Upper bounds for Z$_1$-eigenvalues of generalized Hilbert tensors

In this paper, we introduce the concept of Z$_1$-eigenvalue to infinite dimensional generalized Hilbert tensors (hypermatrix) $mathcal{H}_lambda^{infty}=(mathcal{H}_{i_{1}i_{2}cdots i_{m}})$, $$ mathcal{H}_{i_{1}i_{2}cdots i_{m}}=frac{1}{i_{1}+i_{2}+cdots i_{m}+lambda}, lambdain mathbb{R}setminusmathbb{Z}^-; i_{1},i_{2},cdots,i_{m}=0,1,2,cdots,n,cdots, $$ and proved that its $Z_1$-spectral radius is not larger than $pi$ for $lambda>frac{1}{2}$, and is at most $frac{pi}{sin{lambdapi}}$ for $frac{1}{2}geq lambda>0$. Besides, the upper bound of $Z_1$-spectral radius of an $m$th-order $n$-dimensional generalized Hilbert tensor $mathcal{H}_lambda^n$ is obtained also, and such a bound only depends on $n$ and $lambda$.