In this paper we consider a lower estimate for the ratio I(Ω)
of
the conformal moment of a simple connected
domain Ω in the complex
plane to the moment of
inertia of this domain about its boundary.
Related functionals depending on a simple connected domain Ω
and two
points w_{1},w_{2}
Ω with fixed hyperbolical distance between them
are
estimated. As a consequence a nontrivial
lower estimate for I(Ω) is
obtained.

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