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First of all, the constants $c_i$ will have to depend on the complex structure of $X$ since without prescribing a complex structure one cannot talks about dependence on a basis of the space of holomorphic differentials on $X$. Now, $h$ and $g$ are both Riemannian metrics on a compact manifold and, thus, the identity map $$ (X,g)\to (X,h) $$ is bi-Lipschitz with Lipschitz constants depending, of course, on $g$ and $h$. Suppose that $(X, h), (Y, h)$ are Riemannian manifolds and $f: (X, g)\to (Y, h)$ is an $L$-Lipschitz map. Then for every closed geodesic $c$ in $(X,g)$, the $h$-length $\ell_h(c)$ of $f(c)$ is at most $L\ell_g(c)$. Hence, if $c^*$ is a shortest closed geodesic in the free homotopy class of $f(c)$, we have $$ \ell_h(c^*)\le L\ell_g(c). $$ For $L$-bi-Lipschitz maps one obtains the reverse inequality by considering $f^{-1}$. Thus, for every free homotopy class $[c]$ in $X$ you obtain the inequality $$ L^{-1}\ell_g([c]) \le \ell_h([f(c)])\le L \ell_g([c]), $$ where $\ell_\bullet([\bullet])$ stands for the length of a shortest loop in the given free homotopy class.