# A counterexample to Junghenn's Principles of Analysis $4.29$(d)

Problem: The following is Exercise $$4.29$$(d) in Hugo Junghenn's Principles of Analysis:
Use Jensen's inequality to verify the following for a probability measure $$\mu:$$
$$\|f\|_1\log(\|f\|_1)\leq\log(\|f\log(f)\|_1\quad\text{where }f>0.$$ I tried the problem for a while and now I think that the claim is incorrect. I have cooked up the following counterexample.

Consider the probability space $$((0,1),\mathcal B,\mu)$$, where $$\mathcal B$$ is the Borel $$\sigma$$-field of $$(0,1)$$ and $$\mu$$ is the Lebesgue measure. Next, let $$f(x)=x^2$$ for $$x\in(0,1)$$. Then $$f>0$$ everywhere and we have $$\|f\|_1\log(\|f\|_1)=\frac{1}{3}\log\left(\frac{1}{3}\right).$$ On the other hand, using integration by parts and the fact that $$x^3\log(x)\to0$$ as $$x\searrow0$$, we see that $$\|f\log(f)\|_1=\int_0^1 x^2|\log(x^2)|\,dx=2\int_0^1x^2|\log(x)|\,dx=\frac{2}{9}.$$ But then $$\log(2/9)<3^{-1}\log(3^{-1})$$, hence the inequality does not hold.

However, I do believe that the author meant to ask to prove $$\|f\|_1\log(\|f\|_1)\leq\|f\log(f)\|_1,$$ which is an easy consequence of Jensen's inequality taking the convex function to be $$x\log(x).$$

My Question: Do you agree with my counterexample above? If not, I would like to ask if I am wrong and the inequality does hold, or if there is another, this time correct, counterexample.

Thank you very much for your time and appreciate all the feedback and help.

• You are right. The first log should not have been there on the right side. – Kavi Rama Murthy Sep 16 at 5:22