I hope, the title is not too confusing. My question is the following: We all know the Riemann-Lebesgue-Lemma stating that for $f\in L^1(\mathbb R)$, one has $$ \lim_{k\to\infty} \int f(x)\,e^{ikx}\,dx=0. $$ Intuitively this means that the fast oscillation of $e^{ikx}$ makes the integral small.
Question: Can one show something like $$\int f(x)\,e^{i\phi(x)}\,dx \leq \int f(x)\,e^{i\psi(x)}\,dx, $$ if $\phi>\psi$?
(or maybe, if $\phi-\psi\geq\text{(some constant)}$, or so)
Thanks,
Frank
If it was true, the Fourier transform of an even $L^1(\mathbb{R})$ function would be always increasing on $\mathbb{N}$. This is clearly false, just by considering $f(x)=e^{-x^2}$. Notice that neither the opposite inequality holds, since nothing forces the Fourier transform to be monotonic someway.