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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.

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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.

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# An extension of the Hardy-Littlewood-Polya inequality?

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?