My question is:
Is there monotone property of $\|f\|_p$ when $p$ is increasing, where $\|f\|_p=(\int_a^b f(x)^pdx)^{1/p}$ is the classical $L^p$ norm and $f\in L^p$?
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This proposition is right when $p\rightarrow\infty$ because $\|f\|_\infty=\max|f(x)|$. But I am interested in the case for which $p$ is finite.
My try:
Personally, the $\|f\|_p$ is increasing when $p$ is increasing. I use some common functions, such as $\sin(x)$, $\exp(x)$ etc., and they show that is valid. My idea is to prove the derivative of $\|f\|_p$ to $p$ is positive.
Let $g(p)=(\int_a^bf(x)^pdx)^{1/p}$. We have: $$ g'(p)=\dfrac{(\int_a^bf(x)^pdx)^{1/p-1}}{p}(\int_a^bf(x)^p(\ln f(x)^p-\ln\int_a^bf(t)^pdt)dx) $$ where $g'(p)$ is the derivative of $g(p)$ to $p$. Obviously, we only need to proof the 2nd item of $g'(p)$ is positive. To this end, let $h(x)=\ln(x)$ and $c=\int_a^bf(x)^pdx$, then from Taylor's expansion, we have: $$ h(x)=h(c)+h'(c)(x-c)+h''(\xi)(x-c)^2\leq h(c)+h'(c)(x-c)=h(c)+\dfrac{x-c}{c} $$ By setting $x=f(x)^p$ and integral from $a$ to $b$, we have: $$ \int_a^b\ln f(x)^pdx\leq (b-a)\ln\int_a^b f(t)^pdt+1-(b-a) $$ This is very close to $g'(p)$ by using mid point theorem. But I can not proceed. Does anyone can give me a ingenious proof or show me my proposition is wrong?