On a problem of Avhadiev (Q1888633)

Let \(\Omega\) be a simply connected domain on \(\mathbb{C}\). Let \(\rho_\Omega(w)\) be the conformal radius of \(\Omega\) at \(w\in \Omega\), and let \(d_\Omega(w)\) be the Euclidean distance from \(w\) to the boundary \(\partial\Omega\) of the domain \(\Omega\). Let \[ I_c (\Omega)=\int_\Omega \rho^2_\Omega(x+iy)\,dx\,dy \] be the conformal moment of \(\Omega\), \[ I(\partial\Omega)= \int_\Omega d^2_\Omega (x+iy) \,dx\,dy \] be the moment of interia of \(\Omega\) about \(\partial \Omega\), and let us put \[ I(\Omega)= \frac{I_c(\Omega} {I(\partial \Omega)}. \] In this paper the author gives a better lower estimate for \(I(\Omega)\) then one given by Avhadiev. It is also proved then the function \(f\) minimizing the functional \[ G(f,\alpha)= \frac{\bigl| f'(0)\bigr|^\alpha+\bigl| f^l(r)\bigr|^\alpha(1-r^2)^\alpha} {f^\alpha_\Omega\bigl(f(0)\bigr) +d^\alpha_\Omega\bigl(f(r)\bigr)} \] on the class \(S^0\) of univalent functions on \(U=\{z:|z|<1\}\) maps \(U\) onto an arc biangle bounded domain, bounded by circle arcs centered at the points \(f(0)\) and \(f(r)\) respectively.