Unbiased estimation in one-parameter exponential families for the inverse of the natural parameter with extensions

For one-parameter continuous exponential families, we identify an unbiased estimator of the inverse of the natural parameter $θ$ for cases where $θ> 0$, extending an earlier result of \cite{voinov1985unbiased} applicable to a normal model. We provide various applications for Gamma models, Inverse Gaussian models, distributions obtained by truncation, and ratios of normal means. Moreover, we extend the findings to estimating negative powers $θ^{-k}$, and more generally to complete monotone functions $q(θ)$.