Let $f(z)=\sum_{n=0}^{\infty} \frac{c_n}{n+1}z^n$, where sequence $c_n \in S^1=\{z:|z|=1\}.$ We observe $H^p$ norm $\|f\|_{H_p}$, where $H^p$ is Hardy space, $1 \leq p < \infty$.
Question: For the fix $p \geq 1$, determine $\min_{c_n \in S^1} \|f\|_{H^p}$ and $\max_{c_n \in S^1}\|f\|_{H^p}$ ?

  • $\begingroup$ Looks great but what on Earth makes you think that there is a nice closed form answer for an arbitrary $p$? If you are willing to settle for less than the exact values, it will be helpful to tell us how much you really want. As written, it looks hopeless except for a few trivial cases (maximum for even integer $p$, etc.) $\endgroup$
    – fedja
    May 29, 2017 at 15:48
  • $\begingroup$ Any idea for p=1 or p=3. $\endgroup$ May 29, 2017 at 16:16
  • $\begingroup$ For the exact values, you mean? Of course, not (unless you want to invest time in the computer search and get some meaningless sequences of digits).. How much do you really want? $\endgroup$
    – fedja
    May 29, 2017 at 16:21


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