I'm currently studying the Munkenhoupt class (also called $A_p$ class), and I'm stumbled upon the proof of the following property:
Given $1 < p <\infty$, and $w\in A_p(R^n)$ (note that we have $w(x) > 0$ for all $x\in R^n$). Then we have: if $1<p < r<\infty$, then $C_{r,w}\leq C_{p,w}$
My attempt: I tried many different tricks to apply Holder's inequality to prove the inequality: $\frac{1}{B(x,r)}(\int_{B} w^{\frac{1}{1-r}})^{r-1}\leq \frac{1}{B(x,r)}(\int_{B} w^{\frac{1}{1-p}})^{p-1}$. I was thinking of considering $w^{1/r}w^{-1/r} = w^{1/p}w^{-1/p}$. Then rewrite $1 = w^{(1/r+1/p)}w^{-(1/r+1/p)}$, and take the average integral of the RHS over $B$, then applying the Holder's inequality. Unfortunately, Holder's inequality is not applicable here, since $1/r+1/p$ is not guaranteed to be greater than $1$:(
Can someone please help me with this problem? I will really appreciate any help. I think I couldn't see a clever trick here:P