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 2014 (vol 58 pp. 188)
3. 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 

4. CMB 2011 (vol 54 pp. 630)
5. 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 

6. CMB 2005 (vol 48 pp. 333)
 Alzer, Horst

Monotonicity Properties of the Hurwitz Zeta Function
Let
$$
\zeta(s,x)=\sum_{n=0}^{\infty}\frac{1}{(n+x)^s} \quad{(s>1,\, x>0)}
$$
be the Hurwitz zeta function and let
$$
Q(x)=Q(x;\alpha,\beta;a,b)=\frac{(\zeta(\alpha,x))^a}{(\zeta(\beta,x))^b},
$$
where $\alpha, \beta>1$
and $a,b>0$ are real numbers. We prove:
(i) The function $Q$ is decreasing on $(0,\infty)$ if{}f $\alpha a\beta b\geq \max(ab,0)$.
(ii) $Q$ is increasing on $(0,\infty)$ if{}f $\alpha a\beta b\leq
\min(ab,0)$.
An application of part (i) reveals that for all $x>0$ the function $s\mapsto [(s1)\zeta(s,x)]^{1/(s1)}$ is decreasing on $(1,\infty)$. This settles
a conjecture of Bastien and Rogalski.
Categories:11M35, 26D15 

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 
