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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) $\mathcal{H}_λ^{\infty}=(\mathcal{H}_{i_{1}i_{2}\cdots i_{m}})$, $$ \mathcal{H}_{i_{1}i_{2}\cdots i_{m}}=\frac{1}{i_{1}+i_{2}+\cdots i_{m}+λ},\ λ\in \mathbb{R}\setminus\mathbb{Z}^-;\ i_{1},i_{2},\cdots,i_{m}=0,1,2,\cdots,n,\cdots, $$ and proved that its $Z_1$-spectral radius is not larger than $π$ for $λ>\frac{1}{2}$, and is at most $\fracπ{\sin{λπ}}$ for $\frac{1}{2}\geq λ>0$. Besides, the upper bound of $Z_1$-spectral radius of an $m$th-order $n$-dimensional generalized Hilbert tensor $\mathcal{H}_λ^n$ is obtained also, and such a bound only depends on $n$ and $λ$.