The idea of this post arises when I've known in past weeks a refinement of Young’s inequality, I say Lemma 2.1 from [1]. On the other hand the Wikipedia Young's inequality for products shows generalizations of Young's inequality. As we see the second of those (the linked section) is the generalization using Fenchel–Legendre transforms, involving convex functions $f$ and its respective convex conjugate $g$.
The Wikipedia Convex conjugate also shows this mentioned Fenchel's inequality.
Question. Consider the mentioned generalization of Young's inequality using Fenchel–Legendre transforms. Is it possible a refinement/improvement of this inequality in a similar spirit of cited lemma of [1] $$\text{something}\leq f(a)+g(b)−ab\leq\text{something}$$ for $f(s)$ convex and $g(s)$ its convex conjugate? For what conditions? Many thanks.
Compare with [1] and the first paragraph of the linked Wikipedia, first paragraph of the section Generalization using Fenchel–Legendre transforms.
With the second question I mean that you feel free to add additional conditions for your refinement, in case that you need it. If my question is in the literature feel free to answer this post as a reference request, and I try to search and read the statement from the literature.
References:
[1] J. M. Aldaz, A stability version of Hölder's inequality, Journal of Mathematical Analysis and Applications, Volume 343, Issue 2, (2008), pp. 842-852.