On extremal functions in inequalities for entire functions

Let $B_{\sigma}$, $\sigma>0$, be the class of entire functions of exponential type $\leqslant\sigma$ bounded on the real line. For a number $\tau\in\mathbb{R}$ and a sequence $\{c_k\}_{k\in\mathbb{Z}}$ of complex numbers satisfying the condition $\sum_{k\in\mathbb{Z}}|c_k|<+\infty$, the operator $H$ on $B_{\sigma}$ defined by
$$ H(f)(x)=\sum_{k\in\mathbb{Z}}c_k f\biggl(x-\tau+\frac{k\pi}{\sigma}\biggr) $$
is considered. Obviously,
$$ |H(f)(x)|\leqslant \varkappa \|f\|_{\infty}, \qquad x\in\mathbb{R}, \quad f\in B_{\sigma}, \quad \varkappa=\sum_{k\in\mathbb{Z}} |c_k|. $$
The main purpose of the paper is to describe all extremal functions for this inequality. Theorem 1 proved in the paper asserts that if (1) $\overline{c_{s}}c_{s+1}<0$ for some $s\in\mathbb{Z}$ and (2) there exists an $\varepsilon\in\mathbb{C}$ with $|\varepsilon|=1$ such that $\varepsilon c_k (-1)^k\geqslant 0$ for all $k\in\mathbb{Z}$, then the set of all extremal functions for the above inequality coincides with the set of functions of the form $f(t)=\mu e^{i\sigma t}+\nu e^{-i\sigma t}$, $\mu,\nu\in\mathbb{C}$. The proof of Theorem 1 essentially uses Theorem 2, which says that if $f\in B_{\sigma}$ and there exists a point $\xi\in\mathbb{R}$ for which $|f(\xi)|=\|f\|_{\infty}$ and $f(\xi+\pi/\sigma)=-f(\xi)$, then $f(t)=\mu e^{i\sigma t}+\nu e^{-i\sigma t}$, $\mu,\nu\in\mathbb{C}$. Theorem 3 gives general examples of operators satisfying both conditions of Theorem 1. In particular, such is the fractional derivative operator $H(f)(x)=f^{(r,\beta)}(x)$ for $r\geqslant 1$ and $\beta\in\mathbb{R}$.