An estimate of the Bergman distance on Riemann surfaces

Let $M$ be a hyperbolic Riemann surface with the first eigenvalue ${\lambda }_1(M)>0$. Let $\rho$ denote the distance from a fixed point $x_0\in M$ and $r_x$ the injectivity radius at $x$. We show that there exists a numerical constant $c_0>0$ such that if $r_x\ge c_0{\lambda }_1(M{)}^{-3\mathrm{/}4}\rho (x{)}^{-1\mathrm{/}2}$ holds outside some compact set of $M$, then the Bergman distance verifies $d_B(x,x_0)\gtrsim \log [1+\rho (x)]$.