A sharp bound for the ratio of the first two Dirichlet eigenvalues of a domain in a hemisphere of $\mathbb {S}^n$

For a domain $\Omega$ contained in a hemisphere of the $n$–dimensional sphere $\mathbb {S}^n$ we prove the optimal result $\lambda _2/\lambda _1(\Omega ) \le \lambda _2/\lambda _1(\Omega ^{\star })$ for the ratio of its first two Dirichlet eigenvalues where $\Omega ^{\star }$, the symmetric rearrangement of $\Omega$ in $\mathbb {S}^n$, is a geodesic ball in $\mathbb {S}^n$ having the same $n$–volume as $\Omega$. We also show that $\lambda _2/\lambda _1$ for geodesic balls of geodesic radius $\theta _1$ less than or equal to $\pi /2$ is an increasing function of $\theta _1$ which runs between the value $(j_{n/2,1}/j_{n/2-1,1})^2$ for $\theta _1=0$ (this is the Euclidean value) and $2(n+1)/n$ for $\theta _1=\pi /2$. Here $j_{\nu ,k}$ denotes the $k$th positive zero of the Bessel function $J_{\nu }(t)$. This result generalizes the Payne–Pólya–Weinberger conjecture, which applies to bounded domains in Euclidean space and which we had proved earlier. Our method makes use of symmetric rearrangement of functions and various technical properties of special functions. We also prove that among all domains contained in a hemisphere of $\mathbb {S}^n$ and having a fixed value of $\lambda _1$ the one with the maximal value of $\lambda _2$ is the geodesic ball of the appropriate radius. This is a stronger, but slightly less accessible, isoperimetric result than that for $\lambda _2/\lambda _1$. Various other results for $\lambda _1$ and $\lambda _2$ of geodesic balls in $\mathbb {S}^n$ are proved in the course of our work.