On Bohr's inequality for special subclasses of stable starlike harmonic mappings

Wei Jin; Zhihong Liu; Qian Hu; Wenbo Zhang

doi:10.1515/math-2023-0141

Artikel Open Access

On Bohr's inequality for special subclasses of stable starlike harmonic mappings

Wei Jin , Zhihong Liu , Qian Hu und Wenbo Zhang

Veröffentlicht/Copyright: 1. Dezember 2023

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Abstract

The focus of this article is to explore the Bohr inequality for a specific subset of harmonic starlike mappings introduced by Ghosh and Vasudevarao (Some basic properties of certain subclass of harmonic univalent functions, Complex Var. Elliptic Equ. 63 (2018), no. 12, 1687–1703.). This set is denoted as ℬ H 0 ( M ) ≔ { f = h + g ¯ ∈ ℋ 0 : ∣ z h ″ ( z ) ∣ ≤ M − ∣ z g ″ ( z ) ∣ } for z ∈ D , where 0 < M ≤ 1 . It is worth mentioning that the functions belonging to the class ℬ H 0 ( M ) are recognized for their stability as starlike harmonic mappings. With this in mind, this research has a twofold goal: first, to determine the optimal Bohr radius for this specific subclass of harmonic mappings, and second, to extend the Bohr-Rogosinski phenomenon to the same subclass.

Keywords: stable starlike harmonic mappings; Bohr radius; Bohr’s inequality; Bohr-Rogosinski’s inequalities

MSC 2010: 30C35; 30C45; 35Q30

1 Introduction

In recent years, Bohr’s inequality and its various improved versions have once again sparked extensive research among many scholars. The classical statement of Bohr’s inequality asserts that when the power series f ( z ) = ∑ n = 0 ∞ a n z n converges in the unit disk D ≔ { z ∈ C : ∣ z ∣ < 1 } and satisfies ∣ f ( z ) ∣ ≤ 1 for all z ∈ D , the classical form of the Bohr inequality is expressed as follows:

(1.1) ∑ n = 1 ∞ ∣ a n ∣ r n ≤ 1 ,

for ∣ z ∣ = r ≤ 1 ∕ 3 , and the constant 1/3 cannot be improved. This inequality was defined by Bohr [1] in 1914. In fact, Bohr obtained the conclusion for r ≤ 1 ∕ 6 ; later, Weiner, Riesz, and Schur have independently proved it to 1/3. Then, more and more theories about the Bohr radius and Bohr inequality have been extensively studied. For example, the Bohr phenomenon for certain close-to-convex analytic functions was established in [2], and Ahamed [3] extended this to the case of harmonic mappings. Kayumov and Ponnusamy [4] established a more precise version of the Bohr-type inequalities for bounded analytic functions. In their work, they introduced the concept of p -Bohr radius for the class of odd analytic functions and harmonic functions. Later, more and more scholars extended the analytical Bohr’s inequality to the harmonic case [5–7]. Bohr’s phenomenon in subordination and bounded harmonic classes has been widely researched [8,9]. In 2021, Ahamed and Allu [10] introduced the improved Bohr radius for the class of starlike log-harmonic mappings. For more interesting aspects of the Bohr phenomenon, we refer to [8,11–14] and references therein.

Similar to the Bohr radius, there is also the notion of Rogosinski radius [15], which is defined as follows: if f ( z ) = ∑ n = 0 ∞ a n z n is an analytic function on D such that ∣ f ( z ) ∣ < 1 in D , then for every N ≥ 1 , we have ∣ S N ( z ) ∣ = ∑ n = 0 N a n z n < 1 for r < 1 ∕ 2 , where 1/2 is the best possible quantity [15]. The number r = 1 ∕ 2 is called the Rogosinski radius. And the Bohr-Rogosinski inequality, which is considered by Kayumov et al. [16], is given by:

R N f ( z ) = ∣ f ( z ) ∣ + ∑ n = N ∞ ∣ a n ∣ r n ≤ d ( f ( 0 ) , ∂ f ( D ) ) ,

for ∣ z ∣ = r ≤ R N ; here, d ( f ( 0 ) , ∂ f ( D ) ) denotes the Euclidean distance between f ( 0 ) and the boundary of f ( D ) , where R N is the positive root of the equation 2 ( 1 + r ) r N − ( 1 − r ) 2 = 0 . The aforementioned area of research is crucial for exploring the Bohr phenomenon. In recent years, there has been a notable surge of interest in extending the Bohr-Rogosinski inequality into the realm of harmonic mappings, as demonstrated by recent studies such as those referenced in [17]. Consequently, our objective is to delve deeper into this field, with a focus on the investigation of the specific Bohr-Rogosinski radius and the development of refined inequalities.

A complex-valued function f = u + i v is said to be harmonic in a domain Ω ⊂ C if u and v are real-valued harmonic functions in Ω . Let ℋ denote the class of complex-valued harmonic functions f in the unit disk D normalized by f ( 0 ) = 0 = f z ( 0 ) − 1 . Each function f ∈ ℋ can be expressed as f = h + g ¯ , where h and g are analytic in D with the following power series representations:

h ( z ) = z + ∑ n = 2 ∞ a n z n and g ( z ) = ∑ n = 1 ∞ b n z n .

We call h and g the analytic and co-analytic parts of f , respectively. The Jacobian of f = h + g ¯ is defined as J f = ∣ h ′ ∣ 2 − ∣ g ′ ∣ 2 . Lewy’s theorem [18] implies that a harmonic function f is locally univalent and sense-preserving if and only if J f > 0 in D . Note that J f ( z ) > 0 in D is equivalent to the existence of an analytic function ω , called the (second complex) dilatation of f , given by ω ( z ) = g ′ ( z ) ∕ h ′ ( z ) , with ∣ ω ( z ) ∣ < 1 for all z ∈ D , where h ′ ( z ) ≠ 0 in D .

Let S H be the subclass of ℋ consisting of univalent and sense-preserving harmonic mappings on D . This class was introduced and investigated by Clunie and Sheil-Small [19], the class S H contains the standard class S of analytic univalent functions, and they showed that S H is normal, but not compact. They also investigated the subclass S H 0 consisting of functions f = h + g ¯ ∈ S H with g ′ ( 0 ) = 0 . The series expansions of h and g for the subclass are as follows:

(1.2) h ( z ) = z + ∑ n = 2 ∞ a n z n and g ( z ) = ∑ n = 2 ∞ b n z n .

We define the majorant series associated with f = h + g ¯ as:

(1.3) ℳ f ( r ) = ∑ n = 1 ∞ ( ∣ a n ∣ + ∣ b n ∣ ) r n = r + ∑ n = 2 ∞ ( ∣ a n ∣ + ∣ b n ∣ ) r n .

In [20], the authors introduced the following subclass of harmonic univalent mappings.

Definition 1

For M > 0 , let

(1.4) ℬ H 0 ( M ) ≔ { f = h + g ¯ ∈ S H 0 : ∣ z h ″ ( z ) ∣ ≤ M − ∣ z g ″ ( z ) ∣ for z ∈ D } .

We consider Bohr’s radius for the class of stable starlike harmonic mappings in S H 0 , which were introduced in [21].

Definition 2

A (sense-preserving) harmonic mapping f = h + g ¯ is stable starlike (resp. stable convex) harmonic in the unit disk D if all the mappings f λ = h + λ g with ∣ λ ∣ = 1 are starlike (resp. convex) in D .

The remainder of this article is organized as follows: Section 2 presents essential lemmas and conclusions required to establish our main conclusions. These primarily encompass the coefficient estimation theorem, the growth theorem, and the distortion theorem for the subclass ℬ H 0 ( M ) . Section 3 delves into the investigation of Bohr’s phenomenon and the refinement of Bohr’s inequalities within the specific subclass of harmonic mappings, while considering their stability conditions, as discussed in [22,23]. Additionally, this article explores Bohr-Rogosinski inequalities. The research results of this article provide a theoretical basis and new perspectives for further studying the properties of harmonic mappings.

2 Lemmas and preliminaries

It is well known that functions in the subclass ℬ H 0 ( M ) are stable harmonic starlike with respect to origin for M ≤ 1 [4]. Furthermore, the forthcoming results concerning the coefficient estimation and the growth distortion theorem play a vital role in proving our main results.

Lemma 2.1

[20, Theorem 2.2] Let f = h + g ¯ ∈ ℬ H 0 ( M ) and be given by (1.2). Then, for n ≥ 2 ,

∣ a n ∣ ≤ M n ( n − 1 ) ;
∣ b n ∣ ≤ M n ( n − 1 ) ;
∣ a n ∣ + ∣ b n ∣ ≤ M n ( n − 1 ) ;
∣ a n ∣ − ∣ b n ∣ ≤ M n ( n − 1 ) .

All inequalities are sharp.

Lemma 2.2

[20, Theorem 2.3] Let f = h + g ¯ ∈ ℬ H 0 ( M ) and be given by (1.2). Then,

(2.1) ∣ z ∣ − M 2 ∣ z ∣ 2 ≤ ∣ f ( z ) ∣ ≤ ∣ z ∣ + M 2 ∣ z ∣ 2 .

Both inequalities are sharp for the function f M given by f M ( z ) = z + M n ( n − 1 ) z n .

The following result gives the sharp upper bound of J f for functions f in the class B H 0 ( M ) .

Theorem A

[20, Theorem 2.5] Let f = h + g ¯ ∈ ℬ H 0 ( M ) , then J f ( z ) ≤ ( 1 + M ∣ z ∣ ) 2 , with equality for the function f ( z ) = z + M 2 z 2 .

In [23], the authors obtained the Bohr radius for stable convex harmonic mappings.

Theorem B

[23, Theorem 2.1.1(i)] Let f = h + g ¯ ∈ S H 0 be a stable convex harmonic mapping on the unit disk D . Then,

(2.2) ℳ f ( r ) ≤ d ( f ( 0 ) , ∂ f ( D ) ) ,

if ∣ z ∣ ≤ r 0 = 1 ∕ 3 , where r 0 is the unique root in ( 0 , 1 ) of

r 1 − r = 1 2 .

Theorem C

[23, Theorem 2.1.2(i)] Let f = h + g ¯ ∈ S H 0 be a stable convex harmonic mapping on the unit disk D , and let S r be the area of the image f ( D r ) , with D r = ∣ z ∣ = r . Then,

(2.3) ℳ f ( r ) + S r π k ≤ d ( f ( 0 ) , ∂ f ( D ) ) ,

if ∣ z ∣ ≤ r 0 , where r 0 is the unique root in ( 0 , 1 ) of

(2.4) r 1 − r + r 2 k ( 1 − r 2 ) 2 k = 1 2 .

3 Main results

Before proving the main results of this article, initially, we recall the definition of dilogarithm [24]. The polylogarithm function L i k ( z ) is defined for ∣ z ∣ < 1 by:

Li k ( z ) = ∑ n = 1 ∞ z n n k ,

and by analytic continuation to the entire complex plane. In particular, the Euler dilogarithm function Li 2 ( z ) is defined for ∣ z ∣ < 1 by:

Li 2 ( z ) = ∑ n = 1 ∞ z n n 2 = − ∫ 0 z log ( 1 − t ) t d t .

Next, we will find the minimum Euclidean distance between f ( 0 ) and the boundary of f ( D ) .

For f = h + g ¯ ∈ ℬ H 0 ( M ) , by Lemma 2.2, we have

∣ f ( z ) ∣ ≥ ∣ z ∣ − M 2 ∣ z ∣ 2 .

Then, after simple calculation, we can obtain

(3.1) d ( f ( 0 ) , ∂ f ( D ) ) = ∣ lim ∣ z ∣ → 1 ∣ f ( z ) ∣ − ∣ f ( 0 ) ∣ ∣ ≥ lim ∣ z ∣ → 1 ∣ z ∣ − M 2 ∣ z ∣ 2 = 1 − M 2 .

In this section, we obtain the Bohr-type inequality and a series of improved Bohr inequalities for the functions in the subclass ℬ H 0 ( M ) . First, we obtain the Bohr radius for stable starlike harmonic mappings.

Theorem 3.1

Let f = h + g ¯ ∈ ℬ H 0 ( M ) be a stable starlike harmonic mapping with 0 < M ≤ 1 on the unit disk D . Then,

ℳ f ( r ) ≤ d ( f ( 0 ) , ∂ f ( D ) ) ,

for ∣ z ∣ = r ≤ r M , where r M is the unique root of the equation:

(3.2) r + M ( r + ( 1 − r ) log ( 1 − r ) ) − 1 + M 2 = 0

in ( 0 , 1 ) . The radius r M is the best possible.

Proof

Let f = h + g ¯ ∈ ℬ H 0 ( M ) be given by (1.4). For ∣ z ∣ = r , by Lemma 2.1 (3) and (3.1), we obtain

ℳ f ( r ) = r + ∑ n = 2 ∞ ( ∣ a n ∣ + ∣ b n ∣ ) r n ≤ r + ∑ n = 2 ∞ M n ( n − 1 ) r n = r + M ( r + ( 1 − r ) log ( 1 − r ) ) ≤ d ( f ( 0 ) , ∂ f ( D ) ) .

Hence, we have

ℳ f ( r ) ≤ d ( f ( 0 ) , ∂ f ( D ) ) ,

if ∣ z ∣ ≤ r M , where r M is the unique root of the equation:

r + M ( r + ( 1 − r ) log ( 1 − r ) ) − 1 + M 2 = 0 .

In order to show that r M is the best possible radius, we consider the following function f = f M , which is defined by:

(3.3) f M ( z ) = z + M n ( n − 1 ) z n .

For ∣ z ∣ = r M and the function f = f M , a simple computation shows that

ℳ f ( r ) = ∣ z ∣ + ∑ n = 2 ∞ ( ∣ a n ∣ + ∣ b n ∣ ) ∣ z ∣ n = r M + M ( r M + ( 1 − r M ) log ( 1 − r M ) ) = 1 − M 2 = d ( f ( 0 ) , ∂ f ( D ) ) .

Hence, the radius r M is the best possible. This completes the proof.□

Remark 1

The values in Table 1 correspond to the Bohr radius r M for various values of M in Theorem 3.1. Based on the information presented in Figure 1, we can infer that r M falls in the range 0.404289 ≤ r M < 1 for 0 < M ≤ 1 .

Table 1

Roots r M of (3.2) for different values of M

M	0.01	0.1	0.2	0.3	0.5	0.8	1.0
r M	0.985748	0.886128	0.803307	0.735056	0.622586	0.48526	0.404289

$Figure 1 Graph of r M {r}_{M} when M = 0.01 , 0.2 , 0.4 , 0.6 , 0.8 M=0.01,0.2,0.4,0.6,0.8 , and 1.0.$

Figure 1

Graph of r M when M = 0.01 , 0.2 , 0.4 , 0.6 , 0.8 , and 1.0.

We will now derive the refined Bohr-Rogosinski inequality for functions belonging to the subclass ℬ H 0 ( M ) .

Theorem 3.2

Let f = h + g ¯ ∈ ℬ H 0 ( M ) be a stable starlike harmonic mapping on the unit disk D , with 0 < M ≤ 1 . Then,

∣ f ( z m ) ∣ + ∑ n = N ∞ ( ∣ a n ∣ + ∣ b n ∣ ) ∣ z ∣ n ≤ d ( f ( 0 ) , ∂ f ( D ) ) ,

for ∣ z ∣ = r ≤ r m , M N , where r m , M N is the unique root of the equation:

(3.4) r m + M 2 r 2 m + ∑ n = N ∞ M n ( n − 1 ) r n − 1 + M 2 = 0

in ( 0 , 1 ) . The radius r m , M N is the best possible.

Proof

Since f ∈ ℬ H 0 ( M ) , then by Lemmas 2.1 and 2.2, we can obtain

∣ f ( z m ) ∣ + ∑ n = N ∞ ( ∣ a n ∣ + ∣ b n ∣ ) ∣ z ∣ n ≤ r m 1 + M 2 r m + ∑ n = N ∞ M n ( n − 1 ) r n .

Let H 1 ( r ) : [ 0 , 1 ] → R be defined by

H 1 ( r ) ≔ r m + M 2 r 2 m + ∑ n = N ∞ M n ( n − 1 ) r n − 1 + M 2 .

It is easy to see that H 1 ( r ) is continuous on [ 0 , 1 ] and differentiable on ( 0 , 1 ) . A straightforward calculation reveals that

H 1 ( 1 ) = 1 + M 2 + M N − 1 − 1 + M 2 = M N N − 1 > 0 , ( N ≥ 2 ) .

Furthermore, we observe that

H 1 ( 0 ) = − 1 + M 2 < 0 .

It is evident that H 1 ( 1 ) H 1 ( 0 ) < 0 , and thus, according to the intermediate value theorem, H 1 ( r ) has at least one root in ( 0 , 1 ) . We now show that H 1 ( r ) has exactly one root in ( 0 , 1 ) . Performing a direct computation yields

d d r ( H 1 ( r ) ) = m r m − 1 + m M r 2 m − 1 + ∑ n = N ∞ M n − 1 r n − 1 > 0 ,

for all r ∈ ( 0 , 1 ) . Consequently, H 1 ( r ) strictly increases over ( 0 , 1 ) . Therefore, H 1 ( r ) has the unique root in ( 0 , 1 ) .

To show that r m , M N is the best possible, we consider the function f = f M defined by (3.3). In view of (3.1), for f = f M and ∣ z ∣ = r m , M N , we obtain

∣ f M ( z m ) ∣ + ∑ n = N ∞ ( ∣ a n ∣ + ∣ b n ∣ ) ∣ z ∣ n = ( r m , M N ) m + M 2 ( r m , M N ) 2 m + ∑ n = N ∞ M n ( n − 1 ) ( r m , M N ) n = 1 − M 2 = d ( f ( 0 ) , ∂ f ( D ) ) .

This implies that r m , M N is the best possible. Thus, the proof is complete.□

Let m = 1 in Theorem 3.2, we have the following result.

Corollary 3.3

Let f = h + g ¯ ∈ ℬ H 0 ( M ) be a stable starlike harmonic mapping on the unit disk D , with 0 < M ≤ 1 . Then,

∣ f ( z ) ∣ + ∑ n = N ∞ ( ∣ a n ∣ + ∣ b n ∣ ) ∣ z ∣ n ≤ d ( f ( 0 ) , ∂ f ( D ) ) ,

for ∣ z ∣ = r ≤ r 1 , M , where r 1 , M is the unique root of the equation:

(3.5) r + M 2 r 2 + ∑ n = N ∞ M n ( n − 1 ) r n − 1 + M 2 = 0 .

Remark 2

The aforementioned equation is the sharp Bohr-Rogosinski inequality and sharp Bohr-Rogosinski radius as well known. For certain specific values of M and m , a straightforward computation yields the Bohr-Rogosinski radius r m , M N as presented in Table 2. Figure 2 illustrates that the roots r m , M 2 of (3.4) exceed 0.360456 when N = 2 and m = 1 .

Table 2

Roots r m , M N of (3.4) for different values of N and ( m , M )

( m , M )	(1,0.2)	(1,0.4)	(1,0.6)	(1,0.8)	(1,1)
r m , M 2	0.759124	0.621340	0.519008	0.434651	0.360456
( m , M )	(1,0.2)	(1,0.4)	(1,0.6)	(1,0.8)	(1,1)
r m , M 5	0.822941	0.696518	0.591388	0.498657	0.413617
( m , M )	(2,0.2)	(2,0.4)	(2,0.6)	(2,0.8)	(2,1)
r m , M 2	0.855228	0.755342	0.673436	0.600602	0.532128
( m , M )	(2,0.2)	(2,0.4)	(2,0.6)	(2,0.8)	(2,1)
r m , M 5	0.902969	0.828416	0.762589	0.700316	0.638296

$Figure 2 Graph of particular r m , M N {r}_{m,M}^{N} in (3.4) when M = 0.2 , 0.4 , 0.6 , 0.8 M=0.2,0.4,0.6,0.8 , and 1.0. (a) m = 1, N = 2. (b) m = 1, N = 5. (c) m = 2, N = 2. (d) m = 1, N = 5.$

Figure 2

Graph of particular r m , M N in (3.4) when M = 0.2 , 0.4 , 0.6 , 0.8 , and 1.0. (a) m = 1, N = 2. (b) m = 1, N = 5. (c) m = 2, N = 2. (d) m = 1, N = 5.

Theorem 3.4

Let f = h + g ¯ ∈ ℬ H 0 ( M ) be a stable starlike harmonic mapping with 0 < M ≤ 1 on the unit disk D , and let S r be the area of the image f ( D r ) , with D r = ∣ z ∣ = r . Then,

ℳ f ( r ) + S r π k ≤ d ( f ( 0 ) , ∂ f ( D ) ) ,

for ∣ z ∣ = r ≤ r k , M , where r k , M is the unique root of the equation:

(3.6) r + M ( r + ( 1 − r ) log ( 1 − r ) ) ( r 2 + M 2 ( r 2 Li 2 ( r 2 ) − r 2 + r 2 log ( 1 − r 2 ) − log ( 1 − r 2 ) ) ) k − 1 + M 2 = 0

in ( 0 , 1 ) . The radius r k , M is the best possible.

Proof

First, we need to find the bound on S r in the case where f is stable starlike harmonic. For f = h + g ¯ given by (1.4), it is well known that

S r π = 1 π ∬ D J f d A = ∑ n = 1 ∞ n ( ∣ a n ∣ − ∣ b n ∣ ) ( ∣ a n ∣ + ∣ b n ∣ ) r 2 n = r 2 + ∑ n = 2 ∞ n ( ∣ a n ∣ − ∣ b n ∣ ) ( ∣ a n ∣ + ∣ b n ∣ ) r 2 n .

Since f ∈ ℬ H 0 ( M ) , then by Lemma 2.1, we obtain that

S r π ≤ r 2 + ∑ n = 2 ∞ M 2 n ( n − 1 ) 2 r 2 n .

Therefore, it follows that

ℳ f ( r ) + S r π k = r + ∑ n = 2 ∞ ( ∣ a n ∣ + ∣ b n ∣ ) r n + r 2 + ∑ n = 2 ∞ n ( ∣ a n ∣ − ∣ b n ∣ ) ( ∣ a n ∣ + ∣ b n ∣ ) r 2 n k ≤ r + ∑ n = 2 ∞ M n ( n − 1 ) r n + r 2 + ∑ n = 2 ∞ M 2 n ( n − 1 ) 2 r 2 n k = r + M ( r + ( 1 − r ) log ( 1 − r ) ) + ( r 2 + M 2 ( r 2 L i 2 ( r 2 ) − r 2 − ( 1 − r 2 ) log ( 1 − r 2 ) ) ) k ≤ d ( f ( 0 ) , ∂ f ( D ) ) .

Now, we consider the function H 2 ( r ) : [ 0 , 1 ] → R , where H 2 ( r ) is defined by:

H 2 ( r ) ≔ r + ∑ n = 2 ∞ M n ( n − 1 ) r n + r 2 + ∑ n = 2 ∞ M 2 n ( n − 1 ) 2 r 2 n k − 1 + M 2 .

Using the method similar to Theorem 3.2, we obtain that H 2 ( 0 ) H 2 ( 1 ) < 0 and d d r ( H 2 ( r ) ) > 0 in ( 0 , 1 ) . As a result, the function H 2 ( r ) contains exactly one root in ( 0 , 1 ) , denote it as r k , M . Therefore, H 2 ( r k , M ) = 0 , and for all 0 < r < r k , M , H 2 ( r ) < 0 .

To prove that r k , M is the best possible, we consider the function f = f M defined by (3.3). Using (3.1) for f = f M and r > r k , M , it is easy to see that

r + ∑ n = 2 ∞ ( ∣ a n ∣ + ∣ b n ∣ ) r n + r 2 + ∑ n = 2 ∞ n ( ∣ a n ∣ − ∣ b n ∣ ) ( ∣ a n ∣ + ∣ b n ∣ ) r 2 n k > r k , M + ∑ n = 2 ∞ M n ( n − 1 ) ( r k , M ) n + ( r k , M ) 2 + ∑ n = 2 ∞ M 2 n ( n − 1 ) 2 ( r k , M ) 2 n k = 1 − M 2 = d ( f ( 0 ) , ∂ f ( D ) ) ,

which shows that r k , M is best possible. This completes the proof of Theorem 3.4.□

Remark 3

Table 3 and Figure 3 present the values of the Bohr radius r k , M associated with various values of k and M in Theorem 3.4. If k → ∞ and M → 0 , clearly, r k , M → 1 . Consequently, we can discern that the radius lies in the range 0.326888 ≤ r k , M < 1 for varying values of 0 < M ≤ 1 and non-negative integer k .

Table 3

Roots r k , M of (3.6) for different values of k and M

M	0.2	0.4	0.6	0.8	1.0
r 1 , M	0.553262	0.492872	0.435728	0.380841	0.326888
r 2 , M	0.654969	0.586322	0.519333	0.453293	0.386992
r 5 , M	0.754437	0.662210	0.570533	0.484580	0.404174
r 10 , M	0.794532	0.675369	0.573788	0.485259	0.404289

$Figure 3 Graph of particular r k , M {r}_{k,M} of (3.6) when M = 0.2 , 0.4 , 0.6 , 0.8 M=0.2,0.4,0.6,0.8 , and 1.0. (a) k = 1 k=1 , (b) k = 2 k=2 , (c) k = 5 k=5 , and (d) k = 10 k=10 .$

Figure 3

Graph of particular r k , M of (3.6) when M = 0.2 , 0.4 , 0.6 , 0.8 , and 1.0. (a) k = 1 , (b) k = 2 , (c) k = 5 , and (d) k = 10 .

Theorem 3.5

Let f = h + g ¯ ∈ ℬ H 0 ( M ) be given by (1.4) with 0 < M ≤ 1 . Then,

ℳ f ( r ) + ∑ n = 1 ∞ ( ∣ a n ∣ 2 + ∣ b n ∣ 2 ) r n ≤ d ( f ( 0 ) , ∂ f ( D ) ) ,

for ∣ z ∣ = r ≤ r M , where r M is the unique root of the equation:

(3.7) 2 r + M ( r + ( 1 − r ) log ( 1 − r ) ) + 2 M 2 ( r Li 2 ( r ) + Li 2 ( r ) − 3 r + 2 r log ( 1 − r ) − 2 log ( 1 − r ) ) = 1 − M 2

in ( 0 , 1 ) . The radius r M is the best possible.

Proof

Let f = h + g ¯ ∈ ℬ H 0 ( M ) be given by (1.4). In view of Lemmas 2.1 and 2.2, we have

∣ a n ∣ ≤ M n ( n − 1 ) , ∣ b n ∣ ≤ M n ( n − 1 )

and

∣ f ( z ) ∣ ≤ ∣ z ∣ + M 2 ∣ z ∣ 2 .

Now, we have

(3.8) ℳ f ( r ) + ∑ n = 1 ∞ ( ∣ a n ∣ 2 + ∣ b n ∣ 2 ) r n = 2 r + ∑ n = 2 ∞ ( ∣ a n ∣ + ∣ b n ∣ + ∣ a n ∣ 2 + ∣ b n ∣ 2 ) r n ≤ 2 r + ∑ n = 2 ∞ M n ( n − 1 ) + 2 M 2 n 2 ( n − 1 ) 2 r n = 2 r + M ( r + ( 1 − r ) log ( 1 − r ) ) + 2 M 2 ( r Li 2 ( r ) + Li 2 ( r ) − 3 r + 2 r log ( 1 − r ) − 2 log ( 1 − r ) ) ≤ d ( f ( 0 ) , ∂ f ( D ) ) .

Hence, it is easy to see that

ℳ f ( r ) + ∑ n = 1 ∞ ( ∣ a n ∣ 2 + ∣ b n ∣ 2 ) r n ≤ d ( f ( 0 ) , ∂ f ( D ) ) ,

for r ≤ r M , where r M is the unique root of (3.7) in ( 0 , 1 ) .

To show the sharpness of the radius r M , we consider the function f = f M defined by (3.3). For ∣ z ∣ = r M , we write that

ℳ f ( r ) + ∑ n = 1 ∞ ( ∣ a n ∣ 2 + ∣ b n ∣ 2 ) r n = 2 r M + ∑ n = 2 ∞ M n ( n − 1 ) + 2 M 2 n 2 ( n − 1 ) 2 r M n = 2 r M + M ( r M + ( 1 − r M ) log ( 1 − r M ) ) + 2 M 2 ( r M Li 2 ( r M ) + Li 2 ( r M ) − 3 r M + 2 r M log ( 1 − r M ) − 2 log ( 1 − r M ) ) = 1 − M 2 = d ( f ( 0 ) , ∂ f ( D ) ) ,

and hence, the radius r M is the best possible. This completes the proof.□

Remark 4

Table 4 provides the values of the Bohr radius r M corresponding to varying values of M as outlined in Theorem 3.5. As depicted in Figure 4, we discern that the radius lies in the interval 0.223616 ≤ r M < 1 2 for different values of 0 < M ≤ 1 .

Table 4

Roots r M of (3.7) for different values of M

M	0.01	0.2	0.4	0.6	0.8	1.0
r M	0.496738	0.436652	0.377543	0.322634	0.271524	0.223616

$Figure 4 Graph of r M {r}_{M} when M = 0.01 , 0.2 , 0.4 , 0.6 , 0.8 M=0.01,0.2,0.4,0.6,0.8 , and 1.0.$

Figure 4

Graph of r M when M = 0.01 , 0.2 , 0.4 , 0.6 , 0.8 , and 1.0.

Theorem 3.6

Let f = h + g ¯ ∈ ℬ ℋ 0 ( M ) be given by (1.4) for 0 < M ≤ 1 . Then, for any integer p ≥ 1 ,

∣ f ( z ) ∣ p + ∑ n = 2 ∞ ( ∣ a n ∣ + ∣ b n ∣ ) r n + ∑ n = 2 ∞ ( ∣ a n ∣ + ∣ b n ∣ ) 2 r 2 n ≤ d ( f ( 0 ) , ∂ f ( D ) ) ,

for ∣ z ∣ = r ≤ r p , M , where r p , M is the unique root of

(3.9) r + M 2 r 2 p + M ( r − ( r − 1 ) log ( 1 − r ) ) + M 2 ( − 3 r 2 + 2 ( r 2 − 1 ) log ( 1 − r 2 ) + ( 1 + r 2 ) Li 2 ( r 2 ) ) = 1 − M 2

in ( 0 , 1 ) . Here, r p , M is best possible.

Proof

According to Lemma 2.1, we have that

( ∣ a n ∣ + ∣ b n ∣ ) 2 ≤ M 2 n 2 ( n − 1 ) 2 and ∣ f ( z ) ∣ ≤ ∣ z ∣ + M 2 ∣ z 2 ∣ .

It is obvious that

∣ f ( z ) ∣ p + ∑ n = 2 ∞ ( ∣ a n ∣ + ∣ b n ∣ ) ∣ z ∣ n + ∑ n = 2 ∞ ( ∣ a n ∣ + ∣ b n ∣ ) 2 r 2 n ≤ r + M 2 r 2 p + ∑ n = 2 ∞ M n ( n − 1 ) r n + ∑ n = 2 ∞ M 2 n 2 ( n − 1 ) 2 r 2 n = r + M 2 r 2 p + M ( r − ( r − 1 ) log ( 1 − r ) ) + M 2 ( − 3 r 2 + 2 ( r 2 − 1 ) log ( 1 − r 2 ) + ( 1 + r 2 ) Li 2 ( r 2 ) ) ≤ d ( f ( 0 ) , ∂ f ( D ) ) .

Hence, we have

∣ f ( z ) ∣ p + ∑ n = 2 ∞ ( ∣ a n ∣ + ∣ b n ∣ ) r n + ∑ n = 2 ∞ ( ∣ a n ∣ + ∣ b n ∣ ) 2 r 2 n ≤ d ( f ( 0 ) , ∂ f ( D ) ) ,

if ∣ z ∣ ≤ r p , M , where r p , M is the unique root of equation (3.9). Moreover, let H 3 ( r ) : [ 0 , 1 ] → R , where H 3 ( r ) is defined by:

H 3 ( r ) ≔ r + M 2 r 2 p + ∑ n = 2 ∞ M n ( n − 1 ) r n + ∑ n = 2 ∞ M 2 n 2 ( n − 1 ) 2 r 2 n .

By the similar argument being used in the proof of the previous theorems, it is easy to show that H 3 ( 0 ) H 3 ( 1 ) < 0 and d d r ( H 3 ( r ) ) > 0 in ( 0 , 1 ) . Hence, H 3 ( r ) being continuous and monotone increasing, r p , M is the unique root of the equation H 3 ( r ) = 0 in ( 0 , 1 ) .

Thus, we have

(3.10) r p , M + M 2 ( r p , M ) 2 p + ∑ n = 2 ∞ M n ( n − 1 ) ( r p , M ) n + ∑ n = 2 ∞ M 2 n 2 ( n − 1 ) 2 ( r p , M ) 2 n = r p , M + M 2 ( r p , M ) 2 p + M ( r p , M − ( r p , M − 1 ) log ( 1 − r p , M ) ) + M 2 ( − 3 ( r p , M ) 2 + 2 ( ( r p , M ) 2 − 1 ) log ( 1 − ( r p , M ) 2 ) + ( 1 + ( r p , M ) 2 ) Li 2 ( ( r p , M ) 2 ) ) = 1 − M 2 .

Consider the function f = f M defined by (3.3). Using (3.1) and (3.10) for f = f M and r > r p , M , it can be shown that

∣ f ( r p , M ) ∣ p + ∑ n = 2 ∞ ( ∣ a n ∣ + ∣ b n ∣ ) r n + ∑ n = 2 ∞ ( ∣ a n ∣ + ∣ b n ∣ ) 2 r 2 n > d ( f ( 0 ) , ∂ f ( D ) ) ,

which shows that r p , M is best possible. This completes the proof.□

Remark 5

While examining the roots r p , M of equation (3.9), several intriguing observations come to light.

Table 5 shows radius data for some specific values of M and p in Theorem (3.6).
Figure 5 clearly illustrates different situations of r p , M .
When p = 1 and M = 1 , we obtain that r m i n = 0.358143 . Moreover, when p → ∞ and M → 0 , we obtain that r p , M → 1 .

Table 5

Roots r p , M of (3.6) for different values of k and M

M	0.2	0.4	0.6	0.8	1.0
r 1 , M	0.756680	0.617611	0.515262	0.431503	0.358143
r 5 , M	0.873153	0.781823	0.708167	0.644363	0.585636
r 10 , M	0.893369	0.814365	0.749851	0.693220	0.639704
r 20 , M	0.904360	0.833072	0.755118	0.724470	0.675943

$Figure 5 Graph of particular r p , M {r}_{p,M} in Remark 5: (a) p = 1 p=1 , (b) p = 5 p=5 , (c) p = 10 p=10 , and (d) p = 20 p=20 .$

Figure 5

Graph of particular r p , M in Remark 5: (a) p = 1 , (b) p = 5 , (c) p = 10 , and (d) p = 20 .

Theorem 3.7

Let f = h + g ¯ ∈ ℬ H 0 ( M ) be given by (1.4) with 0 < M ≤ 1 . Then,

ℳ f ( r ) + J f ( z ) ∣ z ∣ ≤ d ( f ( 0 ) , ∂ f ( D ) ) ,

for ∣ z ∣ = r ≤ r M , where r M is the unique root of the equation:

(3.11) r + M ( r + ( 1 − r ) log ( 1 − r ) ) + r ( 1 + M r ) = 1 − M 2

in ( 0 , 1 ) . The radius r M is the best possible.

Proof

According to Theorem A, if f = h + g ¯ ∈ ℬ H 0 ( M ) with equality for the function f ( z ) = z + M 2 z 2 , then

(3.12) J f ( z ) ≤ 1 + M ∣ z ∣ .

Therefore, considering Lemmas 2.1 and 2.2, along with equation (3.12), this leads to

(3.13) ℳ f ( r ) + J f ( z ) ∣ z ∣ ≤ r + ∑ n = 2 ∞ ( ∣ a n ∣ + ∣ b n ∣ ) r n + ∣ z ∣ ( 1 + M ∣ z ∣ ) ≤ r + ∑ n = 2 ∞ M n ( n − 1 ) r n + r ( 1 + M r ) = r + M ( r + ( 1 − r ) log ( 1 − r ) ) + r ( 1 + M r ) .

It is easy to see that M f ( r ) + J f ( z ) ∣ z ∣ ≤ d ( f ( 0 ) , ∂ f ( D ) ) holds for ∣ z ∣ = r ≤ r M , where r M is the unique root of the equation in (3.11).

For the function f = f M and ∣ z ∣ = r M , a simple computation using (3.1) shows that

r M + ∑ n = 2 ∞ ( ∣ a n ∣ + ∣ b n ∣ ) r M n + ∣ z ∣ ( 1 + M ∣ z ∣ ) = r M + ∑ n = 2 ∞ M n ( n − 1 ) r M n + r M ( 1 + M r M ) = r M + M ( r M + ( 1 − r M ) log ( 1 − r M ) ) + r M ( 1 + M r M ) = 1 − M 2 = d ( f ( 0 ) , ∂ f ( D ) ) .

Therefore, the radius r M is the best possible. This completes the proof.□

Remark 6

Table 6 presents the values of the Bohr radius r M for various values of M as specified in Theorem 3.7. As depicted in Figure 6, it is evident that the radius falls in the range 0.214551 ≤ r M < 1 2 for different values of 0 < M ≤ 1 .

Table 6

Roots r of (3.11) for different values of M

M	0.01	0.2	0.4	0.6	0.8	1.0
r M	0.495521	0.421716	0.359355	0.306125	0.258559	0.214551

$Figure 6 Graph of r M {r}_{M} when M = 0.01 , 0.2 , 0.4 , 0.6 , 0.8 M=0.01,0.2,0.4,0.6,0.8 , and 1.0.$

Figure 6

Graph of r M when M = 0.01 , 0.2 , 0.4 , 0.6 , 0.8 , and 1.0.

Acknowledgement

The authors would like to thank the referee for his/her valuable comments for improving this article.

Funding information: This work was supported by the National Natural Science Foundation of China (Grant No. 11961013).
Author contributions: All authors contributed to each part of this work equally and read and approved the final manuscript.
Conflict of interest: The authors state no conflict of interest.
Data availability statement: No data, models, or code was generated or used during the study.

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Received: 2023-07-09

Revised: 2023-09-13

Accepted: 2023-10-11

Published Online: 2023-12-01

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https://doi.org/10.1515/math-2023-0141

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stable starlike harmonic mappings; Bohr radius; Bohr’s inequality; Bohr-Rogosinski’s inequalities

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