Background: In Electrodynamics, the scalar permittivity $\epsilon(\omega)$ relates the Electric displacement field $\vec D$ to the electric field $\vec E$ as $\vec D=\epsilon\vec E$ when assuming a linear, isotropic medium. Causality requires that in time-space the Fourier transform $\tilde\epsilon(t)$ must vanish for $t<0$ (i.e. the future cannot influence $\vec D$), resulting in $\epsilon(\omega)$ being analytic and that the Kramers-Kronig relation (basically using the Hilbert transform) can be used to relate the real and imaginary parts of $\epsilon$. The imaginary part describes absorption and must thus not be negative1, does this put any additional restraints on $\Re \epsilon$?
So in summary:
For $\epsilon(\omega)$ analytical (because the Fourier transform $\tilde\epsilon(t)$ vanishes for $t<0$) and $\forall\omega>0:\Im\epsilon(\omega)\ge0$, what are properties of $\Re\epsilon(\omega)$ for $\omega>0$?
1) or positive, depending on convention as in whether the time dependence is $e^{+i\omega t}$ or $e^{-i\omega t}$
edit also, since $\vec E$ and $\vec D$ are real valued in time-space, $\epsilon(-\omega)=\epsilon(\omega)^*$