1. CMB 2016 (vol 59 pp. 225)
 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. 30)
 Chung, Jaeyoung

On an Exponential Functional Inequality and its Distributional Version
Let $G$ be a group and $\mathbb K=\mathbb C$ or $\mathbb
R$.
In this article, as a generalization of the result of Albert
and Baker,
we investigate the behavior of bounded
and unbounded functions $f\colon G\to \mathbb K$ satisfying the inequality
$
\Biglf
\Bigl(\sum_{k=1}^n x_k
\Bigr)\prod_{k=1}^n f(x_k)
\Bigr\le \phi(x_2, \dots, x_n),\quad \forall\, x_1, \dots,
x_n\in G,
$
where $\phi\colon G^{n1}\to [0, \infty)$. Also, as a distributional
version of the above inequality we consider the stability of
the functional equation
\begin{equation*}
u\circ S  \overbrace{u\otimes \cdots \otimes u}^{n\text {times}}=0,
\end{equation*}
where $u$ is a Schwartz distribution or Gelfand hyperfunction,
$\circ$ and $\otimes$ are the pullback and tensor product of
distributions, respectively, and $S(x_1, \dots, x_n)=x_1+ \dots
+x_n$.
Keywords:distribution, exponential functional equation, Gelfand hyperfunction, stability Categories:46F99, 39B82 

3. 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 

4. CMB 2011 (vol 56 pp. 218)
 Yang, Dilian

Functional Equations and Fourier Analysis
By exploring the relations among functional equations, harmonic analysis and representation theory,
we give a unified and very accessible approach to solve three important functional equations 
the d'Alembert equation, the Wilson equation, and the d'Alembert long equation 
on compact groups.
Keywords:functional equations, Fourier analysis, representation of compact groups Categories:39B52, 22C05, 43A30 

5. CMB 2011 (vol 55 pp. 214)
 Wang, DaBin

Positive Solutions of Impulsive Dynamic System on Time Scales
In this paper, some criteria for the existence of positive solutions of a class
of systems of impulsive dynamic equations on time scales are obtained by
using a fixed point theorem in cones.
Keywords:time scale, positive solution, fixed point, impulsive dynamic equation Categories:39A10, 34B15 

6. CMB 2011 (vol 55 pp. 424)
 Yang, Jianbin; Li, Song

Convergence Rates of Cascade Algorithms with Infinitely Supported Masks
We investigate the solutions of refinement equations of the form
$$
\phi(x)=\sum_{\alpha\in\mathbb
Z^s}a(\alpha)\:\phi(Mx\alpha),
$$ where the function $\phi$
is in $L_p(\mathbb R^s)$$(1\le p\le\infty)$, $a$ is an infinitely
supported sequence on $\mathbb Z^s$ called a refinement mask, and
$M$ is an $s\times s$ integer matrix such that
$\lim_{n\to\infty}M^{n}=0$. Associated with the mask $a$ and $M$ is
a linear operator $Q_{a,M}$ defined on $L_p(\mathbb R^s)$ by
$Q_{a,M} \phi_0:=\sum_{\alpha\in\mathbb
Z^s}a(\alpha)\phi_0(M\cdot\alpha)$. Main results of this paper are
related to the convergence rates of $(Q_{a,M}^n
\phi_0)_{n=1,2,\dots}$ in $L_p(\mathbb R^s)$ with mask $a$ being
infinitely supported. It is proved that under some appropriate
conditions on the initial function $\phi_0$, $Q_{a,M}^n \phi_0$
converges in $L_p(\mathbb R^s)$ with an exponential rate.
Keywords:refinement equations, infinitely supported mask, cascade algorithms, rates of convergence Categories:39B12, 41A25, 42C40 

7. CMB 2011 (vol 54 pp. 580)
 Baoguo, Jia; Erbe, Lynn; Peterson, Allan

Kiguradzetype Oscillation Theorems for Second Order Superlinear Dynamic Equations on Time Scales
Consider the second order superlinear dynamic equation
\begin{equation*}
(*)\qquad
x^{\Delta\Delta}(t)+p(t)f(x(\sigma(t)))=0\tag{$*$}
\end{equation*}
where $p\in C(\mathbb{T},\mathbb{R})$, $\mathbb{T}$ is a time scale,
$f\colon\mathbb{R}\rightarrow\mathbb{R}$ is
continuously differentiable and satisfies $f'(x)>0$, and $xf(x)>0$ for
$x\neq 0$. Furthermore, $f(x)$ also satisfies a superlinear condition, which
includes the nonlinear function $f(x)=x^\alpha$ with $\alpha>1$, commonly
known as the EmdenFowler case. Here the coefficient function $p(t)$ is
allowed to be negative for arbitrarily large values of $t$. In addition to
extending the result of Kiguradze for \eqref{star1} in the real case $\mathbb{T}=\mathbb{R}$, we
obtain analogues in the difference equation and $q$difference equation cases.
Keywords:Oscillation, EmdenFowler equation, superlinear Categories:34K11, 39A10, 39A99 

8. 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 

9. CMB 2005 (vol 48 pp. 505)
 Bouikhalene, Belaid

On the Generalized d'Alembert's and Wilson's Functional Equations on a Compact group
Let $G$ be a compact group. Let $\sigma$ be a continuous involution
of $G$. In this paper, we are
concerned by the following functional equation
$$\int_{G}f(xtyt^{1})\,dt+\int_{G}f(xt\sigma(y)t^{1})\,dt=2g(x)h(y), \quad
x, y \in G,$$ where $f, g, h \colonG \mapsto \mathbb{C}$, to be
determined, are complex continuous functions on $G$ such that $f$ is
central. This equation generalizes d'Alembert's and Wilson's
functional equations. We show that the solutions are expressed by
means of characters of irreducible, continuous and unitary
representations of the group $G$.
Keywords:Compact groups, Functional equations, Central functions, Lie, groups, Invariant differential operators. Categories:39B32, 39B42, 22D10, 22D12, 22D15 
