I am having some trouble proving an inequality involving the trace norm or the operator $f{\cal{F}}_Ng$ where $f, g$ are diagonal matrices, f is positive semidefinite and $\cal{F}_N$ is the Discrete Fourier transform matrix given by $$ ({\cal{F}_N})_{n,m}= \frac{1}{\sqrt{N}}e^{-\frac{2\pi i}{N}nm} $$ a unitary Vandermonde matrix. The inequality I am trying to show is this: $$ \|f{\cal{F}_N} g{\cal{F}_{N}}^{\ast}\|_{tr}\leq \frac{1}{\sqrt{N}} \|f\|_{tr}\|g\|_{tr} $$ where $(\cdot)^{\ast}$ denotes the Hermitian adjoint.
Has anybody got an idea? Thanks very much for your help.