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The main result of this paper supports a conjecture by C. P\'erez and E. Relaabout a very recent result of theirs on self-improving theory. Also, we extendthe conclusions of their theorem to the range $p<1$.  As an application of our result, we give a unified vision of weightedimproved Poincar\'e-type inequalities in the Euclidean setting, which gathersboth weighted improved classical and fractional Poincar\'e inequalities withinan approach which avoids any representation formula. We improve some alreadyknown results.  Finally, we also explore analog inequalities in the context of metric spacesby means of the already known self-improving results.

A note on generalized Poincar\'e-type inequalities with applications to weighted improved Poincar\'e-type inequalities