If $m(X)<\infty$ and $f$ bounded and measurable, then $1\leq q<p<\infty$ implies $\|f\|_{p}^{p}\leq\|f\|_{q}^{q}\|f\|_{\infty}^{p-q}.$

by G the Stackman   Last Updated June 12, 2019 07:20 AM

$\textbf{The Problem:}$ Let $m(X)<\infty$ and $f$ bounded and measurable. For $1\leq q<p<\infty$ prove that $\|f\|_{p}^{p}\leq\|f\|_{q}^{q}\|f\|_{\infty}^{p-q}.$

$\textbf{My Thoughts:}$ At first sight I was trying to somehow bring in Holder's inequality here, but I got nowhere. So I observed that the result would follow if it is true that $$\left(\frac{f(x)}{\|f\|_\infty}\right)^p\leq\left(\frac{f(x)}{\|f\|_\infty}\right)^q\text{ for almost every }x\in X.$$ Now I fix $x\in X$ for which the above is true, and observe that $$\frac{f(x)}{\|f\|_\infty}\leq1.$$ Since the function $$\frac{f(x)}{\|f\|_\infty}\mapsto\left(\frac{f(x)}{\|f\|_\infty}\right)^z$$ decreases monotonically, the result follows.

Is my line of reasoning on the right track?

My apologies if my wording is bad. If it is though, please point out, and I will do my best to correct it.

Thank you for your time and appreciate any feedback.

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