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Let $y,z\in(0,1)^n$ satisfy $||y||_1 = ||z||_1=1$.

Then $$ \frac{||z||_3}{||z||_2} \le K_n ||z/y||_\infty \frac{||y||_3}{||y||_2} $$ where $z/y\in\mathbb R^n$ is the coordinate-wise quotient of $z$ and $y$ and $K_n>0$ is some constant depending only on $n$. What's the best $K_n$ that makes this true? What if $2,3$ are replaced by general $p,q$?

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It's easy to see that $1 \le K_n \le \sqrt{n}$. Do you really need the best constant, or just some improvement on this? – Mark Meckes Mar 3 '11 at 14:59
For such a strange question you should give some motivation and say what you know. – Bill Johnson Mar 3 '11 at 16:15
My motivation is to obtain a Berry-Esseen type bound for the case where the coefficients are generated by a certain kind of partition. This probably isn't making a whole lot of sense; I'll write post a link to a short writeup soon. – Aryeh Kontorovich Mar 3 '11 at 17:27
@Mark -- the most interesting question is whether the best $K_n$ is actually a constant (which I suspect is not the case). The next question is -- does it grow as $n$ to some powr or slower (logarithmically, say)? The $\sqrt n$ bound is unfortunately too crude for my needs. – Aryeh Kontorovich Mar 3 '11 at 22:29
Here is a short writeup providing some motivation: – Aryeh Kontorovich Mar 3 '11 at 23:14

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