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Questions tagged [rearrangement-inequality]

Proofs of inequalities by using Rearrangement inequality or Chebyshov inequality.

4
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68 views

Does $(\sum_{i=1}^n a_i^{1.5})^2 - \sum_{i=1}^n a_i \; \sum_{i=1}^n a_i a_{i+1} > 0$ hold for $n \leq 8$?

Let positive reals $\{a_i\}$, where not all $a_i$ are equal. Does $$ f(\{a_i\}) = (\sum_{i=1}^n a_i^{1.5})^2 - \sum_{i=1}^n a_i \; \sum_{i=1}^n a_i a_{i+1} > 0 $$ hold for $n \le 8$? It is ...
2
votes
0answers
101 views

Does $(\sum_{i=1}^n a_i^{1.5})^2 - \sum_{i=1}^n a_i \; \sum_{i=1}^n a_i a_{i+1} > 0$ hold for $n\le 16$?

Let positive reals $\{a_i\}$, where not all $a_i$ are equal. Does $$ f(\{a_i\}) = (\sum_{i=1}^n a_i^{1.5})^2 - \sum_{i=1}^n a_i \; \sum_{i=1}^n a_i a_{i+1} > 0 $$ hold for $n \le 16$? It is ...
1
vote
0answers
31 views

symmetrized rearrangement on sphere.

I am trying to undestand the Corollary 2.2 from Osgood, Phillips and Sarnak (see http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.486.558&rep=rep1&type=pdf), that is, if $u \in W^{1}(S^...
1
vote
0answers
21 views

Generalization of the Pólya–Szegő inequality

Let $d \in \mathbb{N}, 1\leq p \leq \infty$, $f \in C^{\infty}_0 (\mathbb{R}^d)$. It is known that there is a function $g$ such that $g(x)=g_0(|x|)$ for some non-increase function $g_0:[0, \infty) ...
0
votes
0answers
36 views

Optimization of sum of squares over permutations

Suppose I have fixed, positive values $n_1, \cdots, n_\ell$ and $T$. I'm looking for an algorithm to optimize \begin{align*} f(\boldsymbol{n}) = T\left(\sum_{j=1}^{\ell}\left(\sum_{i=1}^{j}n_i\right)^...