A new uniform lower bound on Weil-Petersson distance

In this paper we study the injectivity radius based at a fixed point along Weil-Petersson geodesics. We show that the square root of the injectivity radius based at a fixed point is $0.3884$-Lipschitz on Teichm"uller space endowed with the Weil-Petersson metric. As an application we reprove that the square root of the systole function is uniformly Lipschitz on Teichm"uller space endowed with the Weil-Petersson metric, where the Lipschitz constant can be chosen to be $0.5492$. Applications to asymptotic geometry of moduli space of Riemann surfaces for large genus will also be discussed.