The condition $\sum_{i=1}^n1/p_i=1$ and positivity imply $p_i\geq1$. We prove the <Generalized Holder inequality> by induction on $n$. The case $n=2$ is the usual <Holder inequality>. For the induction step, set
$$
\frac1q=\sum_{i=1}^{n-1}\frac1{p_i}=1-\frac1{p_n}.
$$
The exponents $q$ and $p_n$ are conjugate, so Hölder followed by the induction hypothesis gives
$$
\begin{aligned}
\left\|\prod_{i=1}^nf_i\right\|_1
&\leq\left\|\prod_{i=1}^{n-1}f_i\right\|_q\|f_n\|_{p_n}\\
&\leq\prod_{i=1}^n\|f_i\|_{p_i}.
\end{aligned}
$$
This proves the claim.

Solution

Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 105 / 3 / a / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2024/iii/paper-105/3/a/solution