Inequalities for exponential sums

We study the classes
\begin{gather*} {\mathscr E}_n:= \biggl\{f\colon f(t)=\sum_{j=1}^n{a_j e^{\lambda_jt}}, \ a_j, \lambda_j\in {\mathbb C} \biggr\}, \\ {\mathscr E}_n^+:= \biggl\{f\colon f(t)=\sum_{j=1}^n{a_j e^{\lambda_jt}}, \ a_j, \lambda_j\in {\mathbb C}, \ \operatorname{Re}(\lambda_j) \geqslant 0 \biggr\}, \\ {\mathscr E}_n^-:= \biggl\{f\colon f(t)=\sum_{j=1}^n{a_j e^{\lambda_jt}}, \ a_j, \lambda_j\in {\mathbb C}, \ \operatorname{Re}(\lambda_j)\leqslant 0 \biggr\}, \end{gather*}
and
$$ {\mathscr T}_n:= \biggl\{f\colon f(t)=\sum_{j=1}^n{a_j e^{i\lambda_jt}}, \ a_j\in {\mathbb C}, \ \lambda_1<\lambda_2<\dots<\lambda_n \biggr\}. $$
A highlight of this paper is the asymptotically sharp inequality
$$ |f(0)|\leqslant (1+\varepsilon_n)3n\|f(t)e^{-9nt/2}\|_{L_2[0,1]}, \qquad f\in {\mathscr T}_n , $$
where $\varepsilon_n$ converges to $0$ rapidly as $n$ tends to $\infty$. The inequality
$$ \sup_{0 \not \equiv f\in {\mathscr T}_n}{ \frac{|f(0)|}{\|f\|_{L_2{[0,1]}}}} \geqslant n $$
is also established. Our results improve an old result due to Halász and a recent result due to Kós. We prove several other related order-sharp results in this paper.
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