Bohr Type Inequalities for Functions with a Multiple Zero at the Origin
Recently, there has been a number of good deal of research on the Bohr’s phenomenon in various settings including a refined formulation of his classical version of the inequality. Among them, in Paulsen et al. (Proc Lond Math Soc 85(2):493–512, 2002) the authors considered cases in which the above f...
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| Abstract | Recently, there has been a number of good deal of research on the Bohr’s phenomenon in various settings including a refined formulation of his classical version of the inequality. Among them, in Paulsen et al. (Proc Lond Math Soc 85(2):493–512, 2002) the authors considered cases in which the above functions have a multiple zero at the origin. In this article, we present a refined version of Bohr’s inequality for these cases and give a partial answer to a question from Paulsen et al. (2002) for the revised setting. |
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| AbstractList | Recently, there has been a number of good deal of research on the Bohr’s phenomenon in various settings including a refined formulation of his classical version of the inequality. Among them, in Paulsen et al. (Proc Lond Math Soc 85(2):493–512, 2002) the authors considered cases in which the above functions have a multiple zero at the origin. In this article, we present a refined version of Bohr’s inequality for these cases and give a partial answer to a question from Paulsen et al. (2002) for the revised setting. |
| Author | Wirths, K.-J. Ponnusamy, S. |
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| Cites_doi | 10.1090/proc/14634 10.1155/S1073792804143444 10.1090/S0002-9939-04-07553-7 10.1007/978-3-319-78247-8 10.5186/aasfm.2019.4416 10.1016/j.indag.2018.09.008 10.1090/S0002-9939-97-04270-6 10.1007/BF03321051 10.1016/j.jmaa.2018.05.038 10.1016/j.crma.2018.01.010 10.1016/j.jmaa.2018.01.035 10.1090/crmp/051/12 10.7146/math.scand.a-10653 10.1007/s40315-017-0206-2 10.1112/plms/s2-13.1.1 10.1112/S0024611502013692 10.1007/BF01475487 |
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| Keywords | Schwarz lemma Bohr’s inequality Cauchy–Schwarz inequality 30B10 Primary 30A10 30C55 41A58 Analytic functions Secondary 30C45 Multiple zero |
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| References | PaulsenVIPopescuGSinghDOn Bohr’s inequalityProc. Lond. Math. Soc.2002852493512191205910.1112/S0024611502013692 BhowmikBDasNBohr phenomenon for subordinating families of certain univalent functionsJ. Math. Anal. Appl.2018462210871098377428010.1016/j.jmaa.2018.01.035 BohrHA theorem concerning power seriesProc. Lond. Math. Soc.191413215157749410.1112/plms/s2-13.1.1 Kayumov, I.R., Ponnusamy, S.: On a powered Bohr inequality. Ann. Acad. Sci. Fenn. Ser. A I Math. 44, 301–310 (2019) BoasHPKhavinsonDBohr’s power series theorem in several variablesProc. Am. Math. Soc.19971251029752979144337110.1090/S0002-9939-97-04270-6 Bombieri, E.: Sopra un teorema di H. Bohr e G. Ricci sulle funzioni maggioranti delle serie di potenze. Boll. Un. Mat. Ital. 17(3), 276–282 (1962) KayumovIRPonnusamySBohr inequality for odd analytic functionsComput. Methods Funct. Theory201717679688371252610.1007/s40315-017-0206-2 Tomić, M.: Sur un théorème de H. Bohr. Math. Scand. 11, 103–106 (1962) SidonSÜber einen Satz von Herrn BohrMath. Z.1927261731732154488810.1007/BF01475487 PaulsenVISinghDBohr’s inequality for uniform algebrasProc. Am. Math. Soc.20041321235773579208407910.1090/S0002-9939-04-07553-7 GarciaSRMashreghiJRossWTFinite Blaschke Products and Their Connections2018ChamSpringer10.1007/978-3-319-78247-8 Ponnusamy, S., Vijayakumar, R., Wirths, K.-J.: New inequalities for the coefficients of unimodular bounded functions. Results Math., to appear RogosinskiWOn the coefficients of subordinate functionsProc. Lond. Math. Soc.1943482488286250028.35502 Abu Muhanna, Y., Ali, R.M., Ponnusamy, S.: On the Bohr inequality. In: Govil N.K. et al. (eds.) Progress in Approximation Theory and Applicable Complex Analysis? Springer Optimization and Its Applications, vol. 117, pp. 265–295 (2016) Fournier, R., Ruscheweyh, S.: On the Bohr radius for simply connected plane domains. Hilbert spaces of analytic functions, CRM Proc. Lecture Notes, vol. 51, pp. 165–171. Amer. Math. Soc., Providence (2010) BénéteauCDahlnerAKhavinsonDRemarks on the Bohr phenomenonComput. Methods Funct. Theory200441119208166110.1007/BF03321051 Ricci, G.: Complementi a un teorema di H. Bohr riguardante le serie di potenze. Rev. Un.Mat. Argentina 17, 185–195 (1955/1956) Evdoridis, S., Ponnusamy, S., Rasila, A.: Improved Bohr’s inequality for mappings defined on simply connected domains (in preparation) Landau, E., Gaier, D.: Darstellung und Begründung einiger neuerer Ergebnisse der Funktionentheorie. KayumovIRPonnusamySImproved version of Bohr’s inequalityComptes Rendus Mathematique20183563272277376759510.1016/j.crma.2018.01.010 KayumovIRPonnusamySBohr’s inequalities for the analytic functions with lacunary series and harmonic functionsJ. Math. Anal. Appl.2018465857871380933410.1016/j.jmaa.2018.05.038 BombieriEBourgainJA remark on Bohr’s inequalityInt. Math. Res. Not.20048043074330212662710.1155/S1073792804143444 Carlson, F.: Sur les coefficients d’une fonction bornée dans le cercle unité (French). Ark. Mat. Astr. Fys. 27A(1), 8 (1940) AlkhaleefahSAKayumovIRPonnusamySOn the Bohr inequality with a fixed zero coefficientProc. Am. Math. Soc.20191471252635274402108610.1090/proc/14634 Evdoridis, S., Ponnusamy, S., Rasila, A.: Improved Bohr’s inequality for locally univalent harmonic mappings. Indag. Math. (N.S.) 30, 201–213 (2019) 330_CR21 330_CR22 VI Paulsen (330_CR19) 2002; 85 330_CR1 VI Paulsen (330_CR20) 2004; 132 330_CR7 330_CR9 330_CR17 IR Kayumov (330_CR15) 2018; 356 330_CR18 330_CR10 330_CR11 W Rogosinski (330_CR23) 1943; 48 330_CR12 IR Kayumov (330_CR16) 2018; 465 SA Alkhaleefah (330_CR2) 2019; 147 C Bénéteau (330_CR3) 2004; 4 HP Boas (330_CR5) 1997; 125 H Bohr (330_CR6) 1914; 13 IR Kayumov (330_CR14) 2017; 17 B Bhowmik (330_CR4) 2018; 462 E Bombieri (330_CR8) 2004; 80 S Sidon (330_CR24) 1927; 26 330_CR25 SR Garcia (330_CR13) 2018 |
| References_xml | – reference: Tomić, M.: Sur un théorème de H. Bohr. Math. Scand. 11, 103–106 (1962) – reference: BohrHA theorem concerning power seriesProc. Lond. Math. Soc.191413215157749410.1112/plms/s2-13.1.1 – reference: Abu Muhanna, Y., Ali, R.M., Ponnusamy, S.: On the Bohr inequality. In: Govil N.K. et al. (eds.) Progress in Approximation Theory and Applicable Complex Analysis? Springer Optimization and Its Applications, vol. 117, pp. 265–295 (2016) – reference: AlkhaleefahSAKayumovIRPonnusamySOn the Bohr inequality with a fixed zero coefficientProc. Am. Math. Soc.20191471252635274402108610.1090/proc/14634 – reference: Carlson, F.: Sur les coefficients d’une fonction bornée dans le cercle unité (French). Ark. Mat. Astr. Fys. 27A(1), 8 (1940) – reference: KayumovIRPonnusamySBohr inequality for odd analytic functionsComput. Methods Funct. Theory201717679688371252610.1007/s40315-017-0206-2 – reference: PaulsenVISinghDBohr’s inequality for uniform algebrasProc. Am. Math. Soc.20041321235773579208407910.1090/S0002-9939-04-07553-7 – reference: Fournier, R., Ruscheweyh, S.: On the Bohr radius for simply connected plane domains. Hilbert spaces of analytic functions, CRM Proc. Lecture Notes, vol. 51, pp. 165–171. Amer. Math. Soc., Providence (2010) – reference: BhowmikBDasNBohr phenomenon for subordinating families of certain univalent functionsJ. Math. Anal. Appl.2018462210871098377428010.1016/j.jmaa.2018.01.035 – reference: Evdoridis, S., Ponnusamy, S., Rasila, A.: Improved Bohr’s inequality for locally univalent harmonic mappings. Indag. Math. (N.S.) 30, 201–213 (2019) – reference: RogosinskiWOn the coefficients of subordinate functionsProc. Lond. Math. Soc.1943482488286250028.35502 – reference: Bombieri, E.: Sopra un teorema di H. Bohr e G. Ricci sulle funzioni maggioranti delle serie di potenze. Boll. Un. Mat. Ital. 17(3), 276–282 (1962) – reference: KayumovIRPonnusamySBohr’s inequalities for the analytic functions with lacunary series and harmonic functionsJ. Math. Anal. Appl.2018465857871380933410.1016/j.jmaa.2018.05.038 – reference: Landau, E., Gaier, D.: Darstellung und Begründung einiger neuerer Ergebnisse der Funktionentheorie. – reference: SidonSÜber einen Satz von Herrn BohrMath. Z.1927261731732154488810.1007/BF01475487 – reference: GarciaSRMashreghiJRossWTFinite Blaschke Products and Their Connections2018ChamSpringer10.1007/978-3-319-78247-8 – reference: BombieriEBourgainJA remark on Bohr’s inequalityInt. Math. Res. Not.20048043074330212662710.1155/S1073792804143444 – reference: BénéteauCDahlnerAKhavinsonDRemarks on the Bohr phenomenonComput. Methods Funct. Theory200441119208166110.1007/BF03321051 – reference: KayumovIRPonnusamySImproved version of Bohr’s inequalityComptes Rendus Mathematique20183563272277376759510.1016/j.crma.2018.01.010 – reference: BoasHPKhavinsonDBohr’s power series theorem in several variablesProc. Am. Math. Soc.19971251029752979144337110.1090/S0002-9939-97-04270-6 – reference: PaulsenVIPopescuGSinghDOn Bohr’s inequalityProc. Lond. Math. Soc.2002852493512191205910.1112/S0024611502013692 – reference: Ponnusamy, S., Vijayakumar, R., Wirths, K.-J.: New inequalities for the coefficients of unimodular bounded functions. Results Math., to appear – reference: Ricci, G.: Complementi a un teorema di H. Bohr riguardante le serie di potenze. Rev. Un.Mat. Argentina 17, 185–195 (1955/1956) – reference: Evdoridis, S., Ponnusamy, S., Rasila, A.: Improved Bohr’s inequality for mappings defined on simply connected domains (in preparation) – reference: Kayumov, I.R., Ponnusamy, S.: On a powered Bohr inequality. Ann. Acad. Sci. Fenn. Ser. A I Math. 44, 301–310 (2019) – volume: 147 start-page: 5263 issue: 12 year: 2019 ident: 330_CR2 publication-title: Proc. Am. Math. Soc. doi: 10.1090/proc/14634 – ident: 330_CR22 – volume: 80 start-page: 4307 year: 2004 ident: 330_CR8 publication-title: Int. Math. Res. Not. doi: 10.1155/S1073792804143444 – volume: 132 start-page: 3577 issue: 12 year: 2004 ident: 330_CR20 publication-title: Proc. Am. Math. Soc. doi: 10.1090/S0002-9939-04-07553-7 – volume-title: Finite Blaschke Products and Their Connections year: 2018 ident: 330_CR13 doi: 10.1007/978-3-319-78247-8 – ident: 330_CR17 doi: 10.5186/aasfm.2019.4416 – ident: 330_CR10 doi: 10.1016/j.indag.2018.09.008 – volume: 125 start-page: 2975 issue: 10 year: 1997 ident: 330_CR5 publication-title: Proc. Am. Math. Soc. doi: 10.1090/S0002-9939-97-04270-6 – ident: 330_CR11 – volume: 4 start-page: 1 issue: 1 year: 2004 ident: 330_CR3 publication-title: Comput. Methods Funct. 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Soc. doi: 10.1112/plms/s2-13.1.1 – volume: 85 start-page: 493 issue: 2 year: 2002 ident: 330_CR19 publication-title: Proc. Lond. Math. Soc. doi: 10.1112/S0024611502013692 – volume: 26 start-page: 731 issue: 1 year: 1927 ident: 330_CR24 publication-title: Math. Z. doi: 10.1007/BF01475487 |
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| Title | Bohr Type Inequalities for Functions with a Multiple Zero at the Origin |
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