Geodesic
Estimates for the
Izvestiya. Mathematics , Tome 86 (2022) no. 5, pp. 839-851.

Voir la notice de l'article provenant de la source Math-Net.Ru

We obtain estimates for the integrals of derivatives of rational functions in multiply connected domains in the plane. A sharp order of growth is found for the integral of the modulus of the derivative of a finite Blaschke product in the unit disc. We also extend the results of Dolzhenko about the integrals of the derivatives of rational functions to a wider class of domains, namely, to domains bounded by rectifiable curves without zero interior angles, and show the sharpness of the results obtained.
Keywords: rational function, Blaschke product, Hardy space
Mots-clés : conformal map, John domain. 
@article{IM2_2022_86_5_a1,
     author = {A. D. Baranov and I. R. Kayumov},
     title = {Estimates for the},
     journal = {Izvestiya. Mathematics },
     pages = {839--851},
     publisher = {mathdoc},
     volume = {86},
     number = {5},
     year = {2022},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/IM2_2022_86_5_a1/}
}
TY  - JOUR
AU  - A. D. Baranov
AU  - I. R. Kayumov
TI  - Estimates for the
JO  - Izvestiya. Mathematics 
PY  - 2022
SP  - 839
EP  - 851
VL  - 86
IS  - 5
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/IM2_2022_86_5_a1/
LA  - en
ID  - IM2_2022_86_5_a1
ER  - 
%0 Journal Article
%A A. D. Baranov
%A I. R. Kayumov
%T Estimates for the
%J Izvestiya. Mathematics 
%D 2022
%P 839-851
%V 86
%N 5
%I mathdoc
%U http://geodesic.mathdoc.fr/item/IM2_2022_86_5_a1/
%G en
%F IM2_2022_86_5_a1
A. D. Baranov; I. R. Kayumov. Estimates for the. Izvestiya. Mathematics , Tome 86 (2022) no. 5, pp. 839-851. http://geodesic.mathdoc.fr/item/IM2_2022_86_5_a1/

[1] S. N. Mergelyan, “On an integral related to analytic functions”, Izv. Akad. Nauk SSSR Ser. Mat., 15:5 (1951), 395–400 (Russian) | MR | Zbl

[2] W. Rudin, “The radial variation of analytic functions”, Duke Math. J., 22:2 (1955), 235–242 | DOI | MR | Zbl

[3] G. Piranian, “Bounded functions with large circular variation”, Proc. Amer. Math. Soc., 19:6 (1968), 1255–1257 | DOI | MR | Zbl

[4] N. G. Makarov, “Probability methods in the theory of conformal mappings”, Algebra i Analiz, 1:1 (1989), 3–59 ; English transl. Leningrad Math. J., 1:1 (1990), 1–56 | MR | Zbl

[5] R. Banuelos and C. N. Moore, “Mean growth of Bloch functions and Makarov's law of the iterated logarithm”, Proc. Amer. Math. Soc., 112:3 (1991), 851–854 | DOI | MR | Zbl

[6] A. Aleman and D. Vukotić, “On Blaschke products with derivatives in Bergman spaces with normal weights”, J. Math. Anal. Appl., 361:2 (2010), 492–505 | DOI | MR | Zbl

[7] D. Protas, “Blaschke products with derivative in function spaces”, Kodai Math. J., 34:1 (2011), 124–131 | DOI | MR | Zbl

[8] D. Protas, “Derivatives of Blaschke products and model space functions”, Canad. Math. Bull., 63:4 (2020), 716–725 | DOI | MR | Zbl

[9] E. P. Dolženko, “Rational approximations and boundary properties of analytic functions”, Mat. Sb., 69(111):4 (1966), 497–524 ; English transl. Amer. Math. Soc. Transl. Ser. 2, 74, Amer. Math. Soc., Providence, RI, 1968, 61–90 | MR | Zbl | DOI

[10] V. V. Peller, “Hankel operators of class $\mathfrak S_p$ and their applications (rational approximation, Gaussian processes, the problem of majorizing operators)”, Mat. Sb., 113(155):4(12) (1980), 538–581 ; English transl. Math. USSR-Sb., 41:4 (1982), 443–479 | MR | Zbl | DOI

[11] S. Semmes, “Trace ideal criteria for Hankel operators, and applications to Besov spaces”, Integral Equations Operator Theory, 7:2 (1984), 241–281 | DOI | MR | Zbl

[12] A. A. Pekarskii, “Inequalities of Bernstein type for derivatives of rational functions, and inverse theorems of rational approximation”, Mat. Sb., 124(166):4(8) (1984), 571–588 ; English transl. Sb. Math., 52:2 (1985), 557–574 | MR | Zbl | DOI

[13] A. A. Pekarskii, “New proof of the Semmes inequality for the derivative of the rational function”, Mat. Zametki, 72:2 (2002), 258–264 ; English transl. Math. Notes, 72:2 (2002), 230–236 | DOI | MR | Zbl | DOI

[14] V. I. Dančenko, “An integral estimate for the derivative of a rational function”, Izv. Akad. Nauk SSSR Ser. Mat,, 43:2 (1979), 277–293 ; English transl. Izv. Math., 14:2 (1980), 257–273 | MR | Zbl | DOI

[15] V. I. Danchenko, “Several integral estimates of the derivatives of rational functions on sets of finite density”, Mat. Sb., 187:10 (1996), 33–52 ; English transl. Sb. Math., 187:10 (1996), 1443–1463 | DOI | MR | Zbl | DOI

[16] E. Dyn'kin, “Inequalities for rational functions”, J. Approx. Theory, 91:3 (1997), 349–367 | DOI | MR | Zbl

[17] E. Dyn'kin, “Rational functions in Bergman spaces”, Complex analysis, operators, and related topics, Oper. Theory Adv. Appl., 113, Birkhäuser, Basel, 2000, 77–94 | DOI | MR | Zbl

[18] A. Baranov and R. Zarouf, “A Bernstein-type inequality for rational functions in weighted Bergman spaces”, Bull. Sci. Math., 137:4 (2013), 541–556 | DOI | MR | Zbl

[19] A. Baranov and R. Zarouf, “The differentiation operator from model spaces to Bergman spaces and Peller type inequalities”, J. Anal. Math., 137:1 (2019), 189–209 | DOI | MR | Zbl

[20] A. Baranov and R. Zarouf, “$H^\infty$ interpolation and embedding theorems for rational functions”, Integral Equations Operator Theory, 91:3 (2019), 18 | DOI | MR | Zbl

[21] O. Martio and J. Sarvas, “Injectivity theorems in plane and space”, Ann. Acad. Sci. Fenn. Ser. A I Math., 4:2 (1979), 383–401 | DOI | MR | Zbl

[22] Ch. Pommerenke, Boundary behaviour of conformal maps, Grundlehren Math. Wiss., 299, Springer-Verlag, Berlin, 1992 | DOI | MR | Zbl