An isoperimetric inequality for eigenvalues of the bi-harmonic operator

} In this article, we put forward a Neumann eigenvalue problem for the bi-harmonic operator $Delta^2$ on a bounded smooth domain $Om$ in the Euclidean $n$-space ${bf R}^n$ ($nge2$) and then prove that the corresponding first non-zero eigenvalue $Upsilon_1(Om)$ admits the isoperimetric inequality of Szego-Weinberger type: $Upsilon_1(Om)le Upsilon_1(B_{Om})$, where $B_{Om}$ is a ball in ${bf R}^n$ with the same volume of $Om$. The isoperimetric inequality of Szego-Weinberger type for the first nonzero Neumann eigenvalue of the even-multi-Laplacian operators $Delta^{2m}$ ($mge1$) on $Om$ is also exploited.