M. Riesz-Schur-type inequalities for entire functions of exponential type

We prove a general M. Riesz-Schur-type inequality for entire functions of exponential type. If $f$ and $Q$ are two functions of exponential types $\sigma > 0$ and $\tau \geqslant 0$, respectively, and if $Q$ is real-valued and the real zeros of $Q$, not counting multiplicities, are bounded away from each other, then
$$ |f(x)|\le (\sigma+\tau) (A_{\sigma+\tau}(Q))^{-1/2}\|Q f\|_{\mathrm C(\mathbb R)},\qquad x\in \mathbb R, $$
where
$$ A_s(Q) \stackrel{\mathrm{def}}{=}\inf_{x\in\mathbb R} \bigl([Q'(x)]^2+s^2 [Q(x)]^2\bigr). $$
We apply this inequality to the weights $Q(x)\stackrel{\mathrm{def}}{=} \sin (\tau x)$ and $Q(x) \stackrel{\mathrm{def}}{=} x$ and describe the extremal functions in the corresponding inequalities.
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