A note on the first eigenvalue of spherically symmetric manifolds
  We give lower and upper bounds for the first eigenvalue of geodesic balls in spherically symmetric manifolds. These lower and upper bounds are $C^{0}$-dependent on the metric coefficients. It gives better lower bounds for the first eigenvalue of spherical caps than those from Betz-Camera-Gzyl. 
Cleon S. Barroso, G. Pacelli Bessa