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In the setting of reversible continuous-time Markov chains, the $CD_\Upsilon$condition has been shown recently to be a consistent analogue to theBakry-\'Emery condition in the diffusive setting in terms of proving Li-Yauinequalities under a finite dimension term and proving the modified logarithmicSobolev inequality under a positive curvature bound. In this article we examinethe case where both is given, a finite dimension term and a positive curvaturebound. For this purpose we introduce the $CD_\Upsilon(\kappa,F)$ condition,where the dimension term is expressed by a so called $CD$-function $F$. Wederive functional inequalities relating the entropy to the Fisher information,which we will call entropy-information inequalities. Further, we deduceapplications of entropy-information inequalities such as ultracontractivitybounds, exponential integrability of Lipschitz functions, finite diameterbounds and a modified version of the celebrated Nash inequality.

Entropy-information inequalities under curvature-dimension conditions for continuous-time Markov chains