Valuation bound for coefficients of an integral formal logarithm ($v_K(c_n)\ge -e{v}_p(n)$)

For an integral one-dimensional formal group law over a finite extension of the p-adic field, the inverse of ${\partial }_{Y}F(T,0)$ has integral coefficients because its constant term is one. Integrating this invariant differential gives ${\log }_F(T)=T+{\sum }_{n\ge 2}{c}_n{T}^n$ with $n{c}_n$ integral. For $v_K(\pi )=1$ and $e=v_K(p)$ the displayed bound follows. The estimate $v_p(n)\le (n-1)/(p-1)$ makes the nonlinear part strictly contracting on ${\pi }^s{\mathcal{O}}_K$ for $s>e/(p-1)$, yielding the deep logarithm subgroup of a formal group.