On a problem of Avhadiev

In this paper we consider a lower estimate for the ratio $I(\Omega)$ of the conformal moment of a simple connected domain $\Omega$ in the complex plane to the moment of inertia of this domain about its boundary. Related functionals depending on a simple connected domain $\Omega$ and two points $\omega_1,\omega_2\in\Omega$ with fixed hyperbolical distance between them are estimated. As a consequence a nontrivial lower estimate for $I(\Omega)$ is obtained.