1
vote
1answer
69 views

symmetric polynomial inequality?

I put $n\ge 2$ balls of various sizes into an urn. I draw two balls (without replacement) from the urn. With each draw, I draw any given ball with probability proportional to its size. Can you ...
1
vote
1answer
116 views

Equal-variables problem in three variables

This question resembles Vasile Cirtoajeā€™s equal-variables results, as explained here (although those results may be useless in the present problem). Let $s,p$ be two positive numbers with ...
7
votes
0answers
161 views

Multivariate polynomial with all coefficients positive

Let $n\geq 3$ be an integer. Consider the following polynomials : $$ f(x_1,x_2, \ldots ,x_n)=\bigg(\frac{1}{n}\sum_{k=1}^n x_k^n\bigg)^{2n-2}- \bigg(\prod_{k=1}^n \frac{x_k^{2n-2}+\big(\prod_{j\neq ...
1
vote
2answers
161 views

Symmetric inequality on positive numbers whose product is one

Despite many attempts, no one at StackOverflow has succeeded in solving that old question about proving a deceptively simple-looking inequality. I propose now a weaker and slightly simpler inequality ...
1
vote
2answers
148 views

Three inequalities with sums of fractions over two positive integers

In a proof, I arrive at three inequalities for all $p,q \geqslant 0$: \begin{align} \frac{p+1}{q+1} + \frac{q+1}{p+1} &\geqslant 1 + \frac{p}{2q+1} + \frac{q}{2p+1} + \frac{1}{p+q+1};\cr ...