SIMO Retiree Blog

(This is Yan Sheng.) Last time we studied a family of inequalities involving absolute values on linear polynomials, and asked the question of how lazy we can be to prove inequalities of that form. This week, we will be applying the same approach to a different family: 3-variable homogeneous symmetric inequalities of low degree. A polynomial $P(x,y,z)$ in 3 variables is called homogeneous of degree $d$ if each of its terms is of degree $d$, and symmetric if $P(x,y,z)=P(x',y',z')$ for all permutations $x',y',z'$ of $x,y,z$. Inequalities involving homogeneous symmetric polynomials include the AM-GM inequality$$\left(\frac{x+y+z}3\right)^3\ge xyz,$$and the Schur inequality$$x^r(x-y)(x-z)+y^r(y-x)(y-z)+z^r(z-x)(z-y)\ge0$$for integers $r\ge0$. For the rest of this post, write $\mathcal P^+_d$ for the family of all homogeneous symmetric polynomials $P(x,y,z)$ of degree $d$ such that $P(x,y,z)\ge0$ holds for all $x,y,z\ge0$. Our Main Problem is to describe $\math...

(This is Yan Sheng.) What does it mean to "properly understand" some mathematical result? For me, I find it the most satisfying when I can answer the following two questions: What is the minimal set of special cases that I need to verify to prove it? How could I have come up with it myself? In this and the next blog post, I'll describe two different situations in olympiad inequalities that I've tried to understand better recently, by answering the two questions above. Theorem (Popoviciu 1965): Let $f:[a,b]\to\mathbb R$ be a convex function. Then for any $x,y,z\in[a,b]$, we have$$\begin{align*}&f(x)+f(y)+f(z)+3f\left(\frac{x+y+z}3\right)\\&\ge2\left(f\left(\frac{x+y}2\right)+\left(\frac{y+z}2\right)+\left(\frac{z+x}2\right)\right).\end{align*}$$ What an interesting statement! It's not immediately clear how to prove it with Jensen's inequality, and it makes me wonder what other similar inequalities hold for convex functions. Let's try provin...