  We prove that there is a constant $c > 0$ depending only on $M \geq 1$ and
$\mu \geq 0$ such that $$\int_y^{y+a}{|g(t)| \, dt} \geq \exp (-c/(a\delta))\,,
a \in (0,\pi]\,,$$ for every $g$ of the form $$g(t) = \sum_{j=0}^n{a_j
e^{i\lambda_jt}}, a_j \in {\Bbb C}, \enskip |a_j| \leq Mj^\mu\,, \enskip
|a_0|=1\,, \enskip n \in {\Bbb N} \,,$$ where the exponents $\lambda_j \in
{\Bbb C}$ satisfy $$\text{\rm Re}(\lambda_0) = 0\,, \qquad \text{\rm
Re}(\lambda_j) \geq j\delta > 0\,, j=1,2,\ldots\,,$$ and for every subinterval
$[y,y+a]$ of the real line. Establishing inequalities of this variety is
motivated by problems in physics.
