Let $p>2$ and $$\|b\|^p_p:=\int_0^{2\pi} |b(e^{it})|^p dt.$$ Assume that $a$ is a bounded holomorphic on the unit disk and assume that $a(0)=0$. What is the best constant $C_p$ in the inequality $$\|a\|_p\le C_p \|\Re(a)\|_p?$$
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$\begingroup$ Did you write $a$ for $b$ or vice-versa? $\endgroup$– Brendan McKayCommented Nov 22, 2016 at 2:41
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$\begingroup$ @Brendan McKay $b$ could be $a$ and $\Re (a)$. $\endgroup$– djoleCommented Nov 22, 2016 at 8:35
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The sharp constant is $C_p = 1/\cos\pi/(2p')$, where $p'=p/(p-1)$.