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Letbe a Riemann surface obtained by removingdistinct points from a hyperbolic Riemann surface  R . For any closed geodesicon  R , there is the shortest closed geodesiconwhich is freely homotopic toon  R . In our previous paper [Yamamoto. An estimate for the hyperbolic length of closed geodesics on Riemann surfaces. Complex Var Elliptic Equ. 2019;64(9):1582–1607], we gave an upper bound for the hyperbolic length ofondepending only onand the hyperbolic length ofon  R  in the case when  R  is of finite type. The purpose of this paper is to improve our previous result.

An estimate for the hyperbolic lengths of closed geodesics on Riemann surfaces II