Systole and small eigenvalues of hyperbolic surfaces

Let $S$ be a closed orientable hyperbolic surface with Euler characteristic$\chi$, and let ${\lambda }_k(S)$ be the $k$-th positive eigenvalue for the Laplacian on $S$. According to famous result of Otal and Rosas, ${\lambda }_{-\chi }>\frac{1}{4}$. In this article, we prove that if thesystole of $S$ is greater than 3,46, then ${\lambda }_{-\chi -1}>\frac{1}{4}$.This inequality is also true for geometrically finite orientable hyperbolic surfaces without cusps with the same assumption on the systole.