For any Riemannian metric $g$ on $S^2$ with volume $4\pi$, a celebrated isoperimetric inequality established by Hersch in 1970 states that the sum of the reciprocals of the first three nonzero eigenvalues is bounded below by 3/2. In this talk, we present a refinement of Hersch's inequality, alongside an improved version of Nadirashvili's isoperimetric inequality for the second eigenvalue. These refined inequalities impose stricter constraints on the joint distribution of the first two nonzero eigenvalues on $S^2$. Additionally, we will discuss related numerical results concerning this distribution. This is joint work with Qingwei Zeng.