Bohr Inequalities in Some Classes of Analytic Functions
Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Complex analysis, Tome 153 (2018), pp. 69-83.

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

The paper is a review of the latest results of I. R. Kayumov and S. Ponnusami on the Bohr inequality. An exact estimate in the strong Bohr inequality is obtained and the Bohr–Rogosinsky radius for a certain class of subordinations is examined. All results are exact.
Keywords: Bohr inequality, Bohr radius, Rogosinski radius, analytic functions, harmonic maps.
@article{INTO_2018_153_a4,
     author = {A. A. Ismagilov and A. V. Kayumova and I. R. Kayumov and S. Ponnusamy},
     title = {Bohr {Inequalities} in {Some} {Classes} of {Analytic} {Functions}},
     journal = {Itogi nauki i tehniki. Sovremenna\^a matematika i e\"e prilo\v{z}eni\^a. Temati\v{c}eskie obzory},
     pages = {69--83},
     publisher = {mathdoc},
     volume = {153},
     year = {2018},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/INTO_2018_153_a4/}
}
TY  - JOUR
AU  - A. A. Ismagilov
AU  - A. V. Kayumova
AU  - I. R. Kayumov
AU  - S. Ponnusamy
TI  - Bohr Inequalities in Some Classes of Analytic Functions
JO  - Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory
PY  - 2018
SP  - 69
EP  - 83
VL  - 153
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/INTO_2018_153_a4/
LA  - ru
ID  - INTO_2018_153_a4
ER  - 
%0 Journal Article
%A A. A. Ismagilov
%A A. V. Kayumova
%A I. R. Kayumov
%A S. Ponnusamy
%T Bohr Inequalities in Some Classes of Analytic Functions
%J Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory
%D 2018
%P 69-83
%V 153
%I mathdoc
%U http://geodesic.mathdoc.fr/item/INTO_2018_153_a4/
%G ru
%F INTO_2018_153_a4
A. A. Ismagilov; A. V. Kayumova; I. R. Kayumov; S. Ponnusamy. Bohr Inequalities in Some Classes of Analytic Functions. Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Complex analysis, Tome 153 (2018), pp. 69-83. http://geodesic.mathdoc.fr/item/INTO_2018_153_a4/

[1] Goluzin G. M., “O podchinennykh odnolistnykh funktsiyakh”, Tr. MIAN SSSR, v. 38, Izd-vo AN SSSR, M., 1951, 68–71 | MR

[2] Abu-Muhanna Y., “Bohr's phenomenon in subordination and bounded harmonic classes”, Complex Var. Elliptic Equ., 55:11 (2010), 1071–1078 | DOI | MR | Zbl

[3] Abu-Muhanna Y., Ali R. M., “Bohr's phenomenon for analytic functions into the exterior of a compact convex body”, J. Math. Anal. Appl., 379:2 (2011), 512–517 | DOI | MR | Zbl

[4] Abu-Muhanna Y., Ali R. M., Ng Z. C., Hasni S. F. M., “Bohr radius for subordinating families of analytic functions and bounded harmonic mappings”, J. Math. Anal. Appl., 420:1 (2014), 124–136 | DOI | MR | Zbl

[5] Ali R. M., Abu-Muhanna Y., Ponnusamy S., “On the Bohr inequality”, Progress in Approximation Theory and Applicable Complex Analysis, Springer Optimization and Its Applications, v. 117, 2016, 265–295 | MR

[6] Ali R. M., Barnard R. W., Solynin A. Yu., “A note on the Bohr's phenomenon for power series”, J. Math. Anal. Appl., 449:1 (2017), 154–167 | DOI | MR | Zbl

[7] Aizenberg L., “Generalization of results about the Bohr radius for power series”, Stud. Math., 180 (2007), 161–168 | DOI | MR | Zbl

[8] Anderson J. M., Clunie J. G., Pommerenke C., “On Bloch functions and normal functions”, J. Reine Angew. Math., 270 (1974), 12–37 | MR | Zbl

[9] Avkhadiev F. G., Wirths K.-J., Schwarz–Pick type inequalities, Birkhäuser-Verlag, Basel–Boston–Berlin, 2009 | MR | Zbl

[10] Bénéteau C., Dahlner A., Khavinson D., “Remarks on the Bohr phenomenon”, Comput. Methods Funct. Theory, 4:1 (2004), 1–19 | DOI | MR

[11] Boas H. P., “Majorant series”, Korean Math. Soc., 37:2 (2000), 321–337 | MR | Zbl

[12] Boas H. P., Khavinson D., “Bohr's power series theorem in several variables”, Proc. Am. Math. Soc., 125:10 (1997), 2975–2979 | DOI | MR | Zbl

[13] Boas H. P., Khavinson D., “Vita: Friedrich Wilhelm Wiener”, Math. Intelligencer, 22 (2000), 73–75 | DOI | MR | Zbl

[14] Bohr H., “A theorem concerning power series”, Proc. London Math. Soc., 13:2 (1914), 1–5 | DOI | MR | Zbl

[15] Colonna F., “The Bloch constant of bounded harmonic mappings”, Indiana Univ. Math. J., 38 (1989), 829–840 | DOI | MR | Zbl

[16] Duren P. L., Univalent functions, Springer-Verlag, New York, 1983 | MR | Zbl

[17] Chen Sh., Ponnusamy S., Wang X., “Landau theorem and Marden constant for harmonic $\nu$-bloch mappings”, Bull. Austr. Math. Soc., 84 (2011), 19–32 | DOI | MR | Zbl

[18] Chen Sh., Ponnusamy S., Wang X., “Coefficient estimates and Landau–Bloch theorem for planar harmonic mappings”, Bull. Malaysian Math. Sci. Soc., 34:2 (2011), 255–265 | MR | Zbl

[19] Chen Sh., Ponnusamy S., Wang X., “Landau–Bloch constants for functions in $\alpha$-Bloch spaces and Hardy spaces”, Complex Anal. Oper. Theory, 6 (2012), 1025–1036 | DOI | MR | Zbl

[20] Kayumov I. R., Ponnusamy S., “Bohr inequality for odd analytic functions”, Comput. Methods Funct. Theory, 17:4 (2017), 679–688 | DOI | MR | Zbl

[21] Kayumov I. R., Ponnusamy S., Bohr's inequality for analytic functions $\smash[b]{\sum\limits_k} b_k z^{kp+m}$ and harmonic functions, arXiv: 1708.05578 [math.CV] | MR

[22] Kayumov I. R., Ponnusamy S., Bohr–Rogosinski radius for analytic functions, arXiv: 1708.05585 [math.CV]

[23] Kayumov I. R., Ponnusamy S., Shakirov N., “Bohr radius for locally univalent harmonic mappings”, Math. Nachr., 291:11-12 (2018), 1757–1768 | DOI | MR | Zbl

[24] Landau E., Gaier D., Darstellung und Begrüundung einiger neuerer Ergebnisse der Funktionentheorie, Springer-Verlag, Berlin, 1986 | MR

[25] Liu G., Ponnusamy S., On harmonic $\nu$-Bloch and $\nu$-Bloch-type mappings, arXiv: 1707.01570 [math.CV] | MR

[26] Pommerenke C., Boundary behaviour of conformal maps, Springer, New York, 1992 | MR | Zbl

[27] Popescu G., “Multivariable Bohr inequalities”, Trans. Am. Math. Soc., 359:11 (2007), 5283–5317 | DOI | MR | Zbl

[28] Rogosinski W., “Über Bildschranken bei Potenzreihen und ihren Abschnitten”, Math. Z., 17 (1923), 260–276 | DOI | MR | Zbl

[29] Schur I., Szegő G., “Über die Abschnitte einer im Einheitskreise beschränkten Potenzreihe”, Sitz.-Ber. Preuss. Acad. Wiss. Berlin Phys.-Math. Kl., 1925, 545–560 | MR | Zbl