Bonnesen-style inequality for the first eigenvalue on a complete surface of constant curvature

By Cheeger’s isoperimetric constants, some lower bounds and upper bounds of λ 1 , the first eigenvalue on a complete surface of constant curvature, are given. Some Bonnesen-style inequalities and reverse Bonnesen-style inequalities for the first eigenvalue are obtained. Those Bonnesen-style inequalities obtained are stronger than the well-known Osserman’s results and the upper bound is stronger than Osserman’s results (Osserman in Proceedings of the International Congress of Mathematicians, Helsinki, 1978 ).