On a powered Bohr inequality
By Ilgiz R. Kayumov, Saminathan Ponnusamy. Annales Academiae Scientiarum Fennicae: Mathematica, 2018
92 citations recorded by the source corpus.
The object of this paper is to study the powered Bohr radius $\rho_p$, $p \in (1,2)$, of analytic functions $f(z)=\sum_{k=0}^{\infty} a_kz^k$ and such that $|f(z)|<1$ defined on the unit disk $|z|<1$. More precisely, if $M_p^f (r)=\sum_{k=0}^\infty |a_k|^p r^k$, then we show that $M_p^f (r)\leq 1$ for $r \leq r_p$ where $r_\rho$ 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.
Subfields: Algebra and Number Theory, Geometry and Topology
View Paper: https://www.semanticscholar.org/paper/6c852c9bb76f257a1d58678e3a0f97bc8208fdeb
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Citations in the corpus
- Bohr phenomenon for certain subclasses of harmonic mappings (2020, background): “For more intriguing aspects of Bohr phenomenon we refer to [3, 6, 8, 15, 22, 23, 26] and the references therein.”
- Bohr radius for certain classes of starlike and convex univalent functions (2020, background): “For more information about Bohr phenomenon stated above and further related intriguing aspects, we refer the reader to the articles (see [2, 3, 8, 9, 12, 25]).”
- Sharp Bohr type inequality (2020, background): “y provide new theory and applications of Bohr’s results on his work on Dirichlet series. For example, several multidimensional generalizations of this result are obtained in [3–6,11,21]. Moreover, in [28], the authors answer the open problem about the powered Bohr radius posed by Djakov and Ramanuja”