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1. CMB Online first
Telescoping estimates for smooth series We derive telescoping majorants and minorants for some classes
of series and give applications of these results.
Keywords:telescoping series, Stietjes constant, Hardy's formula, Stirling's formula Categories:26D15, 40A25, 97I30 |
2. CMB 2011 (vol 55 pp. 355)
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 |
3. CMB 2011 (vol 54 pp. 630)
Mixed Norm Type Hardy Inequalities Higher dimensional mixed norm type
inequalities involving certain integral operators are
characterized in terms of the corresponding lower dimensional
inequalities.
Keywords:Hardy inequality, reverse Hardy inequality, mixed norm, Hardy-Steklov operator Categories:26D10, 26D15 |
4. CMB 2010 (vol 53 pp. 327)
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 |
5. CMB 2005 (vol 48 pp. 333)
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 $\alpha a-\beta b\geq \max(a-b,0)$.
(ii) $Q$ is increasing on $(0,\infty)$ if{}f $\alpha a-\beta b\leq
\min(a-b,0)$.
An application of part (i) reveals that for all $x>0$ the function $s\mapsto [(s-1)\zeta(s,x)]^{1/(s-1)}$ is decreasing on $(1,\infty)$. This settles
a conjecture of Bastien and Rogalski.
Categories:11M35, 26D15 |
6. CMB 1999 (vol 42 pp. 478)
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 |