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 Mar 9 answered A beautiful inequality for convex functions Mar 9 comment A beautiful inequality for convex functions @Julien: I didn't show the motivation of the construction of $g$ in the linked answer, but the basic idea is quite similar to your approach, and the same idea can be applied to constructing $h$. Do I need to post an answer here to show all the details(probably it does not satisfy the "more intuitive" requirement in the earlier version of your question)? Mar 9 comment A beautiful inequality for convex functions @Julien: To see the history of all the editions of a post, you may simply click "edited +time" in the last line of the post. Mar 9 comment A beautiful inequality for convex functions @Julien: I almost forgot all the details in my answer to the linked question, and I just looked back at it briefly. It seems that the same argument there does not work, because it uses the fact that the supremum of convex functions is still convex, which fails to be true for infimum in general. However, I think the method in the remark of my answer there(or the first version of my answer) still works for constructing $h$, and this method shares the same basic idea with your own approach. Mar 9 comment A beautiful inequality for convex functions Here is a closely related question. Jan 9 awarded Generalist Dec 3 comment Operators on $C([0,1])$ that is compact or not. @user62138: You may use a similar argument as part b) in my answer. If $(f_n)$ is bounded in $L^p$ for some \$1