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In this paper, we consider a birth-death process with generator $ \mathcal{L} $ and reversible invariant probability measure $ \pi. $ We identify explicitly the Lipschitzian norm of the solution of the Poisson equation $ -\mathcal{L} G = g-\pi（g） $ for $ |g|\le\varphi $. This leads to some transportation-information inequalities, concentration inequalities and Cheeger-type isoperimetric inequalities. Several examples are provided to illustrate the results.
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