Questions tagged [karamata-inequality]

Inequalities, which we can prove by the Karamata's inequal;ity

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$a^{ab}+b^{bc}+c^{cd}+d^{da}\geq a^{2a^2b^2}+b^{2b^2c^2}+c^{2c^2d^2}+d^{2d^2a^2}$ with some conditions

I'm interested by the following problem : Let $a,b,c,d$ be real positive numbers suc that $a\geq1$,$b\leq 1$ , $c\leq 1$ , $d\leq 1$ with conditions that I precise below then we have : $$a^{ab}+...
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General Method to solve Power Sum Inequality

This is a general method to have power sum inequality . We work with $x_i> 1$ $n$ real numbers . We want to show this kind of inequality : Let $x_i> 1$ be $n$ real positive numbers and $...
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69 views

$N-1$ equal value principle

Let us start with an example, consider the following sum: $$H_n=\sum_{i=1}^{n}\frac{1}{n-1+x_i}$$ With $x_1x_2...x_n=1$, We're willing to show that $H_n \le 1$. To do this we start with the following ...
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68 views

Karamata + Jensen = Niculescu's inequality (version of 1991)

In fact it's my point of view but I think that this version of Niculescu's inequality is a mixture between Karamata and Jensen's inequality . So we have : Let $f(x)$ be a convex strictly ...