1. CMB Online first
 Atıcı, Ferhan M.; Yaldız, Hatice

Convex Functions on Discrete Time Domains
In this paper, we introduce the definition of a convex real
valued function $f$ defined on the set of integers, ${\mathbb{Z}}$. We
prove that $f$ is convex on ${\mathbb{Z}}$ if and only if $\Delta^{2}f
\geq 0$ on ${\mathbb{Z}}$. As a first application of this new concept,
we state and prove discrete HermiteHadamard inequality using
the basics of discrete calculus (i.e. the calculus on ${\mathbb{Z}}$).
Second, we state and prove the discrete fractional HermiteHadamard
inequality using the basics of discrete fractional calculus.
We close the paper by defining the convexity of a real valued
function on any time scale.
Keywords:discrete calculus, discrete fractional calculus, convex functions, discrete HermiteHadamard inequality Categories:26B25, 26A33, 39A12, 39A70, 26E70, 26D07, 26D10, 26D15 

2. CMB Online first
 Lin, Minghua

A determinantal inequality involving partial traces
Let $\mathbf{A}$ be a density matrix in $\mathbb{M}_m\otimes
\mathbb{M}_n$. Audenaert [J. Math. Phys. 48 (2007) 083507] proved
an inequality for Schatten $p$norms:
\[
1+\\mathbf{A}\_p\ge \\tr_1 \mathbf{A}\_p+\\tr_2 \mathbf{A}\_p,
\]
where $\tr_1, \tr_2$ stand for the first and second partial
trace, respectively. As an analogue of his result, we prove a
determinantal inequality
\[
1+\det \mathbf{A}\ge \det(\tr_1 \mathbf{A})^m+\det(\tr_2 \mathbf{A})^n.
\]
Keywords:determinantal inequality, partial trace, block matrix Categories:47B65, 15A45, 15A60 

3. CMB 2015 (vol 59 pp. 204)
 Spektor, Susanna

Restricted Khinchine Inequality
We prove a Khintchine type inequality under the assumption that
the sum of
Rademacher random variables equals zero. We also show a new
tailbound for a hypergeometric random variable.
Keywords:Khintchine inequality, Kahane inequality, Rademacher random variables, hypergeometric distribution. Categories:46B06, 60E15, 52A23, 46B09 

4. CMB 2015 (vol 58 pp. 486)
 Duc, Dinh Thanh; Nhan, Nguyen Du Vi; Xuan, Nguyen Tong

Inequalities for Partial Derivatives and their Applications
We present various weighted integral inequalities for partial
derivatives acting on products and compositions of functions
which are applied to establish some new Opialtype inequalities
involving functions of several independent variables. We also
demonstrate the usefulness of our results in the field of partial
differential equations.
Keywords:inequality for integral, Opialtype inequality, HÃ¶lder's inequality, partial differential operator, partial differential equation Categories:26D10, 35A23 

5. CMB 2014 (vol 58 pp. 432)
 Yang, Dachun; Yang, Sibei

Secondorder Riesz Transforms and Maximal Inequalities Associated with Magnetic SchrÃ¶dinger Operators
Let $A:=(\nablai\vec{a})\cdot(\nablai\vec{a})+V$ be a
magnetic SchrÃ¶dinger operator on $\mathbb{R}^n$,
where $\vec{a}:=(a_1,\dots, a_n)\in L^2_{\mathrm{loc}}(\mathbb{R}^n,\mathbb{R}^n)$
and $0\le V\in L^1_{\mathrm{loc}}(\mathbb{R}^n)$ satisfy some reverse
HÃ¶lder conditions.
Let $\varphi\colon \mathbb{R}^n\times[0,\infty)\to[0,\infty)$ be such that
$\varphi(x,\cdot)$ for any given $x\in\mathbb{R}^n$ is an Orlicz function,
$\varphi(\cdot,t)\in {\mathbb A}_{\infty}(\mathbb{R}^n)$ for all $t\in (0,\infty)$
(the class of uniformly Muckenhoupt weights) and its uniformly critical upper type index
$I(\varphi)\in(0,1]$. In this article, the authors prove that
secondorder Riesz transforms $VA^{1}$ and
$(\nablai\vec{a})^2A^{1}$ are bounded from the
MusielakOrliczHardy space $H_{\varphi,\,A}(\mathbb{R}^n)$, associated with $A$,
to the MusielakOrlicz space $L^{\varphi}(\mathbb{R}^n)$. Moreover, the authors
establish the boundedness of $VA^{1}$ on $H_{\varphi, A}(\mathbb{R}^n)$. As applications, some
maximal inequalities associated with $A$ in the scale of $H_{\varphi,
A}(\mathbb{R}^n)$ are obtained.
Keywords:MusielakOrliczHardy space, magnetic SchrÃ¶dinger operator, atom, secondorder Riesz transform, maximal inequality Categories:42B30, 42B35, 42B25, 35J10, 42B37, 46E30 

6. CMB 2014 (vol 58 pp. 561)
 MartinezMaure, Yves

Plane Lorentzian and Fuchsian Hedgehogs
Parts of the BrunnMinkowski theory can be extended to hedgehogs, which are
envelopes of families of affine hyperplanes parametrized by their Gauss map.
F. Fillastre introduced Fuchsian convex bodies, which are the
closed convex sets of LorentzMinkowski space that are globally invariant
under the action of a Fuchsian group. In this paper, we undertake a study of
plane Lorentzian and Fuchsian hedgehogs. In particular, we prove the
Fuchsian analogues of classical geometrical inequalities (analogues which
are reversed as compared to classical ones).
Keywords:Fuchsian and Lorentzian hedgehogs, evolute, duality, convolution, reversed isoperimetric inequality, reversed Bonnesen inequality Categories:52A40, 52A55, 53A04, 53B30 

7. CMB 2011 (vol 55 pp. 555)
 Michalowski, Nicholas; Rule, David J.; Staubach, Wolfgang

Weighted $L^p$ Boundedness of Pseudodifferential Operators and Applications
In this paper we prove weighted norm inequalities with weights in
the $A_p$ classes, for pseudodifferential operators with symbols in
the class ${S^{n(\rho 1)}_{\rho, \delta}}$ that fall outside the
scope of CalderÃ³nZygmund theory. This is accomplished by
controlling the sharp function of the pseudodifferential operator by
HardyLittlewood type maximal functions. Our weighted norm
inequalities also yield $L^{p}$ boundedness of commutators of
functions of bounded mean oscillation with a wide class of operators
in $\mathrm{OP}S^{m}_{\rho, \delta}$.
Keywords:weighted norm inequality, pseudodifferential operator, commutator estimates Categories:42B20, 42B25, 35S05, 47G30 

8. CMB 2011 (vol 55 pp. 708)
9. CMB 2011 (vol 55 pp. 597)
 Osękowski, Adam

Sharp Inequalities for Differentially Subordinate Harmonic Functions and Martingales
We determine the best constants $C_{p,\infty}$ and $C_{1,p}$,
$1 < p < \infty$, for which the following holds. If $u$, $v$ are
orthogonal harmonic functions on a Euclidean domain such that $v$ is
differentially subordinate to $u$, then
$$ \v\_p \leq C_{p,\infty}
\u\_\infty,\quad
\v\_1 \leq C_{1,p} \u\_p.
$$
In particular, the inequalities are still sharp for the conjugate
harmonic functions on the unit disc of $\mathbb R^2$.
Sharp probabilistic versions of these estimates are also studied.
As an application, we establish a sharp version of the classical logarithmic inequality of Zygmund.
Keywords: harmonic function, conjugate harmonic functions, orthogonal harmonic functions, martingale, orthogonal martingales, norm inequality, optimal stopping problem Categories:31B05, 60G44, 60G40 

10. CMB 2011 (vol 55 pp. 537)
11. CMB 2011 (vol 55 pp. 88)
 Ghanbari, K.; Shekarbeigi, B.

Inequalities for Eigenvalues of a General Clamped Plate Problem
Let $D$ be a
connected bounded domain in $\mathbb{R}^n$. Let
$0<\mu_1\leq\mu_2\leq\dots\leq\mu_k\leq\cdots$ be the eigenvalues
of the following Dirichlet
problem:
$$
\begin{cases}\Delta^2u(x)+V(x)u(x)=\mu\rho(x)u(x),\quad x\in
D
u_{\partial D}=\frac{\partial u}{\partial n}_{\partial
D}=0,
\end{cases}
$$
where $V(x)$ is a nonnegative potential,
and $\rho(x)\in C(\bar{D})$ is positive.
We prove the following inequalities:
$$\mu_{k+1}\leq\frac{1}{k}\sum_{i=1}^k\mu_i+\Bigl[\frac{8(n+2)}{n^2}\Bigl(\frac{\rho_{\max}}
{\rho_{\min}}\Bigr)^2\Bigr]^{1/2}\times
\frac{1}{k}\sum_{i=1}^k[\mu_i(\mu_{k+1}\mu_i)]^{1/2},
$$
$$\frac{n^2k^2}{8(n+2)}\leq
\Bigl(\frac{\rho_{\max}}{\rho_{\min}}\Bigr)^2\Bigl[\sum_{i=1}^k\frac{\mu_i^{1/2}}{\mu_{k+1}\mu_i}\Bigr]
\times\sum_{i=1}^k\mu_i^{1/2}.
$$
Keywords:biharmonic operator, eigenvalue, eigenvector, inequality Category:35P15 

12. CMB 2011 (vol 54 pp. 630)
13. 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 

14. CMB 2009 (vol 53 pp. 153)
15. CMB 2008 (vol 51 pp. 161)
 Agarwal, Ravi P.; OteroEspinar, Victoria; Perera, Kanishka; Vivero, Dolores R.

Wirtinger's Inequalities on Time Scales
This paper is devoted to the study of Wirtingertype
inequalities for the Lebesgue $\Delta$integral on an arbitrary time scale $\T$.
We prove a general inequality for a class of absolutely continuous
functions on closed subintervals of an adequate subset of $\T$.
By using this expression and by assuming that $\T$ is bounded,
we deduce that
a general inequality is valid for every absolutely continuous function on $\T$
such that its $\Delta$derivative belongs to $L_\Delta^2([a,b)\cap\T)$ and at most it vanishes
on the boundary of $\T$.
Keywords:time scales calculus, $\Delta$integral, Wirtinger's inequality Category:39A10 

16. CMB 2006 (vol 49 pp. 256)
 Neelon, Tejinder

A BernsteinWalsh Type Inequality and Applications
A BernsteinWalsh type inequality for $C^{\infty }$ functions of several
variables is derived, which then is applied to obtain analogs and
generalizations of the following classical theorems: (1) BochnakSiciak
theorem: a $C^{\infty }$\ function on $\mathbb{R}^{n}$ that is real
analytic on every line is real analytic; (2) ZornLelong theorem: if a
double power series $F(x,y)$\ converges on a set of lines of positive
capacity then $F(x,y)$\ is convergent; (3) AbhyankarMohSathaye theorem:
the transfinite diameter of the convergence set of a divergent series is
zero.
Keywords:BernsteinWalsh inequality, convergence sets, analytic functions, ultradifferentiable functions, formal power series Categories:32A05, 26E05 

17. CMB 2006 (vol 49 pp. 185)
 Averkov, Gennadiy

On the Inequality for Volume and Minkowskian Thickness
Given a centrally symmetric convex body $B$ in $\E^d,$ we denote
by $\M^d(B)$ the Minkowski space ({\em i.e.,} finite dimensional
Banach space) with unit ball $B.$ Let $K$ be an arbitrary convex
body in $\M^d(B).$ The relationship between volume $V(K)$ and the
Minkowskian thickness ($=$ minimal width) $\thns_B(K)$ of $K$ can
naturally be given by the sharp geometric inequality $V(K) \ge
\alpha(B) \cdot \thns_B(K)^d,$ where $\alpha(B)>0.$ As a simple
corollary of the RogersShephard inequality we obtain that
$\binom{2d}{d}{}^{1} \le \alpha(B)/V(B) \le 2^{d}$ with equality
on the left attained if and only if $B$ is the difference body of
a simplex and on the right if $B$ is a crosspolytope. The main
result of this paper is that for $d=2$ the equality on the right
implies that $B$ is a parallelogram. The obtained results yield
the sharp upper bound for the modified BanachMazur distance to the
regular hexagon.
Keywords:convex body, geometric inequality, thickness, Minkowski space, Banach space, normed space, reduced body, BanachMazur compactum, (modified) BanachMazur distance, volume ratio Categories:52A40, 46B20 

18. CMB 2006 (vol 49 pp. 82)
 Gogatishvili, Amiran; Pick, Luboš

Embeddings and Duality Theorem for Weak Classical Lorentz Spaces
We characterize the weight functions
$u,v,w$ on $(0,\infty)$ such that
$$
\left(\int_0^\infty f^{*}(t)^
qw(t)\,dt\right)^{1/q}
\leq
C \sup_{t\in(0,\infty)}f^{**}_u(t)v(t),
$$
where
$$
f^{**}_u(t):=\left(\int_{0}^{t}u(s)\,ds\right)^{1}
\int_{0}^{t}f^*(s)u(s)\,ds.
$$
As an application we present a~new simple characterization of
the associate space to the space $\Gamma^ \infty(v)$, determined by the
norm
$$
\f\_{\Gamma^ \infty(v)}=\sup_{t\in(0,\infty)}f^{**}(t)v(t),
$$
where
$$
f^{**}(t):=\frac1t\int_{0}^{t}f^*(s)\,ds.
$$
Keywords:Discretizing sequence, antidiscretization, classical Lorentz spaces, weak Lorentz spaces, embeddings, duality, Hardy's inequality Categories:26D10, 46E20 

19. CMB 2004 (vol 47 pp. 206)
20. CMB 2001 (vol 44 pp. 376)
21. 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 

22. CMB 1999 (vol 42 pp. 87)
 Kittaneh, Fuad

Some norm inequalities for operators
Let $A_i$, $B_i$ and $X_i$ $(i=1, 2, \dots, n)$ be operators on a
separable Hilbert space. It is shown that if $f$ and $g$ are
nonnegative continuous functions on $[0,\infty)$ which satisfy the
relation $f(t)g(t) =t$ for all $t$ in $[0,\infty)$, then
$$
\Biglvert \,\Bigl\sum^n_{i=1} A^*_i X_i B_i \Bigr^r \,\Bigrvert^2
\leq \Biglvert \Bigl( \sum^n_{i=1} A^*_i f (X^*_i)^2 A_i \Bigr)^r
\Bigrvert \, \Biglvert \Bigl( \sum^n_{i=1} B^*_i g (X_i)^2 B_i
\Bigr)^r \Bigrvert
$$
for every $r>0$ and for every unitarily invariant norm. This result
improves some known CauchySchwarz type inequalities. Norm
inequalities related to the arithmeticgeometric mean inequality and
the classical Heinz inequalities are also obtained.
Keywords:Unitarily invariant norm, positive operator, arithmeticgeometric mean inequality, CauchySchwarz inequality, Heinz inequality Categories:47A30, 47B10, 47B15, 47B20 

23. CMB 1997 (vol 40 pp. 169)