Extending Muirhead’s Inequality

Abstract

We present a method to prove an inequality concerning a linear combination of symmetric monomial functions. This is based on Muirhead’s inequality combining with a graph theoretical setting. As an application we prove some interesting inequalities motivated from extremal combinatorics.

Appendix

Write $a_i$ for $\alpha _{p_i}$, and let $q_i=1-p_i$.

Fact 1

Let $t\ge 1$ be an integer, and let $l_j$ be the line $y=(r-1)x+j$. Then the walk $W_{p_i}$ hits the line $l_t$ with probability $a_i^t$, and the walk $W'$ hits the line $l_{rt}$ with probability $\beta ^t$.

Proof

Suppose that the probability that the walk $W_{p_i}$ hits the line $l_t$ is given by $X^t$ for some $X\in (0,1)$. After the first step, the walk is at (0, 1) with probability $p_i$, and at (1, 0) with probability $q_i$. From (0, 1) the probability for the walk hitting $l_t$ is $X^{t-1}$, and from (1, 0) the probability is $X^{t-1+r}$. Then it follows

$$\begin{aligned} X^t=p_iX^{t-1}+q_iX^{t-1+r}, \end{aligned}$$

that is,

$$\begin{aligned} X=p_i+q_iX^r. \end{aligned}$$

Thus $X=a_i$, and the walk hits the line $l_t$ with probability $a_i^t$.

Next suppose that the probability that the walk $W'$ hits the line $l_{rt}$ is given by $Y^t$ for some $Y\in (0,1)$. After the first r steps, it is at $(x,r-x)$ with probability

$$\begin{aligned} \sum _{I\in \left( {\begin{array}{c}[r]\\ x\end{array}}\right) }\prod _{i\in I}p_i\prod _{j\in [r]\setminus I}q_j, \end{aligned}$$

where $[r]=\{1,2,\ldots ,r\}$. From $(x,r-x)$ the probability for the walk hitting $l_{rt}$ is $Y^{x+t-1}$. This yields

$$\begin{aligned} Y^t=\sum _{x=0}^rY^{x+t-1} \sum _{I\in \left( {\begin{array}{c}[r]\\ x\end{array}}\right) }\prod _{i\in I}p_i\prod _{j\in [r]\setminus I}q_j, \end{aligned}$$

that is,

$$\begin{aligned} Y=\sum _{x=0}^rY^{x} \sum _{I\in \left( {\begin{array}{c}[r]\\ x\end{array}}\right) }\prod _{i\in I}p_i\prod _{j\in [r]\setminus I}q_j =\prod _{i=1}^r(p_i+q_iY). \end{aligned}$$

Thus $Y=\beta$, and the walk hits the line $l_{rt}$ with probability $\beta ^t$. $\square$

Define a polynomial f(x) by

$$\begin{aligned} f(X):=-X+\prod _{i=1}^r(p_i+q_iX). \end{aligned}$$

By definition $f(\beta )=0$.

Fact 2

If $0<y<1$ and $f(y)\le 0$, then $\beta \le y$.

Proof

This follows because $f(0)>0$, $f(1)=0$, $f'(1)=-1+\sum _{i=1}^r q_i> -1+r\cdot \frac{1}{r}=0$ (here we used $p_i<1-\frac{1}{r}$), and $f''(x)>0$ for $x>0$. $\square$

Fact 3

The inequality $a:=a_1\cdots a_r\le \beta$ follows from (1).

Proof

By Fact 7 it suffices to show $f(a)\le 0$. Since $a_i=p_i+q_ia_i^r$ we have

$$\begin{aligned} a=a_1\cdots a_r=\prod _{i=1}^r(p_i+q_ia_i^r), \end{aligned}$$

and

$$\begin{aligned} f(a)=-a+\prod _{i=1}^r(p_i+q_ia) =-\prod _{i=1}^r(p_i+q_ia_i^r)+\prod _{i=1}^r(p_i+q_ia). \end{aligned}$$

So we need to show

$$\begin{aligned} \prod _{i=1}^r(p_i+q_ia)\le \prod _{i=1}^r(p_i+q_ia_i^r). \end{aligned}$$

(14)

Solving $a_i=p_i+(1-p_i)a_i^r$ for $p_i$ gives

$$\begin{aligned} p_i=\frac{a_i-a_i^r}{1-a_i^r}. \end{aligned}$$

Then

$$\begin{aligned} p_i+q_ia&= \frac{a_i-a_i^r}{1-a_i^r} +\left( 1-\frac{a_i-a_i^r}{1-a_i^r}\right) a=\frac{1}{1-a_i^r}\left( a_i-a_i^r+(1-a_i)a\right) ,\\ p_i+q_ia_i^r&=a_i =\frac{1}{1-a_i^r}\left( a_i(1-a_i^r)\right) . \end{aligned}$$

Noting that $0<a_i<1$ we see that (14) is equivalent to

$$\begin{aligned}&\prod _{i=1}^r(a_i-a_i^r+(1-a_i)a) \le \prod _{i=1}^ra_i({1-a_i^r})\\ \Longleftrightarrow \quad&\prod _{i=1}^r\left( 1-a_i^{r-1}+(1-a_i)\frac{a}{a_i}\right) \le \prod _{i=1}^r(1-a_i^r)\\ \Longleftrightarrow \quad&\prod _{i=1}^r\left( 1+a_i+\cdots +a_i^{r-2} +\frac{a}{a_i}\right) \le \prod _{i=1}^r\left( 1+a_i+\cdots +a_i^{r-1} \right) , \end{aligned}$$

and multiplying both sides by $a=a_1\cdots a_r$ we get (1). $\square$

Fig. 1

The down map sending $\tilde{\lambda }=77544200$ to $\tilde{\mu }=66654110$

Fig. 2

Example for Lemma 4 and Lemma 5

Fig. 3

Hasse diagram of $\Lambda (\beta )$ and $\Lambda (\beta ')$ as posets for $r=7$ and $d=15$

Fig. 4

The reduced graph for t and $t'$ ($r=8$ and $d=29$)

Fig. 5

An optimal flow in the reduced graph ($r=8$ and $d=29$)