Kyungpook Mathematical Journal 2022; 62(2): 243-256
Published online June 30, 2022
Copyright © Kyungpook Mathematical Journal.
Bohr’s Phenomenon for Some Univalent Harmonic Functions
Chinu Singla∗, Sushma Gupta and Sukhjit Singh
Received: May 21, 2021; Revised: October 6, 2021; Accepted: October 7, 2021
In 1914, Bohr proved that there is an
Keywords: Bohr radius, harmonic univalent functions, convex in one direction
The Bohr inequality, first introduced in 1914 by Harald Bohr in his seminal work  and subsequently improved independently by M. Riesz, I. Shur and F. Wiener, essentially states that if
In 2010, Abu Muhanna  investigated some Bohr radius problems using the concept of subordination. For two analytic functions
Theorem 1.1. If
In the recent years, a number of research articles (for example see [2, 6, 7]) are published and many hidden facts of this subject are brought into broad daylight. In particular, Bhowmik and Das  successfully extended the Bohr inequalities of type (1.2) for certain harmonic functions. A complex valued function
Theorem 1.2. Let
In this article, our aim is to establish the Bohr's phenomenon and compute Bohr radius for some subclasses of univalent harmonic functions. We also propose to improvise Theorem 1.1. and 1.2. stated above.
We close this section by setting certain notations for subsequent use in this paper. We denote by
2. Main Results
We begin this section by stating following lemma which immediately follows from the work of Bhowmik and Das .
Lemma 2.1. Let
Using this lemma, we now improvise Theorem 1.1 by taking univalent analytic function in
Theorem 2.2. If
In a similar manner, we restate Theorem 1.2. as under;
Theorem 2.3. Let
In next theorem, we establish Bohr's phenomenon for univalent harmonic functions
Theorem 2.4. Let
Corollary 2.5. Let
By plotting the graph of
ϕ(r)w.r.t rfor n=3.
We observe that if
Lemma 2.1 and Theorem 2.4 together lead us to the following result for the subordination class
Corollary 2.6. Let
Next theorem shows the existence of Bohr's phenomenon for
Theorem 2.7. Let
Remark 2.8. We observe that if we take
In the following theorem we establish Bohr's phenomenon for univalent harmonic functions convex in one direction.
Theorem 2.9. Let
We can drop the condition of univalency of
Our last theorem gives Bohr radius for convex univalent harmonic functions in
Theorem 2.11. Let
3. Proof of Theorems
We begin this section by stating a lemma which is easy to prove.
Lemma 3.1. Let
Proof of Theorem 2.4. From
Now, applying Lemma 3.1, we get
To see that this
Figure 2. Image of
Proof of Theorem 2.7. From
Thus we have
On integrating from 0 to r, we get
and this implies
Now, we have
From (3.5), we get
Therefore, we get
Multiplying both sides with
Proof of Theorem 2.9. Let
Now to prove Theorem 2.10., we first state the following result of Sheil-Small .
Lemma 3.2. If
Proof of Theorem 2.10.
Figure 3. Image of
To prove Theorem 2.11., we need following result of Clunie and Sheil-Small .
Lemma 3.3. If a harmonic function
Proof of Theorem 2.11. In view of Lemma 3.3.,
This gives for
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