A lower bound for restricted exponential sums

Let $M(f;r)=\max_{|z|=r}|f(z)|$. Define $\mathcal{F}_n$, $\mathcal{G}_n$ to be the classes of exponential sums of the form $\sum_{k=1}^n \lambda_k e^{\lambda_k z}$ and $\sum_{k=1}^n e^{\lambda_k z}$, respectively, with $|\lambda_1|=\dots=|\lambda_n|=1$. For $r\in \left(0,\tfrac{1}{2}\right)$, we prove that $\inf_{f\in \mathcal{F}_n} M(f;r)\asymp n r^{n-1}\big/(n-1)!$, and establish the Turán—Govil type bound $\inf_{g\in \mathcal{G}_n} M(g';r)\big/M(g;r)\asymp r^{n-1}\big/(n-1)!$. Approximations of entire functions of exponential type $\sigma\le 1$ on compact sets $K\subset \mathbb{C}$ by sums $f_n\in \mathcal{F}_n$, as well as representations of harmonics of a trigonometric polynomial $T_n(t)$ in the form of sums of its translations, $T_n(t-t_k)$, are also considered. In particular, we obtain a new Fejér type estimate for the leading harmonic $\tau_{2n}(t)$ of nonnegative polynomials $T_{2n}(t)$ of even degree $2n$.