First, let us recall the sequence version of the rearrangement inequality.
Rearrangement inequality for nonnegative finite sequences: Let $N\in\mathbb{N}$ with $x_1\leq x_2\leq\cdots\leq x_N$ and $y_1\leq y_2\leq\cdots\leq y_N$ nondecreasing sequences of nonnegative real numbers. If $\sigma$ is a permutation of $\{1,2,\cdots,N\}$ then we have $$\sum_{i=1}^Nx_{N+1-i}y_i\leq\sum_{i=1}^Nx_{\sigma(i)}y_i\leq\sum_{i=1}^Nx_iy_i.$$
(Actually, this is valid for finite real-valued sequences, not just nonnegative ones. But I wish to restrict my attention to the nonnegative case for now.)
A rearrangement inequality for measurable nonnegative functions on $[0,1]$ may be conjectured as follows. If $f:[0,1]\to[-\infty,\infty]$ is measurable then we define its distribution function $\mu_f:[0,\infty]\to[0,\infty]$ by the rule $$\mu_f(t)=\mu\left\{x\in[0,1]:|f(x)|>t\right\}$$ where $\mu$ is the standard Lebesgue measure. Two functions are called equimeasurable of they have the same distribution function. It is known, in particular, that $f:[0,1]\to[-\infty,\infty]$ is equimeasurable with its nonincreasing rearrangement $f^*:[0,1]\to[0,\infty]$ defined by $$f^*(x)=\inf\left\{t:\mu_f(t)\leq x\right\}.$$ Hence, it is also equimeasurable with what we might call its nondecreasing rearrangement $f_*:[0,1]\to[0,\infty]$ defined by the rule $$f_*(x)=f^*(1-x).$$
Observe that the rearrangement inequality for nonnegative finite sequences is equivalent to the following: If $f:\{1,2,\cdots,N\}\to[0,\infty]$ and $g:\{1,2,\cdots,N\}\to[0,\infty]$ are measurable functions, then $$\sum_{i=1}^Nf_*(i)g^*(i)\leq \sum_{i=1}^Nf(i)g^*(i)\leq\sum_{i=1}^Nf^*(i)g^*(i),$$ where the definitions above are extended in the obvious way to measurable functions $f,g:\{1,\cdots,N\}\to[0,\infty]$. I conjecture the following.
Conjecture. Let $f,g:[0,1]\to[0,\infty]$ be nonnegative measurable functions. Then $$\int_0^1f_*(x)g^*(x)\;dx\leq \int_0^1f(x)g^*(x)\;dx\leq\int_0^1f^*(x)g^*(x)\;dx.$$
The last inequality is just Hardy-Littlewood. But I am curious if it is known whether the first inequality holds, i.e. whether $$\int_0^1f_*(x)g^*(x)\;dx\leq \int_0^1f(x)g^*(x)\;dx.$$
Thanks!