Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 31 No. 5s (2024) 

 

540 
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A Class of Analytic Functions with respect to Symmetric Points Involving 

Multiplicative Derivative 

 

Kadhavoor R. Karthikeyan1, Seetharam Varadharajan2 

1Department of Applied Mathematics and Science, College of Engineering, National University of Science & 

Technology, CPO Seeb 111, Al Hail, Muscat, Oman. Email: karthikeyan1979@nu.edu.om 

2Mathematics Section, Department of Information Technology, University of Technology and Applied Sciences - Al 

Mussanah, Oman. Email: svrajanram@gmail.com 

 

Article History: 

Received: 18-05-2024 

Revised: 20-06-2024 

Accepted: 11-07-2024 

Abstract:  

Here we explore the behaviour and deviations of the geometric properties of a class of 

univalent functions when the classical derivative is replaced with a multiplicative 

derivative. The primary question that we will be addressing here is that given a more 

versatile calculus of Newton and Euler, why we need a study involving such a restrictive 

calculus so called as multiplicative calculus. Precisely, we introduce and study a new 

subclass of analytic function with respect to symmetric points using multiplicative 

derivative. We obtain the estimates for the initial coefficients and Fekete-Szegő 

inequalities of the same. We have included some examples to establish the inclusion and 

closure properties of our defined class. Further, we obtain the logarithmic and inverse 

coefficients for the defined function class. 

Keywords: multiplicative calculus, starlike function, convex function, close-to-convex 

function, subordination. 

1. Introduction 

For 𝒰 = {𝜔 ∈ ℂ; |𝜔| < 1}, we let 𝒜 to denote the class of functions analytic with normalization 

𝜑(0) = 0 = 𝜑′(0) − 1. We denote the classes of starlike and convex function by 𝒮∗(𝛾) and 𝒞(𝛾) 

respectively.  It is well-known that 𝒮∗(𝛾) and C(γ) satisfies the condition 

𝑅𝑒 (
𝜔𝜑′(𝜔)

𝜑(𝜔)
) > 𝛾      𝑎𝑛𝑑  𝑅𝑒 (1 +

𝜔𝜑′′(𝜔)

𝜑′(𝜔)
) > 𝛾, (𝜔 ∈ 𝒰; 0 ≤ 𝛾 < 1), 

respectively. Let 𝒫 signify the category of functions that are analytic in 𝒰 with 𝑝(0) = 1 and 

𝑅𝑒{𝑝(𝜔)} > 0 for all 𝜔 ∈ 𝒰. Let 𝒮 denote the class of functions 𝜑 ∈ 𝒜 which are univalent in 𝒰. The 

class 𝒮 is not preserved under even the most basic operations like addition or subtraction. However, 

the class is preserved under 𝑘 −root transformation. It is well known that if 𝜑 ∈ 𝒜 is in 𝒮, then 

$[𝜑(𝜔𝑘)]
1

𝑘 , (𝑘 is a positive integer) is also in 𝒮. Refer to [9, pg. 18] for the formal definition of 𝑘-

symmetric function.  For every integer 𝑘, let 𝜑𝑘(𝜔) be defined by the following equality 

𝜑𝑘(𝜔) =
1

𝑘
∑

𝜑(𝜀𝜈 𝜔)

𝜀𝜈

𝑘−1

𝜈=0

, (𝜑 ∈ 𝒜).                            (1.1) 

From (1.1), we see that 𝜑𝑘(𝜔) satisfies the linearity conditions. Sakaguchi [22] defined the class 

𝒮𝑠
∗(𝛾), the class of function starlike with respect to symmetric points as follows 



Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 31 No. 5s (2024) 

 

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𝑅𝑒 (
2𝜔𝜑′(𝜔)

𝜑(𝜔) − 𝜑(−𝜔)
) > 𝛾, (𝜔 ∈ 𝒰; 0 ≤ 𝛾 < 1) . 

The functions belonging to the class 𝒮𝑠
∗(𝛾) are univalent (see [22]).  Extending the Sakaguchi class of 

starlike function, the class of starlike functions with respect to 𝑘 −symmetric points denoted by 𝒮𝑠
𝑘(𝛾) 

was introduced and is known to satisfy the analytic characterization 

𝑅𝑒 (
2𝜔𝜑′(𝜔)

𝜑𝑘(𝜔)
) > 𝛾, (𝑘 = 1, 2, 3, … ) ,                                (1.2) 

where 𝜑𝑘(𝜔) =
1

𝑘
∑

𝜑(𝜀𝜈 𝜔)

𝜀𝜈
𝑘−1
𝜈=0 , (𝜑 ∈ 𝒜).  For developments and study of various subclasses of 

analytic functions with respect to symmetric points, refer to [12, 13, 23, 24, 25, 26, 27, 28]. 

Bashirov, Kurpinar and őzyapĭ in [5] (also see [6, 7, 21]) studied the properties of a calculus titled 

Multiplicative calculus which has been a useful mathematical tool in economics and finance. For a 

positive real valued function 𝜑:ℛ →   ℛ, the multiplicative derivative 𝜑∗ is defined as follows 

𝜑∗(𝑥) = lim
ℎ→0

(
𝜑(𝑥 + ℎ)

𝜑(𝑥)
)

1
ℎ

= 𝑒
𝜑′(𝑥)
𝜑(𝑥) = 𝑒[ln𝜑(𝑥)]′ 

where 𝜑′(𝑥) is the ordinary derivative. The ∗-derivative of 𝜑 at 𝜔 belonging to a small neighbourhood 

of a domain in a complex plane where 𝜑 is non-vanishing differentiable, is given by 

𝜑∗(𝜔) = 𝑒
𝜑′(𝜔)
𝜑(𝜔)   𝑎𝑛𝑑 𝜑∗(𝑛)(𝜔) = 𝑒

[
𝜑′(𝜔)
𝜑(𝜔)

]

(𝑛)

,   𝑛 = 1, 2, …. 

Influenced by the definition of multiplicative derivative, recently Karthikeyan and 

Murugusundaramoorthy in [10] introduced and studied a class of analytic functions ℛ(𝜒) satisfying 

the subordination condition 

𝜔 𝑒
ω2φ′(ω)

φ(ω)
 

φ(ω)
≺ 𝜒(𝜔)                                                                                              (1.3) 

  where 𝜒 ∈ 𝒫 and 𝜒(𝒰) is symmetric with respect to the real axis which has a series expansion of the 

form                                                         

𝜒(𝜔) = 1 + 𝐿1𝜔 + 𝐿2𝜔
2 + 𝐿3𝜔

3 + ⋯ , (𝐿1 >  0;   𝜔 ∈ 𝒰).                      (1.4)      

The class is non-empty and possess good geometrical implications but it does not reduce to well-

known subclasses of 𝒮. For the detailed analysis and closure properties of the class ℛ(𝜒), refer to [10, 

11]. 

Throughout this paper, we let 

Γ𝑛,𝑘 =
1

𝑘
∑ [exp

2𝜋 𝑖

𝑘
]
(𝑛−1)𝜈𝑘−1

𝜈=0

 

and  

𝜑𝑘(𝜔) = 𝜔 + ∑ Γ𝑛,𝑘𝑎𝑛𝜔𝑛

∞

𝑛=2

. 



Communications on Applied Nonlinear Analysis 

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Motivated by ℛ(𝜒), here we will introduce and study a new subclass of starlike functions with respect 

to symmetric points. The definition of the new function class are as follows. 

Definition 1.1. Let 𝑘 be chosen such that Γ𝑛,𝑘 = (𝑛 − 1). A function 𝜑 ∈ 𝒜 is said to be in ℳ𝑘(𝜒), 

if it satisfies the following condition 

ω F∗(ω)

𝑒 φk(ω)
≺ 𝜒(𝜔),      (ω ∈ 𝒰;  e =\exp(1))                              (1.5) 

where 𝐹∗(𝜔) = 𝑒
ωφ′(ω)

φ(ω) , 𝜑𝑘(𝜔) =
1

𝑘
∑

𝜑(𝜀𝜈 𝜔)

𝜀𝜈
𝑘−1
𝜈=0 ,  𝜒 ∈ 𝒫  and 𝜒(𝒰) is defined as in (1.4). 

The class  ℳ𝑘(𝜒) has been defined by replacing the classical derivative with a multiplicative derivative 

in (1.2). Note that 𝑘 = 1 is not admissible in ℳ𝑘(𝜒), so the class of function 𝜑 ∈ 𝒜 satisfying 

𝑅𝑒 (
𝜔 𝐹∗(𝜔)

𝑒 𝜑(𝜔)
) > 0, (𝐹∗(𝜔) = 𝑒

ωφ′(ω)
φ(ω) ) 

fails to exist. The reason for imposing 𝑘 ≠  1 in ℳ𝑘(𝜒)  is that we would be unable to work within the 

existing framework, since the requirement of the condition of 𝐿1 to be non-zero would be violated. 

Alternatively, we will now define a class which would be defined for 𝑘 = 1. 

Definition 1.2. Let 𝑘 be chosen such that Γ𝑛,𝑘 ≠ 0. A function 𝜑 ∈ 𝒜 is said to be in ℒ𝑘(𝜒), if it 

satisfies the following condition 

ω 𝑒
𝜔2φ′(ω)

φ(ω)

 φk(ω)
≺ 𝜒(𝜔),                               (ω ∈ 𝒰),                              (1.6) 

where 𝜑𝑘(𝜔) is defined as in (1.1), 𝜒 ∈ 𝒫 and 𝜒(𝒰)$ is defined as in (1.4). 

From the study of [10], we find that classes involving the multiplicative derivative does not have any 

well-known classes as its special cases. But these classes had very good geometric behaviour when 

compared to various other subclasses of analytic functions. 

Letting 𝑘 = 2 and 𝜒(𝜔) =
1+𝜔

1−𝜔
 in (1.5), we get the following familiar analytic characterizations 

    𝑅𝑒 (
2𝜔 𝑒

𝜔𝜑′(𝑤)
𝜑(𝜔)

−1
 

𝜑(𝜔)−𝜑(−𝜔)
) > 0.                                                (1.7) 

Notice that the expression in (1.7) is similar to the analytic characterization of 𝒮𝑠
∗(0). Also letting 𝑘 =

1 in definition 1.2, the class ℒ𝑘(𝜒) reduces to the class ℛ(𝜒) studied by Karthikeyan and 

Murugusundaramoorthy in [10]. 

2. Coefficients Inequalities Of Functions In 𝓜𝒌(𝝌) And 𝓛𝒌(𝝌) 

Now we will find the solution to the Fekete-Szegő problem for 𝜑 ∈  ℳ𝑘(𝜒). 

Lemma 2.1 [15] If 𝑑(𝜔) = 1 + ∑ 𝑑𝑘
∞
𝑘=1 𝜔𝑘 ∈ 𝒫, and 𝜌 is complex number, then 

|𝑑2 − 𝜌 𝑑1
2| ≤ 2max{1; |2𝜌 − 1|}, 



Communications on Applied Nonlinear Analysis 

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and the result is sharp. 

Theorem 2.1. If 𝜑(𝜔) ∈  ℳ𝑘(𝜒), then we have 

|𝑎2| ≤
𝐿1

|1 − Γ2,𝑘|
                                                                                                            (2.1) 

| 𝑎3| ≤
𝐿1

|2 − Γ3,𝑘|
max {1; |

𝐿2

𝐿1
− 𝐿1

(Γ2,𝑘
2 − Γ2,𝑘 −

1
2)

(1 − Γ2,𝑘)
2 |}                                   (2.2) 

and for all 𝜌 ∈ ℂ 

|𝑎3 − 𝜌 𝑎2
2| ≤

𝐿1

|2 − Γ3,𝑘|
max {1; |

𝐿2

𝐿1
− 𝐿1

(Γ2,𝑘
2 − Γ2,𝑘 −

1
2)

(1 − Γ2,𝑘)
2 −

𝐿1𝜌(2 − Γ3,𝑘)

(1 − Γ2,𝑘)
2 |}.       (2.3)                   

The inequality is sharp for each 𝜌 ∈ ℂ. 

Proof. As 𝜑 ∈  ℳ𝑘(𝜒), by (1.5) we have 

𝜔 𝐹∗(𝜔)

𝑒 𝜑𝑘(𝜔)
= 𝜒[𝑤(𝜔)].                                                                                          (2.4) 

Thus, let 𝜗 ∈  𝒫 be of the form 𝜗(𝜔) = 1 + ∑ 𝜗𝑘𝜔
𝑘 ∞

𝑘=1  and defined by 

𝜗(𝜔) =
1 + 𝑤(𝜔)

1 − 𝑤(𝜔)
, 𝜔 ∈ 𝒰 . 

On computation, the right hand side of (2.4) 

𝜒[𝑤(𝜔)] = 1 +
𝜗1𝐿1

2
𝜔 +

𝐿1

2
[𝜗2 −

𝜗1
2

2
(1 −

𝐿2

𝐿1
)]𝜔2 + ⋯.                (2.5) 

The left hand side of (2.4) will be of the form 

𝜔 𝐹∗(𝜔)

𝑒 𝜑𝑘(𝜔)
= 1 + 𝑎2[1 − Γ2,𝑘]𝜔 + [(Γ2,𝑘

2 − Γ2,𝑘 −
1

2
) 𝑎2

2 + (2 − Γ3,𝑘)𝑎3]𝜔2 + ⋯.   (2.6) 

From (2.5) and (2.6), we obtain  

 𝑎2 =
1

(1 − Γ2,𝑘)
[
𝜗1𝐿1

2
]                                                                                (2.7) 

and  

 𝑎3 =
𝐿1

2(2 − Γ3,𝑘)
[𝜗2  −

𝜗1
2

2
(1 −

𝐿2

𝐿1
− 𝐿1

(Γ2,𝑘
2 − Γ2,𝑘 −

1
2)

(1 − Γ2,𝑘)
2 )]                    (2.8)  

Equations (2.1) can be obtained by applying the well-known result of |𝜗1| ≤  2 in (2.7). Applying 

Lemma 2.1 in (2.8), we get (2.2). 

 



Communications on Applied Nonlinear Analysis 

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Now to prove the Fekete-Szegő inequality for the class ℳ𝑘(𝜒), we consider 

|𝑎3 − 𝜌 𝑎2
2| = |

𝐿1

2(2 − Γ3,𝑘)
[𝜗2  −

𝜗1
2

2
(1 −

𝐿2

𝐿1
− 𝐿1

(Γ2,𝑘
2 − Γ2,𝑘 −

1
2)

(1 − Γ2,𝑘)
2 )] −

𝜌 𝜗1
2 𝐿1

2

4(1 − Γ2,𝑘)
2| 

= |
𝐿1

2(2 − Γ3,𝑘)
[𝜗2  −

𝜗1
2

2
(1 −

𝐿2

𝐿1
− 𝐿1

(Γ2,𝑘
2 − Γ2,𝑘 −

1
2)

(1 − Γ2,𝑘)
2 +

𝐿1𝜌(2 − Γ3,𝑘)

(1 − Γ2,𝑘)
2 )]|. 

Using the triangle inequality and Lemma 2.1 in the above equality, we can obtain (2.3). 

Let 𝑘 = 2 in Theorem 2.1, we have the following. 

Corollary 2.1. Let 𝜑 ∈ ℳ2(𝜒). Then, 

|𝑎2| ≤  𝐿1,                           |𝑎3 | ≤  𝐿1 max {1; |
𝐿2

𝐿1 
+

𝐿1

2
|}  

and for a complex number 𝜌, 

|𝑎3 − 𝜌 𝑎2
2| ≤  𝐿1 max {1; |

𝐿2

𝐿1 
+

𝐿1

2
(1 − 2𝜌)| } 

The inequality is sharp for each 𝜌 ∈ ℂ. 

Proof.  By the definition of φk(ω), we have 

  𝜑𝑘(𝜔) =
1

𝑘
∑

𝜑(𝜀𝜈 𝜔)

𝜀𝜈

𝑘−1

𝜈=0

= 𝜔 + ∑ Γ𝑛,𝑘 𝑎𝑛

∞ 

𝑛=2

𝜔𝑛, 

where Γ𝑛,𝑘 =
1

𝑘
∑ [exp (

2𝜋𝑖

𝑘
) ]

(𝑛−1)𝜈
𝑘−1
𝜈=0 . It can be easily seen that 

Γ2,2 =
1

2
∑[exp(𝜋𝑖)]𝜈
1

𝜈=0

= 0, Γ3,2 =
1

2
∑[exp(𝜋𝑖)]2𝜈

1

𝜈=0

= 1. 

Substituting the above expression in (2.1), (2.2) and (2.3), we obtain the assertion of the corollary. 

Fixing 𝜒(𝜔) to be well-known conic regions, we can obtain several applications of our result. But here 

we will restrict to pointing out the case when 𝜒(𝜔) is known to be extremal. 

Letting 𝜒(𝜔) =
1+𝜔

1−𝜔
 in Corollary 2.1, we get 

Corollary 2.2. Let 𝜑 ∈ 𝒜 satisfy the condition 

 

 𝑅𝑒 (
2𝜔 𝑒

𝜔𝜑′(𝜔)
𝜑(𝜔)

−1

𝜑(𝜔) − 𝜑(−𝜔)
) > 0. 

Then, 

|𝑎2| ≤  2,      |𝑎3| ≤  4 



Communications on Applied Nonlinear Analysis 

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and for a complex number 𝜌, 

|𝑎3 − 𝜌 𝑎2
2| ≤ 2max{1;  2|1 − 𝜌|} 

Theorem 2.2. If 𝜑(𝜔) ∈ ℒ𝑘(𝜒), then we have 

|𝑎2| ≤
1

|Γ2,𝑘|
|𝐿1 + 1|.                                                                    (2.9) 

|𝑎3| ≤
𝐿1

|Γ3,𝑘|
{max {1, |

𝐿2

𝐿1
− 𝐿1|} + |

1

Γ2,𝑘
+ 1| +

3

2|𝐿1|
}        (2.10) 

and for all 𝜌 ∈ ℂ 

|𝑎3 − 𝜌 𝑎2
2 | ≤

𝐿1

|Γ3,𝑘|
 

[max {1, |
𝐿2

𝐿1
− 𝐿1 +

𝜌𝐿1Γ3,𝑘

Γ2,𝑘
2 |} + |

1

Γ2,𝑘
+ 1 −

2𝜌Γ3,𝑘

Γ2,𝑘
2 | +

1

2|𝐿1|
|3 − −

2𝜌Γ3,𝑘

Γ2,𝑘
2 |]   (2.11)  

The inequality is sharp for each 𝜌 ∈ ℂ. 

Proof. Expanding the left hand side of (1.6) and simplifying the expansion we get 

𝜔 𝑒
𝜔2𝜑′(𝜔)

𝜑(𝜔)

 𝜑𝑘(𝜔)
=  1 + (1 − 𝑎2Γ2,𝑘)𝜔 + (

1

2
+ 𝑎2 − 𝑎2Γ2,𝑘 + 𝑎2

2Γ2,𝑘
2 − 𝑎3Γ3,𝑘)𝜔2 + ⋯.      (2.12) 

Given 𝜑 ∈ ℒ𝑘   (𝜒), so the right hand side of the expansion of (1.6) is the same as (2.5). 

From (2.12) and (2.5), we obtain  

𝑎2 = −
1

Γ2,𝑘
[
𝜗1𝐿1

2
− 1]                                                                                          (2.13) 

and  

      𝑎3 = −
1

Γ3,𝑘
{
𝐿1

2
 [𝜗2 −

𝜗1
2

2
(1 −

𝐿2

𝐿1
+ 𝐿1 )] +

𝜗1𝐿1

2
[

1

Γ2,𝑘
+ 1] −

3

2
}.            (2.14) 

Equations (2.9) can be obtained by applying the well-known result of |𝜗1| ≤  2 in (2.13) Applying 

Lemma 2.1 in (2.14), we get (2.10). 

Now to prove the Fekete-Szegő inequality for the class ℒ𝑘(𝜒), we consider 

|𝑎3 − 𝜌 𝑎2
2| = |

1

Γ3,𝑘
{
𝐿1

2
 [𝜗2 −

𝜗1
2

2
(1 −

𝐿2

𝐿1
+ 𝐿1 )] +

𝜗1𝐿1

2
[

1

Γ2,𝑘
+ 1] −

3

2
}

+
𝜌

Γ2,𝑘
2 [

𝜗1
2 𝐿1

2

4
− 𝜗1𝐿1 + 1]|. 

Using the triangle inequality and Lemma 2.1 in the above equality, we can obtain (2.11). 

 



Communications on Applied Nonlinear Analysis 

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Corollary 2.3. [10] If 𝜑(𝜔) ∈  ℒ1(𝜒), then we have 

|𝑎2| ≤  1 + 𝐿1. 

|𝑎3| ≤ 𝐿1 [max {1, |
𝐿2

𝐿1
− 𝐿1|} +

3

2|𝐿1|
+ 2] 

and for all 𝜌 ∈ ℂ 

|𝑎3 − 𝜌 𝑎2
2| ≤  𝐿1 [max {1, |

𝐿2

𝐿1
− 𝐿1(1 − 𝜌)|} +

1

2|𝐿1|
|3 − 2𝜌| + 2|1 − 𝜌|]. 

The inequality is sharp for each 𝜌 ∈ ℂ. 

Letting 𝜒(𝜔) =
1+𝜔

1−𝜔
 in Corollary 2.3, we get 

Corollary 2.4 Let 𝜑 ∈ 𝒜 satisfy the condition 

𝑅𝑒 (
𝜔𝑒

𝜔2𝜑′(𝜔)
𝜑(𝜔)

𝜑(𝜔)
) > 0. 

Then, 

|𝑎2| ≤  3,   |𝑎3| ≤
15

2
 

and for a complex number 𝜌, 

|𝑎3 − 𝜌 𝑎2
2| ≤  2 [max{1, |2𝜌 − 1|} + 2|1 − 𝜌| +

1

4
|3 − 2𝜌|] . 

The inequality is sharp for each 𝜌 ∈ ℂ. 

3. Coefficient Estimates For The Inverse Functions. 

In this section, we will find the coefficient estimates for the inverse functions of 𝜑 belonging to the 

classes ℳ𝑘(𝜒) and ℒ𝑘(𝜒). Refer to [14 18] for its relevance and application in the field of univalent 

function theory. The following result would help us to obtain the coefficient estimates for 𝜑−1 

(provided it exists), form the coefficient estimates of 𝜑. 

Lemma 3.1. [9, p. 56]  If the function 𝜑 ∈ 𝒜 and  𝜑−1 = 𝑔(𝑤) given by 

𝑔(𝑤) = 𝑤 + ∑ 𝑏𝑘

∞

𝑘=2

𝑤𝑘                                                     (3.1) 

are inverse functions, then for 𝑘 ≥  2 



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𝑏𝑘

=
(−1)𝑘+1

𝑘!

[
 
 
 
 

𝑘𝑎2 1            0                ⋯                                        0
2𝑘𝑎3 (𝑘 + 1)𝑎2 2                   ⋯                                  0

3𝑘𝑎4

⋮
(𝑘 − 1)𝑘𝑎𝑘

(2𝑘 + 1)𝑎3

⋮
[𝑘(𝑘 − 2) + 1]𝑎𝑘−1

(𝑘 + 1)𝑎2    ⋯                                   0
⋮                      ⋮                          (𝑘 − 2)
[𝑘(𝑘 − 3) + 2]𝑎𝑘−2 ⋯ (2𝑘 − 2)𝑎2 ]

 
 
 
 

            (3.2) 

The elements of the determinant in (3.2) are given by  

Θij = {
[(i − j + 1)k + j − 1]ai−j+2,         if   𝑖 + 1 ≥ 𝑗 

0,                                                       if   𝑖 + 1 < 𝑗
. 

The functions in ℳ𝑘(𝜒) need not be univalent, but since 𝜑′(0) = 1 ≠ 0 for all 𝜑 ∈ ℳ𝑘(𝜒) and 

𝜑(0) = 0, there exist an inverse function in some small disk with center at 𝑤 = 0. 

Theorem 3.1. Let 𝜑 ∈ ℳ𝑘(𝜒) and let 𝜑−1 be the inverse of 𝜑 defined by 

𝜑−1(𝑤) = 𝑤 + ∑ 𝑏𝑘

∞

𝑘=2

𝑤𝑘 , (| 𝑤| < 𝑟;  𝑟 ≥
1

4
), 

then 

| 𝑏2| ≤
𝐿1

| 1 − Γ2,𝑘|
 

|𝑏3| ≤
𝐿1

| 2 − Γ3,𝑘|
max {1; |

𝐿2

𝐿1
−

𝐿1 (Γ2,𝑘
2 − Γ2,𝑘 −

1
2)

(1 − Γ2,𝑘)
2 −

2𝐿1(2 − Γ3,𝑘)

(1 − Γ2,k)
2

 

|} 

and for a complex number 𝜏, 

|b3 − τ a2
2 | ≤

𝐿1

| 2 − Γ3,𝑘|
max {1; |

𝐿2

𝐿1
−

𝐿1 (Γ2,𝑘
2 − Γ2,𝑘 −

1
2)

(1 − Γ2,𝑘)
2 −

(𝜏 − 2)𝐿1(2 − Γ3,𝑘)

(1 − Γ2,k)
2

 

|} 

Proof. From 𝜑(𝜔) = 𝜔 + ∑ 𝑎𝑛𝜔𝑛∞
𝑛=2  and (3.1), we have  

𝑏2 = −𝑎2     𝑎𝑛𝑑    𝑏3 = 2𝑎2
2 − 𝑎3. 

The estimate for | 𝑏2| = | 𝑎2| follows immediately from (2.7). Letting 𝜌 = 2 in (2.3), we get the 

estimate |𝑏3|. 

To find the Fekete-Szegő inequality for 𝜑−1, consider 

|𝑏3 − 𝜏 𝑏2
2| = |2𝑎2

2 − 𝑎3 − 𝜏 𝑎2
2| = |𝑎3 − (𝜏 − 2)𝑎2

2|. 

Changing 𝜌 = (𝜏 − 2) in the (2.3), we get the desired result. 

Analogous to the results obtained in Theorem 3.2, we can easily get the following result. 

Theorem 3.2 Let 𝜑 ∈ ℒ𝑘(𝜒)$ and let 𝜑−1 be the inverse of 𝜑 defined by  

𝜑−1(𝑤) = 𝑤 + ∑ 𝑏𝑘

∞

𝑘=2

𝑤𝑘 , (| 𝑤| < 𝑟;  𝑟 ≥
1

4
), 



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then$$ 

| 𝑏2| ≤
1

| Γ2,𝑘|
[𝐿1 + 1], 

|𝑏3| ≤
𝐿1

| Γ3,𝑘|
[max {1; |

𝐿2

𝐿1
− 𝐿1 +

2𝐿1Γ3,𝑘

Γ2,𝑘
2 |} + |1 +

1

Γ2,𝑘
−

4Γ3,𝑘

Γ2,𝑘
2 | +

1

2| 𝐿1|
|3 −

4Γ3,𝑘

Γ2,𝑘
2 |] 

and for a complex number 𝜏, 

|𝑏3 − 𝜏 𝑎2
2| ≤

𝐿1

| Γ3,𝑘|
[max {1; |

𝐿2

𝐿1
− 𝐿1 +

(𝜏 − 2)𝐿1Γ3,𝑘

Γ2,𝑘
2 |} + |1 +

1

Γ2,𝑘
−

2(𝜏 − 2)Γ3,𝑘

Γ2,𝑘
2 |

+
1

2| 𝐿1|
|3 −

2(𝜏 − 2)Γ3,𝑘

Γ2,𝑘
2 |]. 

4. Logarithmic Coefficients 

Milin in [16] studied the properties of the logarithmic coefficients to obtain the bounds of the Taylor 

coefficients of univalent functions. The Milin conjuncture about the inequalities of the logarithmic 

coefficients garnered the attention several researchers in those period of time, because proving Milin 

conjuncture would imply proving Robertson conjecture and the Bieberbach conjecture. Refer to 

Ponnusamy et al. [18, 19 20] and [1, 2, 3, 4, 17] for the detailed study on properties and significance 

of the logarithmic coefficients. 

The logarithmic coefficients 𝑑𝑛 of a function 𝜑 ∈ 𝒜 such that 
𝜑(𝜔)

𝜔
≠ 0 for all 𝜔 ∈ 𝒰 is defined by  

            log 𝜑(𝜔)  = 2 ∑ 𝑑𝑛𝜔𝑛

∞

𝑛=1

. 

Note that for all functions 𝜑(𝜔) ∈ ℳ𝑘(𝜒) and ℒ𝑘(𝜒), the relation (4.1) is well-defined. 

Theorem 4.1. If 𝜑(𝜔) ∈ ℳ𝑘(𝜒),  with the logarithmic coefficients given by (4.1), then, 

|𝑑1| ≤
𝐿1

2| 1 − Γ2,𝑘|
 

|𝑑2| ≤
𝐿1

2| 2 − Γ3,𝑘|
max {1; |

𝐿2

𝐿1
−

𝐿1 (Γ2,𝑘
2 − Γ2,𝑘 −

1
2)

(1 − Γ2,𝑘)
2 −

𝐿1(2 − Γ3,𝑘)

2(1 − Γ2,𝑘)
2|}. 

For 𝜇 ∈ ℂ, we have 

|𝑑2 − 𝜇𝑑1
2| ≤

𝐿1

2| 2 − Γ3,𝑘|
max {1; |

𝐿2

𝐿1
−

𝐿1 (Γ2,𝑘
2 − Γ2,𝑘 −

1
2)

(1 − Γ2,𝑘)
2 −

𝐿1(1 + 𝜇)(2 − Γ3,𝑘)

2(1 − Γ2,𝑘)
2 |}.      (4.2) 

 

 



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Proof. From 𝜑(𝜔) = 𝜔 + ∑ 𝑎𝑛𝜔𝑛∞
𝑛=2  and equating the first two coefficients of the relation (4.1),  we 

get 

𝑑1 =
𝑎2

2
,    𝑑2 =

1

2
(𝑎3 −

𝑎2
2

2
) . 

Using (2.1) and (2.3) in the above expression, we can find the estimates for 𝑑1 and 𝑑2. To find the 

estimate (4.2), consider 

|𝑑2 − 𝜇 𝑑1
2| =

1

2
[𝑎3 −

1 + 𝜇

2
𝑎2

2] 

Changing 𝜌 =
(1+\𝑚𝑢)

2
 in (2.3) and simplifying, we get the desired result. 

For completeness, we just state the following. 

Theorem 4.2. If 𝜑(𝜔) ∈ 𝐿𝑘(𝜒),  with the logarithmic coefficients given by (4.1), then, 

|𝑑1| ≤
1

2| Γ2,𝑘|
[𝐿1 + 1] 

|𝑑2| ≤
𝐿1

2| Γ3,𝑘|
[max {1; |

𝐿2

𝐿1
− 𝐿1 + 2

𝐿1Γ3,𝑘

Γ2,𝑘
2 |} + |1 +

1

Γ2,k
−

4Γ3,k

Γ2,k
2 | +

1

2| L1|
|3 −

4Γ3,k

Γ2,k
2 |]. 

For 𝜇 ∈ ℂ, we have 

|𝑑2 − 𝜇𝑑1
2| ≤

𝐿1

2| Γ3,𝑘|
[max {1; |

𝐿2

𝐿1
− 𝐿1 +

(1 + 𝜇)𝐿1Γ3,𝑘

Γ2,𝑘
2 |} + |1 +

1

Γ2,k
−

(1 + 𝜇)Γ3,k

Γ2,k
2 |

+
1

2| L1|
|3 −

(1 + 𝜇)Γ3,k

Γ2,k
2 |]. 

Conclusions: The function class ℳ𝑘(𝜒) was defined by replacing the classical derivative with a 

multiplicative derivative in the well-known class of starlike functions with respect to symmetric points. 

The definition ℳ𝑘(𝜒) is not defined for all integers 𝑘. In fact the class exist only for the integers values 

of 𝑘, for which Γ𝑛,𝑘 ≠ (𝑛 − 1). Hence we defined a class ℒ𝑘(𝜒), influenced by the multiplicative 

derivative which will be defined for Γ𝑛,𝑘 = (𝑛 − 1). 

Letting 𝜒 to be a specific conic region and varying parameters involved in the Definitions 1.1 and 1.2, 

the function classes ℳ𝑘(𝜒)  and ℒ𝑘(𝜒) will reduce to classes having good geometry. Our main results 

have wide applications. The classical starlike functions with respect to 𝑘-symmetric points are known 

to be univalent. But the classes ℳ𝑘(𝜒)  and ℒ𝑘(𝜒) are neither a subclass nor a generalized class of 

univalent functions. So, the scope of further research of this paper are to explore the relationship and 

closure properties with the known classes like spirallike, starlike, and convex. 

Conflicts of Interest: Both authors declare that they have no conflict of interest. 

Funding: This research study received no external funding. 

Institutional Review Board Statement: Not applicable. 

Data Availability: No data was used to support this study. 



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