeanChristophe; Harada, Tetsuo; Lee, EunYoung

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 Hardytype inequalities.
This discussion will lead to many questions concerning the
relationship between Hardytype 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, DahChin

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 starshaped 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 LevinCochranLee type inequalities.
Keywords:multidimensional inequalities, geometric mean operators, exponential inequalities, starshaped regions Categories:26D15, 26D10 

7. CMB 1999 (vol 42 pp. 478)
 Pruss, Alexander R.

A Remark On the MoserAubin 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 MoserAubin 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 
