Combine my [first previous question](https://mathoverflow.net/questions/303013/a-muirhead-like-inequality) and [second previous question](https://mathoverflow.net/questions/302956/an-inequality-related-to-power-sum-and-elementary-symmetric-polynomial-and-major) with the [Muirhead inequality](https://en.wikipedia.org/wiki/Muirhead%27s_inequality). I have posed conjectures of two inequalities as follows:



> 
> **Inequality 1:** Let $n>2$ and $1 \le m \le n$ be integers. Let $x\_1, \dots, x\_n$ and $y\_1,\dots, y\_n$ be nonnegative real numbers such that $(x\_1,\dots, x\_n)$ [majorizes](https://en.wikipedia.org/wiki/Majorization) $(y\_1,\dots, y\_n)$. Then for all reals $0 \leq a\_1, a\_2,\cdots,a\_n \leq 1$, 
> 
> 
> $$\sum\limits\_{1 \le p\_1 <\cdots < p\_m \le n}\left( \sum\limits\_{1 \le i\_1 <\cdots < i\_m \le n} x\_{i\_1}^{a\_{p\_1}} \cdots x\_{i\_m}^{a\_{p\_m}} \right) \leq \sum\limits\_{1 \le p\_1 <\cdots < p\_m \le n}\left( \sum\limits\_{1 \le i\_1 <\cdots < i\_m \le n} y\_{i\_1}^{a\_{p\_1}} \cdots y\_{i\_m}^{a\_{p\_m}} \right) $$
> 
> 
> 


**Example for inequality 1 with $n=3, m=2$:**


$${x\_1}^{a\_1}.{x\_2}^{a\_2}+{x\_1}^{a\_1}.{x\_3}^{a\_2}+{x\_2}^{a\_1}.{x\_3}^{a\_2}+{x\_1}^{a\_2}.{x\_2}^{a\_3}+{x\_1}^{a\_2}.{x\_3}^{a\_3}+{x\_2}^{a\_2}.{x\_3}^{a\_3}+{x\_1}^{a\_1}.{x\_2}^{a\_3}+{x\_1}^{a\_1}.{x\_3}^{a\_3}+{x\_2}^{a\_1}.{x\_3}^{a\_3} \leq {y\_1}^{a\_1}.{y\_2}^{a\_2}+{y\_1}^{a\_1}.{y\_3}^{a\_2}+{y\_2}^{a\_1}.{y\_3}^{a\_2}+{y\_1}^{a\_2}.{y\_2}^{a\_3}+{y\_1}^{a\_2}.{y\_3}^{a\_3}+{y\_2}^{a\_2}.{y\_3}^{a\_3}+{y\_1}^{a\_1}.{y\_2}^{a\_3}+{y\_1}^{a\_1}.{y\_3}^{a\_3}+{y\_2}^{a\_1}.{y\_3}^{a\_3}$$



> 
> **Inequality 2:** Let $n>2$ and $1 \le m \le n$ be integers. Let $x\_1, \dots, x\_n$ and $y\_1,\dots, y\_n$ be nonnegative real numbers such that $(x\_1,\dots, x\_n)$ [majorizes](https://en.wikipedia.org/wiki/Majorization) $(y\_1,\dots, y\_n)$. Then for all reals $ a\_1, a\_2,\dots,a\_n \geq 0$, 
> 
> 
> $$\sum\limits\_{1 \le p\_1 <\cdots < p\_m \le n}\left( \sum\limits\_{1 \le i\_1 <\cdots < i\_m \le n} a\_{i\_1}^{x\_{p\_1}} \cdots a\_{i\_m}^{x\_{p\_m}} \right) \geq \sum\limits\_{1 \le p\_1 <\cdots < p\_m \le n}\left( \sum\limits\_{1 \le i\_1 <\cdots < i\_m \le n} a\_{i\_1}^{y\_{p\_1}} \cdots a\_{i\_m}^{y\_{p\_m}} \right)$$
> 
> 
> 


**Example for inequality 2 with $n=3, m=2$:**


$${a\_1}^{x\_1}.{a\_2}^{x\_2}+{a\_1}^{x\_1}.{a\_3}^{x\_2}+{a\_2}^{x\_1}.{a\_3}^{x\_2}+{a\_1}^{x\_2}.{a\_2}^{x\_3}+{a\_1}^{x\_2}.{a\_3}^{x\_3}+{a\_2}^{x\_2}.{a\_3}^{x\_3}+{a\_1}^{x\_1}.{a\_2}^{x\_3}+{a\_1}^{x\_1}.{a\_3}^{x\_3}+{a\_2}^{x\_1}.{a\_3}^{x\_3} \geq 
{a\_1}^{y\_1}.{a\_2}^{y\_2}+{a\_1}^{y\_1}.{a\_3}^{y\_2}+{a\_2}^{y\_1}.{a\_3}^{y\_2}+{a\_1}^{y\_2}.{a\_2}^{y\_3}+{a\_1}^{y\_2}.{a\_3}^{y\_3}+{a\_2}^{y\_2}.{a\_3}^{y\_3}+{a\_1}^{y\_1}.{a\_2}^{y\_3}+{a\_1}^{y\_1}.{a\_3}^{y\_3}+{a\_2}^{y\_1}.{a\_3}^{y\_3}$$



> 
> 
> > 
> > **My question:** I am looking for a proof of two inequalities above.
> > 
> > 
> > 
> 
> 
> 



Combination power elementary symmetric polynomial inequality
These inequalities are true when $m=1$, and they both follow from Karamata's inequality, since $x^a$ is concave when $0\le a\le 1$, and $a^x$ is convex. However they are both *false* for every $m\geq 2$.


**Counterexample to Inequality 1** Set $a\_1=1$ and $a\_i=0$ for $i>1$. Then your inequality now reads $$\sum\_{i=1}^{n-m+1}\binom{n-1}{m-1}\binom{n-i}{m-1}x\_i\le \sum\_{i=1}^{n-m+1}\binom{n-1}{m-1}\binom{n-i}{m-1}y\_i.$$
Notice that since $m\geq 2$ we have $n-m+1\le n-1$ so that $x\_n$ and $y\_n$ don't appear in the inequality. So let $x\_1=x\_2=\cdots=x\_{n-1}=M$ be a very large number and take $x\_n=0$. Then take $y\_1=y\_2=\cdots=y\_{n-1}=M-1$ and $y\_n=n-1$. This is easily seen to be a counterexample since every term on the left hand side is greater than the corresponding term on the right.


**Counterexample to Inequality 2** We will take $a\_1=a\_2=\cdots a\_{n-1}=1$ and $a\_n=2$ as well as $x\_1=nM$ and $x\_i=0$ for $i\geq 2$, and all $y\_i=M$. The left hand side is seen to equal $\binom{n}{m}^2=O(1)$ whereas the right hand side is $O(2^M)$, so the inequality breaks as soon as $M$ is large enough.


