Let $H$ be an $n\times n$ Hilbert matrix, 
$$h\_{ij}=(i+j-1)^{-1}.$$ 


The matrix $p$-norm corresponding to the p-norm for vectors is:
$\left \| A \right \| \_p = \sup \limits \_{x \ne 0} \frac{\left \| A x\right \| \_p}{\left \| x\right \| \_p}$, $p\ge 1$. 


Is there a known (or what is the) formula for $\left \| H \right \| \_p$ in terms of $n$ and $p$? 


I saw a related post for $n=\infty$.
<https://math.stackexchange.com/questions/837832/norm-of-hilbert-matrix-is-it-equal-to-pi>


After reading the post [Spectral norm for a truncated Hilbert matrix](https://mathoverflow.net/questions/125376/spectral-norm-for-a-truncated-hilbert-matrix) I guess the answer to my query is no... so a big challenge. 



The norm of a Finite Hilbert matrix
I think the answer is no, indeed. Even in the particular case $p=2$ the formula is not known (to my best knowledge). Concerning $\|H\|\_{2}$, is known $\|H\|\_{2}\leq\pi$. More precisely, we have the following upper bound
$$\|H\|\_{2}\leq 2w\_n\arcsin\frac{1}{w\_n}$$
where
$$w\_n=2\left[\binom{2n}{n}\right]^{-1/2n},$$
see P. Otte: Upper bounds for the spectral radius of the $n\times n$ Hilbert matrix, Pacific J. Math. 219 No. 2 (2005) 101-109.


In addition, quite precise asymptotic formula for $\|H\|\_{2}$ is known as $n\to\infty$. Namely,
$$\|H\|\_{2}=\pi-\frac{\pi^{5}}{2\ln^{2} n}+O\left(\frac{\ln\ln n}{\ln^{3} n}\right),$$
see N.G. de Bruijn and H.S. Wilf, On Hilbert’s inequality in n dimensions, Bull. Amer. Math. Soc. 68 (1962), 70–73.


