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I am deriving some bounds involving two functions, which are $f(x)=\log(x)$ and $f(x)=-x\times\log(x)$. I found in several books that these two functions are concave. I have draw them down using python to quickly check. For that reason, by applying Jensen's inequality the next inequality should hold:

$f(x)+f(y)\leq f(x+y)$

which holds in case of $f(x)=\log(x)$ but not for $f(x)=-x\times\log(x)$. What am I missing?

Thank you.

jdeJuan
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1 Answers1

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Here is a possible way I think, comment if anything wrong.

Note that, for any $x>1$ we have $(x+y)^{x+y}=(x+y)^x(x+y)^y>x^xy^y$

So, $$(x+y)^{-(x+y)}<x^{-x}y^{-y}$$ $$\implies -(x+y)\log(x+y)<-x\log x-y\log y$$ $$\implies f(x+y)<f(x)+f(y)$$

Sujit Bhattacharyya
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  • It seems that your derivation is correct. However this will mean that the function is convex, which is not. – jdeJuan Mar 06 '19 at 15:35
  • In fact. If $x\times\log x$ is concave, the sum over $x$ will also be concave. Note that $\sum_x -x\times \log x$ is the entropy, and this function is concave. https://en.wikipedia.org/wiki/Entropy_(information_theory)#/media/File:Binary_entropy_plot.svg – jdeJuan Mar 06 '19 at 15:47