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 $w_1,w_2\in\Omega$ with fixed hyperbolical distance between them are estimated. As a consequence a nontrivial lower estimate for $I(\Omega)$ is obtained.
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