  In this paper, we introduce the concept of Z$_1$-eigenvalue to infinite
dimensional generalized Hilbert tensors (hypermatrix)
$\mathcal{H}_\lambda^{\infty}=(\mathcal{H}_{i_{1}i_{2}\cdots i_{m}})$,
  $$
  \mathcal{H}_{i_{1}i_{2}\cdots i_{m}}=\frac{1}{i_{1}+i_{2}+\cdots
i_{m}+\lambda},\
  \lambda\in \mathbb{R}\setminus\mathbb{Z}^-;\
i_{1},i_{2},\cdots,i_{m}=0,1,2,\cdots,n,\cdots,
  $$
  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
$\frac{1}{2}\geq \lambda>0$. 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 $\lambda$.
