Estimates for the first eigenvalue of the one-dimensional $p$-Laplacian

In the present paper, we study the first eigenvalue $\lambda (p)$ of the one-dimensional $p$-Laplacian in the interval $(-1,1)$. We give an upper and lower estimate of $\lambda (p)$ and study its asymptotic behavior as $p\to 1+0$ or $p\to \mathrm{\infty }$.