An exact order of the majorant growth in the Schwarz - Pick inequality for torsional rigidity

The beginning of Schwarz - Pick type inequalities may be found in classical papers of Pick [14], Caratheodory [13], Szasz [19], Bernstein [12] and others. In recent years this program is actively developed, a number of results on inequalities of this type can be found in articles of Ruscheweyh [16; 17], Yamashita [20], Avkhadiev [7-10] etc. (see also [2-4]). These results are concerned with function holomorphic or meromorphic in a domain Ω in the extended complex plane C and 𝑓(Ω) ⊂ Π ⊂ C. In [6] we obtained Schwarz - Pick type inequalities for the torsional rigidity. As known, the Saint-Venant functional for the torsional rigidity in an arbitrary plane Ω can be found as the solution of the generalized problem (see [1; 11; 15]) 𝑃(Ω) = sup 𝑢∈𝐶∞0 (Ω) (︀2 rΩ 𝑢(𝑥)𝑑𝑥)︀2 rΩ |∇𝑢|2𝑑𝑥𝑑𝑦, where (𝑥, 𝑦) ∈ Ω, 𝐶∞0 (Ω) - the space of smooth functions with compact support in Ω. Let Ω ∈ C arbitrary simply connected domain and 0 ∈ C. According toRiemann’s theorem there exists a function such that : Δ → Ω and 𝑓(0) = 0. Let Ω𝑟 the image of the circle Δ𝑟 = { ∈ C : | | 1