Page 147 - 8th European Congress of Mathematics ∙ 20-26 June 2021 ∙ Portorož, Slovenia ∙ Book of Abstracts
P. 147
RENT TOPICS IN COMPLEX ANALYSIS (MS-32)
B0 ⊂ C, the inequality I20(γ) 1 2−γ
rγ (B0, 0) r (B1, a1) r (B2, a2) 2 |a1 − a2|
is true. The sign of equality in this inequality is attained, when the points a0, a1, a2 and the
domains B0, B1, B2 are, respectively, the poles and circular domains of the quadratic differential
Q(w)dw2 = − (4 − γ)w2 + γ dw2.
w2(w2 − 1)2
Theorem 2. [8] Let n ∈ N, n 3, γ ∈ (1, n]. Then, for any system of different points
{ak}kn=1 of the unit circle and for any collection of mutually non-overlapping domains B0, Bk,
a0 = 0 ∈ B0 ⊂ C, ak ∈ Bk ⊂ C, k = 1, n, the following inequality holds
n π n−γ I20 2γ n
sin n 2
rγ(B0, 0) r(Bk, ak)
n .
k=1
References
[1] Dubinin V.N. Condenser capacities and symmetrization in geometric function theory.
Birkhäuser/Springer, Basel, 2014.
[2] Kuz’mina G.V. The method of extremal metric in extremal decomposition problems with
free parameters // J. Math. Sci., 2005, V. 129, No. 3, pp. 3843–3851.
[3] Kovalev L.V. On the problem of extremal decomposition with free poles on a circle //
Dal’nevostochnyi Mat. Sb., 1996, No. 2, pp. 96–98. (in Russian)
[4] Bakhtin A.K., Bakhtina G.P., Zelinskii Yu.B. Topological-algebraic structures and geo-
metric methods in complex analysis. Zb. prats of the Inst. of Math. of NASU, 2008. (in
Russian)
[5] Bakhtin A.K. Separating transformation and extremal problems on nonoverlapping simply
connected domains. J. Math. Sci., 2018, V. 234, No. 1, pp. 1–13.
[6] Bakhtin A.K., Denega I.V. Weakened problem on extremal decomposition of the complex
plane // Matematychni Studii, 2019, V. 51, No. 1, pp. 35–40.
[7] Denega I. Estimates of the inner radii of non-overlapping domains. J. Math. Sci., 2019, V.
242, No. 6, pp. 787–795.
[8] Bakhtin A.K., Denega I.V. Extremal decomposition of the complex plane with free poles
// J. Math. Sci., 2020, V. 246, No. 1, pp. 1–17.
[9] Bakhtin A.K., Denega I.V. Extremal decomposition of the complex plane with free poles
II // J. Math. Sci., 2020, V. 246, No. 5, pp. 602–616.
145
B0 ⊂ C, the inequality I20(γ) 1 2−γ
rγ (B0, 0) r (B1, a1) r (B2, a2) 2 |a1 − a2|
is true. The sign of equality in this inequality is attained, when the points a0, a1, a2 and the
domains B0, B1, B2 are, respectively, the poles and circular domains of the quadratic differential
Q(w)dw2 = − (4 − γ)w2 + γ dw2.
w2(w2 − 1)2
Theorem 2. [8] Let n ∈ N, n 3, γ ∈ (1, n]. Then, for any system of different points
{ak}kn=1 of the unit circle and for any collection of mutually non-overlapping domains B0, Bk,
a0 = 0 ∈ B0 ⊂ C, ak ∈ Bk ⊂ C, k = 1, n, the following inequality holds
n π n−γ I20 2γ n
sin n 2
rγ(B0, 0) r(Bk, ak)
n .
k=1
References
[1] Dubinin V.N. Condenser capacities and symmetrization in geometric function theory.
Birkhäuser/Springer, Basel, 2014.
[2] Kuz’mina G.V. The method of extremal metric in extremal decomposition problems with
free parameters // J. Math. Sci., 2005, V. 129, No. 3, pp. 3843–3851.
[3] Kovalev L.V. On the problem of extremal decomposition with free poles on a circle //
Dal’nevostochnyi Mat. Sb., 1996, No. 2, pp. 96–98. (in Russian)
[4] Bakhtin A.K., Bakhtina G.P., Zelinskii Yu.B. Topological-algebraic structures and geo-
metric methods in complex analysis. Zb. prats of the Inst. of Math. of NASU, 2008. (in
Russian)
[5] Bakhtin A.K. Separating transformation and extremal problems on nonoverlapping simply
connected domains. J. Math. Sci., 2018, V. 234, No. 1, pp. 1–13.
[6] Bakhtin A.K., Denega I.V. Weakened problem on extremal decomposition of the complex
plane // Matematychni Studii, 2019, V. 51, No. 1, pp. 35–40.
[7] Denega I. Estimates of the inner radii of non-overlapping domains. J. Math. Sci., 2019, V.
242, No. 6, pp. 787–795.
[8] Bakhtin A.K., Denega I.V. Extremal decomposition of the complex plane with free poles
// J. Math. Sci., 2020, V. 246, No. 1, pp. 1–17.
[9] Bakhtin A.K., Denega I.V. Extremal decomposition of the complex plane with free poles
II // J. Math. Sci., 2020, V. 246, No. 5, pp. 602–616.
145
