In the field of (elementary) classical inequalities one of the most famous tools is the majorization inequality due to Karamata [1] (also known as Hardy-Littlewood-Polya). In its integral version, it states that:

If $\psi_1, \psi_2 : [a,b] \rightarrow \mathbb{R_{>0}}$ are non-negative measurable functions and, for example, $\psi_1$ "majorates" over $\psi_2$ (usually donated via $\psi_2 \succeq \psi_2$):

$$\int_{a}^{s} \psi_1 \, dx \geq \int_{a}^{s} \psi_2 \, dx, \, \forall s \in [a,b) \> \text{ and } \> \int_{a}^{b} \psi_1 \, dx = \int_{a}^{b} \psi_2 \, dx$$ then for any increasing $\phi : [a,b] \rightarrow \mathbb{R}$ holds: $$\int_{a}^{b} \psi_1 \phi \, dx \geq \int_{a}^{b} \psi_2 \phi \, dx .$$

From here one may derive [2] the majority of the classical inequalities: $AM \geq GM$, Muirhead, Schur, Cauchy-Schwartz, Hölder, Minkowski, Jensen, Chebyshev, etc.

Besides this one, in the literature there is another famous inequality [3], named after Hardy and Littlewood - the generalized rearrangement inequality:

Let $\phi^*$ donates the symmetric (decreasing) rearrangement of a measurable, non-negative, asymptotically vanishing function $\phi$ defined over $\mathbb{R}^n$ - roughly speaking, $\phi^* (x)$ is the "height" at which the "size" of the level set $\{ t : \phi(t) > \phi^* (x) \}$ is equal to $x$ (please, excuse me for not going into great depth with the definitions). Then for any two functions $f, g : \mathbb{R}^n \rightarrow \mathbb{R}_{>0}$ of the above type: $$ \int f g \, dx \leq \int f^* g^* \, dx .$$

My question is whether there is any relation between the above inequalities and, more generally, between the concepts of majorization and symmetric rearrangement? I'm sorry if the answer is obvious - I haven't given it much thought...



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