$L_p$-versions of one conformally invariant inequality

We obtain $L_p$ versions of theorems proved by J. L. Fernández, J. M. Rodríguez in The exponent of convergence of Riemann surfaces, bass Riemann surfaces, Ann. Acad. Sci. Fenn. Ser. A. I. Mathematica 15, 165–182 (1990).
An important role in the proof of our results is approach of V. M. Miklyukov and M. K. Vuorinen. We use the isoperimetric profile of hyperbolic domains.