Application of upper estimates for products of inner radii to distortion theorems for univalent functions

  • I. V. Denega Institute of Mathematics NAS of Ukraine Department of complex analysis and potential theory Kyiv, Ukraine
  • Ya. V. Zabolotnyi Institute of Mathematics NAS of Ukraine Department of complex analysis and potential theory Kyiv, Ukraine

Анотація

In 1934 Lavrentiev solved the problem of maximum of
product of conformal radii of two non-overlapping simply connected domains. In the case of three or more points, many authors considered estimates of a more general Mobius invariant of the form
$$
T_{n}:={\prod\limits_{k=1}^n
r(B_{k},a_{k})}{\bigg(\prod\limits_{1\leqslant k<p\leqslant n}
|a_{k}-a_{p}|\bigg)^{-\frac{2}{n-1}}},
$$
where $r(B,a)$ denotes the inner radius of the domain $B$ with respect to the point $a$ (for an infinitely distant point under the corresponding factor we understand the unit).
In 1951 Goluzin for $n=3$ obtained an accurate evaluation for $T_{3}$.
In 1980 Kuzmina showed
that the problem of the evaluation of $T_{4}$ is
reduced to the smallest capacity problem in the certain continuum
family and obtained the exact inequality for $T_{4}$.
No other ultimate results in this problem for $n \geqslant 5$ are known at present.
In 2021 \cite{Bakhtin2021,BahDen22} effective upper estimates are obtained for $T_{n}$, $n \geqslant 2$.
Among the possible applications of the obtained results in other tasks of the function theory are the so-called distortion theorems.
In the paper we consider an application of upper estimates for products of inner radii to distortion theorems for univalent functions
in disk $U$, which map it onto a star-shaped domains relative to the origin.

Біографії авторів

I. V. Denega, Institute of Mathematics NAS of Ukraine Department of complex analysis and potential theory Kyiv, Ukraine

Institute of Mathematics NAS of Ukraine
Department of complex analysis and potential theory
Kyiv, Ukraine

Ya. V. Zabolotnyi, Institute of Mathematics NAS of Ukraine Department of complex analysis and potential theory Kyiv, Ukraine

Institute of Mathematics NAS of Ukraine
Department of complex analysis and potential theory
Kyiv, Ukraine

Посилання

M.A. Lavrentiev, On the theory of conformal mappings, Tr. Sci. Inst. An. USSR, 5 (1934), 159–245.

G.M. Goluzin, Geometric theory of functions of a complex variable, Amer. Math. Soc. Providence, R.I., 1969.

G.V. Kuz’mina, On the problem of the maximum of the conform radien’s product of non-overlapping domains, J. Soviet Math., 19 (1982), №6, 1715–1726.

A.K. Bakhtin, Y.V. Zabolotnii, Estimates of the products of inner radii for multiconnected domains, Ukr. Mat. Zhurnal, 73 (2021), №1, 9–22.

T.H. Gronwall, Some remarks on conformal representation, Annals Math., 16 (1914–1915), №1, 72–76.

L. Bieberbach, Uber die Koeffizienten derjenigen Potenzreihen, welche eine schlichte Abbildung des Einheitskreises vermitteln, S. B. Preuss. Acad. Wiss., 138 (1916), 940–955.

O. Roth, A distortion theorem for bounded univalent functions, Ann. Acad. Sci. Fenn. Math., 27 (2002), 257–272.

V.N. Dubinin, V.Y. Kim, Distortion theorems for functions regular and bounded in the disk, J. Math. Sci., 150 (2008), 2018–2026.

V.N. Dubinin, Distortion theorem for complex polynomials, Sib. Elektron. Mat. Izv., 15 (2018), 1410–1415.

Seong-A. Kim, D. Minda, Two-point distortion theorems for univalent functions, Pacific J. Math., 163 (1994), №1, 137–157.

J.A. Jenkins, On two-point distortion theorems for bounded univalent regular functions, Kodai. Math. J., 24 (2001), №3, 329–338.

S.I. Kalmykov, About multipoint distortion theorems for rational functions, Siberian Math. J., 61 (2020), №1, 85–94.

W.K. Hayman, Multivalent functions, Cambridge Univ. Press, 1994.

A.K. Bakhtin, G.P. Bakhtina, Yu.B. Zelinskii, Topological-algebraic structures and geometric methods in complex analysis, Zb. prats of the Inst. of Math. of NASU, 2008.

A.K. Bakhtin, I.V. Denega, Generalized M.A. Lavrentiev’s inequality, J. Math. Sci., 262 (2022), №2, 138–153.

A.K. Bakhtin, Y.V. Zabolotnii, Estimates of products of some powers of inner radii for multiconnected domains, Ukr. Mat. Zhurn., 73 (2021), №9, 1155–1169.

E.G. Emelyanov, On the connection of two problems on extremal partitioning, Zap. Nauchn. Sem. LOMI. 160 (1987), 91–98.

Опубліковано
2023-12-18
Як цитувати
Denega, I. V., & Zabolotnyi, Y. V. (2023). Application of upper estimates for products of inner radii to distortion theorems for univalent functions. Математичні студії, 60(2), 138-144. https://doi.org/10.30970/ms.60.2.138-144
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