On the harmonic measure of continua of a fixed diameter

Let $\mathcal{E}$ be the family of all continua $E$ in $\bar{U}\setminus\{0\}$, where $U=\{|z|<1\}$, let $U(E)$ be the connected component of $U\setminus E$ containing the point $z=0$, $\omega_E(z_0)=\omega(z_0,E,U(E))$ be the harmonic measure of $E$ relative to the domain $U(E)$ at the point $z_0\in U(E)$. In the paper one answers affirmatively a question raised by B. Rodkin [K.F. Barth, D.A. Branna, and W.K. Hayman, "Research problems in complx analysis,’’ Bull. London Math. Soc.,l6, No. 5, 490–517, 1984]. Namely, one proves that in the family $\mathcal{E}(d_0)$ of continua $E\in\mathcal{E}$, satisfying the condition $\operatorname{diam}E=d_0$, $\quad0<d_0\leq2$, one has the inequality
$$ \omega_E(0)\geq\frac1\pi\arcsin d_0/2, $$
one indicates all the cases for which equality prevails.