Let $1<p<q<\infty$, and $r\in [p,q]$ whith $\frac{1}{r}= \frac{\alpha}{p}+ \frac{1-\alpha}{q}$. If $f\in L_p\cap L_q$ then
$$\|f\|_r \leq \|f\|_p^\alpha\|f\|_q^{(1-\alpha)}$$
My teacher referred as this equation as an alternative form of the Liapunov's inequality:
$$
\|f\|_r^r\leq \|f\|_p^{\lambda p}\|f\|_q^{(1-\lambda)q} \quad \text{with } r=\lambda p +(1-\lambda)q
$$
My question is: how can I get the first form from the second?
Alternately, can I get the first form by applying the Hölder inequality ( $\|fg\|_1 \leq \|f\|_p \|g\|_{p'}$ where $\frac{1}{p} +\frac{1}{p'}=1$ ) to get something like this proof of the Liapunov inequality?
Best Answer
You get the first from the second by taking $\lambda =\frac {q-r} {r-p}$ and rasing both sides to power $\frac 1 r$. [The cases $r=p$, $r=q$ are to be handled separately. I will leave that to you].