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Search: MSC category 26D15 ( Inequalities for sums, series and integrals )

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1. CMB Online first

Wirths, Karl Joachim
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)

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

3. CMB 2011 (vol 54 pp. 630)

Fiorenza, Alberto; Gupta, Babita; Jain, Pankaj
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)

Luor, Dah-Chin
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)

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 $\alpha a-\beta b\geq \max(a-b,0)$. (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)

Pruss, Alexander R.
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

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