Bohr's inequalities for the analytic functions with lacunary series and harmonic functions

We determine the Bohr radius for the class of all functions f of the form f(z)=zm∑k=0∞akpzkp analytic in the unit disk |z|<1 and satisfy the condition |f(z)|≤1 for all |z|<1. In particular, our result also contains a solution to a recent conjecture of Ali et al. [9] for the Bohr radius for odd...

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Bibliographic Details
Published inJournal of mathematical analysis and applications Vol. 465; no. 2; pp. 857 - 871
Main Authors Kayumov, Ilgiz R., Ponnusamy, Saminathan
Format Journal Article
LanguageEnglish
Published Elsevier Inc 15.09.2018
Subjects
Bohr's inequality
Harmonic functions
p-Symmetric functions
Schwarz lemma
Subordination
Harmonic functions
Subordination
Schwarz lemma
Bohr's inequality
p-Symmetric functions
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ISSN0022-247X
1096-0813
DOI10.1016/j.jmaa.2018.05.038

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Summary:We determine the Bohr radius for the class of all functions f of the form f(z)=zm∑k=0∞akpzkp analytic in the unit disk |z|<1 and satisfy the condition |f(z)|≤1 for all |z|<1. In particular, our result also contains a solution to a recent conjecture of Ali et al. [9] for the Bohr radius for odd analytic functions, solved by the authors in [17]. We consider a more flexible approach by introducing the p-Bohr radius for harmonic functions which in turn contains the classical Bohr radius as special case. Also, we prove several other new results and discuss p-Bohr radius for the class of odd harmonic bounded functions.
ISSN:0022-247X
1096-0813
DOI:10.1016/j.jmaa.2018.05.038