Annales Academiæ Scientiarum Fennicæ
Mathematica
Volumen 44, 2019, 301-310

ON A POWERED BOHR INEQUALITY

Ilgiz R. Kayumov and Saminathan Ponnusamy

Kazan Federal University
Kremlevskaya 18, 420 008 Kazan, Russia; ikayumov 'at' kpfu.ru

Indian Institute of Technology Madras, Department of Mathematics
Chennai–600 036, India; samy 'at' isichennai.res.in, samy 'at' iitm.ac.in

Abstract. The object of this paper is to study the powered Bohr radius ρ_p, p ∈ (1,2), of analytic functions f(z) = ∑_k=0^∞ a_kz^k defined on the unit disk |z| < 1 and such that |f(z)| < 1 for |z| < 1. More precisely, if M_p^f(r) = ∑_k=0^∞|a_k|^pr^k, then we show that M_p^f(r) ≤ 1 for r ≤ r_p where r_ρ is the powered Bohr radius for conformal automorphisms of the unit disk. This answers the open problem posed by Djakov and Ramanujan in 2000. A couple of other consequences of our approach is also stated, including an asymptotically sharp form of one of the results of Djakov and Ramanujan. In addition, we consider a similar problem for sense-preserving harmonic mappings in |z| < 1. Finally, we conclude by stating the Bohr radius for the class of Bieberbach–Eilenberg functions.

2010 Mathematics Subject Classification: Primary 30A10, 30H05, 30C35; Secondary 30C45.

Key words: Bounded analytic functions, p-symmetric functions, Bohr's inequality, subordination, harmonic mappings, Bieberbach–Eilenberg functions.

Reference to this article: I. R. Kayumov and S. Ponnusamy: On a powered Bohr inequality. Ann. Acad. Sci. Fenn. Math. 44 (2019), 301-310.

Full document as PDF file

https://doi.org/10.5186/aasfm.2019.4416