Let $x,y$ be vectors in $\mathbb{R}^n$ and let's use the notation $\hat x$ for the vector $x$ with its components sorted in increasing order.
The Hardy-Littlewood-Polya inequality states that
$$ x\cdot y \leq \hat x\cdot \hat y.$$
Let us also use the notation $xy\in\mathbb{R}^n$ to denote the coordinate-wise product of $x$ and $y$.
I conjecture that
$$
\frac{ ||xy||\_p ||xy||\_r}{||xy||\_q}
\le
\frac{ ||\hat x\hat y||\_p ||\hat x\hat y||\_r}{||\hat x\hat y||\_q}
$$
for all $1\le p\le q\le r$.
For $q=p$ and $q=r$, my conjectured inequality is true by the HLP inequality. Any ideas for a proof?


UPDATE: thank you for the quick answers. The counterexamples indeed work when negative coordinates for x and y are allowed. However, when all the coordinates of x and y are required to be positive, the conjecture seems to hold.


UPDATE 2: so the conjecture is totally false; see below for counterexamples.



An extension of the Hardy-Littlewood-Polya inequality?
Here is a Matlab script that will generate a quick counterexample for you:



```
function [x,y]=testIneq(n, p, q, r)

% x and y are length n vectors
% Try: [x,y]=testIneq(2,1,2,3) to get a counterexample!

flag = 1;
iter = 0;

while (flag)

    iter = iter + 1;
    x = randn(n,1);
    y = randn(n,1);

    xh = sort(x);
    yh = sort(y);

    xy = x .* y;
    xyh = xh .* yh;

    lhs = norm(xy,p) * norm(xy,r) / norm(xy,q);
    rhs = norm(xyh,p) * norm(xyh,r) / norm(xyh,q);

    if (rhs < lhs)
        flag = 0;
        fprintf('Found countex after %d tries\n', iter);
    end
end

```

end


Example: $x =[-2.1384,-0.8396]$, $y =[1.3546,-1.0722]$, with $p=1$, $q=2$, $r=3$.


