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Regular version of the site

Article

On means with power-2 for the derivatives of functions of class S

St Petersburg Mathematical Journal. 2017. Vol. 28. No. 6. P. 855-867.

Let F ∈ S, S be the standard class of conformal mapping of the unit disk \mathbb{D}, and let . Suppose that there exist Jordan domains  G_1 and G, G_1 ⊃ G and G_1 ⊃ G such that  G ⊂ \mathbb{C}\f (\mathbb{D}), \partial{f}(\mathbb{D}) ∩ \partial{G},  contains a Dini-smoth arc \gamma, and G_1 ∩ \partial{f}(\mathbb{D})  (D) ∩ \partial{G}=\gamma. It is established that, in this case, for any r, 0 < r < 1  with F,  does not maximize the expression Z |z|=r 1 |F ′(z)| 2 |dz|  in the class S.