Given $(X,\mathcal{F},m)$ a measure space with $m(X)=1$ and $||f||_p<\infty$ for some $p>0$. Need to show that $\forall q \in (0,p)$ $$\int \log |f| dm\leq \log(||f||_q),$$ $$ \log ||f||_q \leq \frac{\int |f|^qdm -1}{q},$$ $$\lim_{q\to 0} \frac{\int |f|^q dm -1}{q}=\int \log |f| dm$$ and finally $$\lim_{q\to 0} ||f||_q= e^{\int \log (|f|)dm} .$$ My attempt at first inequality since $\log$ is concave we have $$\int \log (|f|)dm\leq \log\left(\int|f|dm\right)$$ by Jensen's inequality. Want to claim $$\int |f| dm \leq ||f||_q$$ but only true for $q>1$. If I assume the previous claim then $$\int \log (|f|)dm\leq \log\left(\int|f|dm\right)\leq \log ||f||_q$$ by properties of $\log$. For the second inequality I have no idea as well as the limits. The limits should follow from the inequalities somehow.


For the first question, apply Jensen's inequality as you have to the function $|f|^q$, then multiply both sides by $1/q$. We get

$$\int \log (|f|)dm = \frac{1}{q}\int \log (|f|^q)dm\leq \frac{1}{q}\log\left(\int|f|^qdm\right) = \log\left(\left(\int|f|^qdm\right)^{1/q}\right)$$

establishing the result.

The second inequality essentially derives from $ x - 1 - \log x \ge 0$, which is true for all $x \in \mathbb{R}_+$. You can prove this with freshman calculus.

The third can be established with l'Hospital's rule:

$$\lim_{q\to 0} \frac{\int |f|^q dm -1}{q}= \lim_{q\to 0} \frac{\int |f|^q \log|f| dm}{1} = \int \log |f| dm$$

Note that both equalities above require justification - Why $\int_X$ and $\frac{\partial}{\partial q}$ commute, and why $\int_X$ and $\lim_{q \to 0}$ commute. I'll leave these to you since they arise from fundamental reasons that you either should know, or should spend time thinking about if you don't.

Lastly, squeeze both sides of the inequality established in the first question to finish.


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