• Corpus ID: 233231531

Note on Improved Bohr Inequality for harmonic mappings

@inproceedings{Ponnusamy2021NoteOI,
  title={Note on Improved Bohr Inequality for harmonic mappings},
  author={Saminathan Ponnusamy and Ramakrishnan Vijayakumar},
  year={2021}
}
Let D = {z ∈ C : |z| < 1} denote the open unit disk and H∞ denote the class of all bounded analytic functions f on the unit disk D with the supremum norm ‖f‖∞ := supz∈D |f(z)|. Also, let B = {f ∈ H∞ : ‖f‖∞ ≤ 1} and B0 = {ω ∈ B : ω(0) = 0}. Note that if |f(z)| = 1 for some z ∈ ∂D, then, by the maximum principle, it follows that f should be unimodular constant functions. So, one can conveniently exclude constant functions. In 1914, the following theorem was proved by Harald Bohr [4] for 0 ≤ r ≤ 1… 
1 Citations

Improved Bohr inequalities for certain classes of harmonic mappings

. The Bohr radius for the class of harmonic functions of the form f ( z ) = h + g in the unit disk D := { z ∈ C : | z | < 1 } , where h ( z ) = P ∞ n =0 a n z n and g ( z ) = P ∞ n =1 b n z n is to

References

SHOWING 1-10 OF 26 REFERENCES

Bohr radius for locally univalent harmonic mappings

We consider the class of all sense‐preserving harmonic mappings f=h+g¯ of the unit disk D , where h and g are analytic with g(0)=0 , and determine the Bohr radius if any one of the following

Bohr–Rogosinski Inequalities for Bounded Analytic Functions

In this paper we first consider another version of the Rogosinski inequality for analytic functions $f(z)=\sum_{n=0}^\infty a_nz^n$ in the unit disk $|z| < 1$, in which we replace the coefficients

On the Bohr Inequality

The Bohr inequality, first introduced by Harald Bohr in 1914, deals with finding the largest radius r, 0 < r < 1, such that ∑n = 0 ∞ | a n | r n ≤ 1 holds whenever | ∑n = 0 ∞ a n z n | ≤ 1 in the

Bohr Inequality for Odd Analytic Functions

We determine the Bohr radius for the class of odd functions f satisfying $$|f(z)|\le 1$$|f(z)|≤1 for all $$|z|<1$$|z|<1, solving the recent problem of Ali et al. (J Math Anal Appl 449(1):154–167,

On a powered Bohr inequality

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

Modifications of Bohr's inequality in various settings

Abstract. The concept of Bohr radius for the class of bounded analytic functions was introduced by Harald Bohr in 1914. His initial result received greater interest and was

Bohr--Rogosinski radius for analytic functions

There are a number of articles which deal with Bohr's phenomenon whereas only a few papers appeared in the literature on Rogosinski's radii for analytic functions defined on the unit disk $|z|<1$. In

A remark on Bohr's inequality

This paper studies the classical problem of determining the maximum growth of the Taylor series majorant of holomorphic functions with bound 1 in the unit disk, as a function of the radius of the

Bohr-type inequalities of analytic functions

The Bohr-type radius of the alternating series associated with the Taylor series of analytic functions is investigated and it is proved that most of the results are sharp.