  We introduce a transport-majorization argument that establishes a
majorization in the convex order between two densities, based on control of the
gradient of a transportation map between them. As applications, we give
elementary derivations of some delicate Fourier analytic inequalities, which in
turn yield geometric "slicing-inequalities" in both continuous and discrete
settings. As a further consequence of our investigation we prove that any
strongly log-concave probability density majorizes the Gaussian density and
thus the Gaussian density maximizes the R\'enyi and Tsallis entropies of all
orders among all strongly log-concave densities.
