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) ${\mathcal{H}}_{\lambda }^{\mathrm{\infty }}=({\mathcal{H}}_{i_1{i}_2\cdots i_m})$, $${\mathcal{H}}_{i_1{i}_2\cdots i_m}=\frac{1}{i_1+i_2+\cdots i_m+\lambda },\lambda \in \mathbb{R}\setminus {\mathbb{Z}}^{-};i_1,i_2,\cdots ,i_m=0,1,2,\cdots ,n,\cdots ,$$ and proved that its $Z_1$-spectral radius is not larger than $\pi$ for $\lambda >\frac{1}{2}$, and is at most $\frac{\pi }{\sin \lambda \pi }$ for $\frac{1}{2}\ge \lambda >0$. 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 $\lambda$.