Article
Kyungpook Mathematical Journal 2022; 62(2): 243256
Published online June 30, 2022
Copyright © Kyungpook Mathematical Journal.
Bohr’s Phenomenon for Some Univalent Harmonic Functions
Chinu Singla∗, Sushma Gupta and Sukhjit Singh
Department of Mathematics, Sant Longowal Institute of Engineering and Technology, Longowal148106, India
email : chinusingla204@gmail.com, sushmagupta1@yahoo.com and sukhjit_d@yahoo.com
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
1. Introduction
The Bohr inequality, first introduced in 1914 by Harald Bohr in his seminal work [3] and subsequently improved independently by M. Riesz, I. Shur and F. Wiener, essentially states that if
for all
In 2010, Abu Muhanna [8] investigated some Bohr radius problems using the concept of subordination. For two analytic functions
Theorem 1.1. If
for all
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 [2] successfully extended the Bohr inequalities of type (1.2) for certain harmonic functions. A complex valued function
Theorem 1.2. Let
for
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 [1].
Lemma 2.1. Let
for
Using this lemma, we now improvise Theorem 1.1 by taking univalent analytic function in
Theorem 2.2. If
for all
In a similar manner, we restate Theorem 1.2. as under;
Theorem 2.3. Let
for
for
In next theorem, we establish Bohr's phenomenon for univalent harmonic functions
Theorem 2.4. Let
for
in
Letting
Corollary 2.5. Let
for
This
By plotting the graph of

n r_{0} 1 0.3485... 2 0.3121... 3 0.1794... 4 0.0959...

Figure 1.
ϕ(r) w.r.tr forn=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
for
Next theorem shows the existence of Bohr's phenomenon for
Theorem 2.7. Let
for
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
for
We can drop the condition of univalency of
Let
for
Our last theorem gives Bohr radius for convex univalent harmonic functions in
Theorem 2.11. Let
for
3. Proof of Theorems
We begin this section by stating a lemma which is easy to prove.
Lemma 3.1. Let
for all
Proof of Theorem 2.4. From
where
Now, applying Lemma 3.1, we get
for all
Thus
Now, we need to verify that inequality (3.3) holds for
Then
To see that this

Figure 2. Image of
z<0.3485 underf_{0}(z).
Proof of Theorem 2.7. From
where
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
As
for
Proof of Theorem 2.9. Let
Since
for
Now to prove Theorem 2.10., we first state the following result of SheilSmall [11].
Lemma 3.2. If
Proof of Theorem 2.10.
and so,
if

Figure 3. Image of
z<0.3134 underK(z).
To prove Theorem 2.11., we need following result of Clunie and SheilSmall [4].
Lemma 3.3. If a harmonic function
then
Proof of Theorem 2.11. In view of Lemma 3.3.,
This gives for
for
for
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