Optimal rate of integration and $\varepsilon$-entropy of a class of analytic functions

The author considers a class $F$ of analytic functions real in the interval $[-1,1]$ and bounded in the unit circle. As an estimate of the optimal quadrature error $R(n)$ over the class $F$ it is shown that
$$ e^{\left(-2\sqrt2+\frac1{\sqrt2}\right)\pi\sqrt n}\le R(n)\le e{-\frac\pi{\sqrt2}n}. $$
With the additional condition that $\max\limits_{x\in[-1,1]}|f(x)|\le B$, an estimate is obtained for the $\varepsilon$-entropy $H_\varepsilon(F)$:
$$ \frac8{27}\frac{(\ln2)^2}{\pi^2}\le\lim\frac{H_\varepsilon(F)}{(\log\frac1\varepsilon)^3}\le\frac2{\pi^2}(\ln2)^2. $$