Monotonicity of functionals associated to product measures via their Fourier transform and applications

Let $\mu$ be a probability measure on $\mathbb{R}$. We give conditions on the Fourier transform of its density for functionals of the form $H(a)={\int }_{{\mathbb{R}}^n}h(⟨a,x⟩){\mu }^n(dx)$ to be Schur monotone. As applications, we put certain known and new results under the same umbrella, given by a condition on the Fourier transform of the density. These results include certain moment comparisons for independent and identically distributed random vectors, when the norm is given by intersection bodies, and the corresponding vector Khinchin inequalities. We also extend the discussion to higher dimensions.