Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 32 No. 2 (2025) 

 

383 
https://internationalpubls.com  

Toeplitz Matrices whose Elements are Coefficients of new Subclasses of 

Analytical Functions 

 

M. Nandeesh1, M. Ruby Salestina2, Archana3, G. Murugusundaramoorthy 𝟒* 

(Nandeesh .M)  1, Department of Mathematics, Bharathi college (Autonomous), Bharathinagara, Maddur Taluk, Mandya 

District, Karnataka, India - 571422. 

E-mail address: nandeesh02@gmail.com ; http://orcid.org/0009-0003-3389-8929 . 

(Ruby Salestina .M)  2, Department of Mathematics, Yuvaraja's College, University of Mysore, Mysuru, Karnataka, India 

- 570005. 

E-mail address: ruby.salestina@gmail.com ; http://orcid.org/0000-0002-3318-2061 . 

(Archana)  3, Department of Mathematics, Bharathi college (Autonomous), Bharathinagara,    Maddur Taluk, Mandya 

District , Karnataka, India – 571422 

E-mail address:  archanap6005@gmail.com   . 

(G. Murugusundaramoorthy)  4 ＊ Department of Mathematics, School of Advanced Science, Vellore Institute of 

Technology, Vellore, Tamil Nadu, India - 632014. 

E-mail address:  gmsmoorthy@yahoo.com ; http://orcid.org/0000-0001-8285-6619. 

 

Article History: 

Received: 03-08-2024 

Revised: 10-09-2024 

Accepted: 20-09-2024    

Abstract:  

In this study, we explore Toeplitz matrices composed of coefficients from new subclasses 

and establish upper limits for the initial four determinants like of these matrices. Our 

findings are innovative and unique, with the only similar results being in recent works by 

Thomas and Halim [1], which pertain to starlike and close - to - convex functions, and by 

Radhika et al. [2], focusing on functions with bounded boundary rotation. Along with we 

have determined the Zalcman, Generalized Zalcman conjecture and Krushkal inequalities 

for some parameters. Keywords: Star-like function, Convex function, Coefficient bounds, 

Univalent functions, Toeplitz matrices, Hankel determinants, Zalcman conjecture, 

Generalized Zalcman conjecture and Krushkal inequalities. 

Keywords: Star - like function, Convex function, Coefficient bounds, Univalent functions, 

Toeplitz matrices, Hankel determinants, Zalcman conjecture, Generalized Zalcman 

conjecture and Krushkal inequalities. 

MSC (2010): 30C45, 33C50, 30C80. 

1. Introduction 

Hankel matrices (and their determinants) hold significant importance in various mathematical fields 

and find numerous practical uses. A closely related concept to Hankel determinants is the Toeplitz 

determinants. Essentially, a Toeplitz matrix can be likened to an inverted Hankel matrix, as Hankel 

matrices have constant entries along their reverse diagonal, while Toeplitz matrices maintain constant 

entries along their diagonal. A comprehensive overview of the applications of Toeplitz matrices in 

both pure and applied mathematics can also be located in reference [7]. They possess excellent 

computational properties and are compatible with a wide range of algorithms and determinant 

computations.  

Let 𝓐 signify the class of functions of the form 

mailto:nandeesh02@gmail.com
http://orcid.org/0009-0003-3389-8929
mailto:ruby.salestina@gmail.com
http://orcid.org/0000-0002-3318-20619
mailto:archanap6005@gmail.com
mailto:gmsmoorthy@yahoo.com


Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 32 No. 2 (2025) 

 

384 
https://internationalpubls.com  

𝑓(𝑧) = 𝑧 +∑  

∞

𝑛=2

 𝑑𝑛𝑧
𝑛. (1.1) 

which are analytic in the open unit disk 𝔻 = {𝑧: 𝑧 ∈ ℂ and |𝑧| < 1}. Further, represent by 𝒮 the class 

of all functions in 𝓐 which are univalent in 𝔻 and normalized by 𝑓(0) = 0 = 𝑓′(0) − 1. Also, an 

significant class of functions will be called 𝒫,𝒫 defines the family of functions 𝜙 with the limitations 

that the image domain of 𝜙 ( 𝜙 𝑖𝑠 𝑎 𝑐onvex function 𝑤𝑖𝑡ℎ 𝑅𝑒(𝜙) > 0 in 𝔻.) is symmetric along the 

real axis and star - like about 𝜙(0) = 1 with 𝜙′(0) > 0. We say that for 𝑓1, 𝑓2 ∈ 𝒜, an 𝑓1 is subordinate 

to 𝑓2 and write 𝑓1(𝑧) ≺ 𝑓2(𝑧), if and only if there exists 𝑤, analytic in 𝔻, such that 𝑤(0) = 0, |𝑤(𝑧)| <

1 for |𝑧| < 1 and 𝑓1(𝑧) = 𝑓2(𝑤(𝑧)). In particular, if 𝑓2 is univalent in 𝔻, then we have the following 

equivalence: 

𝑓1(𝑧) ≺ 𝑓2(𝑧) ⟺ 𝑓1(0) = 𝑓2(0) and 𝑓1(|𝑧| < 1) ⊂ 𝑓2(|𝑧| < 1). (1.2) 

Two of the most important and well - investigated subclass of univalent functions are the class 𝒮∗(𝛼) 

is the class star - like functions of order 𝛼, (0 ≤ 𝛼 < 1) is defined by 

𝒮∗(𝛼) = {𝑓 ∈ 𝓐:Re (
𝑧𝑓′(𝑧)

𝑓(𝑧)
) > 𝛼,  (𝑧 ∈ 𝔻)} . (1.3) 

The class 𝒦(𝛼) ⊂ 𝒮 of convex functions of order 𝛼, (0 ≤ 𝛼 < 1) is defined by 

𝒦(𝛼) = {𝑓 ∈ 𝓐:Re(1 +
𝑧𝑓′′(𝑧)

𝑓′(𝑧)
) > 𝛼,  (𝑧 ∈ 𝔻)} . (1.4) 

The class 𝒱(𝛼) ⊂ 𝒮 of closed - to - convex functions of order 𝛼, (0 ≤ 𝛼 < 1) is defined by 

𝒱(𝛼) = {𝑓 ∈ 𝓐:Re(
𝑧𝑓′(𝑧)

𝑔(𝑧)
) > 𝛼,  (𝑧 ∈ 𝔻)} . (1.5) 

where 𝑔(𝑧) = 𝑧 + ∑𝑛=2
∞  𝑏𝑛𝑧

𝑛 belongs to star - like functions and so on. 

Let 𝒫 be an analytic and univalent function with positive real part in 𝔻, 𝑝(0) = 0, 𝑝′(0) = 

1, Re(𝑝(𝑧)) > 0 and 𝒫 maps the unit disk 𝔻 onto a region of star - like function with respect to 

symmetric points of the real axis. The Taylor series expansion of such that function. 

𝑝(𝑧) = 1 +∑  

∞

𝑛=1

 𝑝𝑛𝑧
𝑛,  |𝑝𝑛| ≤ 2. (1.6) 

where all the coefficients are real and 𝑝1 > 0. Throughout this paper we assume that the function 𝑝 

satisfies the above conditions unless otherwise stated. 

By 𝒮∗(𝑝) and 𝒦(𝑝) we denote the following classes of function 

𝒮∗(𝑝) = {𝑓 ∈ 𝓐:Re (
𝑧𝑓′(𝑧)

𝑓(𝑧)
) ≺ 𝑝(𝑧),  (𝑧 ∈ 𝔻)} . (1.7)

𝒦(𝑝) = {𝑓 ∈ 𝓐:Re (1 +
𝑧𝑓′′(𝑧)

𝑓′(𝑧)
) ≺ 𝑝(𝑧),  (𝑧 ∈ 𝔻)} . (1.8)

 



Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 32 No. 2 (2025) 

 

385 
https://internationalpubls.com  

The classes 𝒮∗(𝑝),𝒦(𝑝) are the extension of classical set of star - like and convex functions (e.g., see 

Ma and Minda [31]). These functions serve as the common source from which these subclasses inherit 

their properties and all took their sources from the class of Caratheòdory function 𝒫. The work of 

Sokól and Stankiewicz [19], introduced a class denoted as 𝒮ℒ∗, which comprises normalized analytic 

functions 𝑓 in 𝔻 satisfying the condition 

|[
𝑧𝑓′(𝑧)

𝑓(𝑧)
]

2

− 1| < 1 

This class is referred to as Sokól - Stankiewicz star-like functions. Additionally, Raza and Malik [17], 

have determined the upper bound of the third Hankel determinant 

𝐻3(1) for the class 𝒮ℒ∗. Furthermore, Sahoo and Patel [18] obtained some upper bound to the second 

Hankel determinant for the class 

ℛ̃ = {𝑓 ∈ 𝓐: |𝑓′(𝑧)2 − 1| < 1,  (𝑧 ∈ 𝔻)}. (1.9) 

Motivated by the above-mentioned works obtained by earlier researchers, Trailokya Panigrahi and 

JanuszSokól [12], introduce the following subclass of analytical function. 

Definition 1.1. A function 𝑓 ∈ 𝓐 is said to be in the class 𝓐𝓡𝝀
∗ , 0 ≤ 𝜆 ≤ 1, if it satisfies the condition 

|[
𝑧𝑓′(𝑧)

(1 − 𝜆)𝑓(𝑧) + 𝜆𝑧
]

2

− 1| < 1,  (𝑧 ∈ 𝔻). (1.10) 

The family 𝓐(𝝀) of new subclasses in analytical functions of type 𝜆; 0 ≤ 𝜆 ≤ 1 provides a transition 

from the class of star - like functions to the class of functions of bounded boundary rotation. To see 

this, we note that for the choice of 𝜆 = 0, we have 𝓐(𝝀) ≡ 𝒮∗(0) ≡ 𝒮∗ the class of star - like functions 

𝑓 ∈ 𝓐, so that ℜ(
𝑧𝑓′

𝑓
) > 0 in 𝔻 for the choice of 𝜆 = 1, we get the family of functions ℛ̃ of functions 

𝑓 ∈ 𝓐, of bounded boundary rotation so that ℜ(𝑓′) > 0 in 𝔻. (For further details see [3].) 

Note that for 𝜆 = 0, the class 𝓐𝓡𝟎
∗ , reduces to the class 𝒮ℒ∗, studied by Raza and Malik [17] and 

while 𝜆 = 1, the class 𝓐𝓡𝟏
∗ , reduces to ℛ̃ studied by Sahoo and Patel [18]. In terms of subordination, 

relation (1.10), can be written 

𝓐(𝝀) =
𝑧𝑓′(𝑧)

(1 − 𝜆)𝑓(𝑧) + 𝜆𝑧
≺ 𝑝(𝑧),  (𝑧 ∈ 𝔻). (1.11) 

In this research paper, we setout on an investigation into the determinants of symmetric Toeplitz 

matrices, where their entries represent the coefficients 𝑎𝑛 of star - like and close - to - convex functions. 

Toeplitz matrices are extensively studied structured matrices with applications in various fields such 

as mathematics, statistics, image processing, quantum mechanics and more (e.g., Ye and Lim [4] ). 

We recall the definition of the Hankel determinant 

𝐻𝑘(𝑛) = |

𝑎𝑛 𝑎𝑛+1 ⋯ 𝑎𝑛+𝑘−1
𝑎𝑛+1 𝑎𝑛+2 ⋯ 𝑎𝑛+𝑘
⋮ ⋮ ⋯ ⋮

𝑎𝑛+𝑘−1 𝑎𝑛+𝑘 ⋯ 𝑎𝑛+2𝑘−2

| . (1.12) 



Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 32 No. 2 (2025) 

 

386 
https://internationalpubls.com  

for example, 

𝐻2(1) = |
𝑎1 𝑎2
𝑎2 𝑎3

| ,  𝐻2(2) = |
𝑎2 𝑎3
𝑎3 𝑎4

| ,  𝐻3(1) = |

𝑎1 𝑎2 𝑎3
𝑎2 𝑎3 𝑎4
𝑎3 𝑎4 𝑎5

| . (1.13) 

and define the symmetric Toeplitz determinant 

𝑇𝑘(𝑛) = |

𝑎𝑛 𝑎𝑛+1 ⋯ 𝑎𝑛+𝑘−1
𝑎𝑛+1 𝑎𝑛 ⋯ 𝑎𝑛+𝑘
⋮ ⋮ ⋯ ⋮

𝑎𝑛+𝑘−1 𝑎𝑛+𝑘 ⋯ 𝑎𝑛

| . (1.14) 

for example, 

𝑇2(2) = |
𝑎2 𝑎3
𝑎3 𝑎2

| ,  𝑇2(3) = |
𝑎3 𝑎4
𝑎4 𝑎3

| ,  𝑇3(2) = |

𝑎2 𝑎3 𝑎4
𝑎3 𝑎2 𝑎3
𝑎4 𝑎3 𝑎2

| ,  𝑇3(1) = |
1 𝑎2 𝑎3
𝑎2 1 𝑎2
𝑎3 𝑎2 1

| (1.15) 

For 𝑓 ∈ 𝓐, the problem of finding the best possible bounds for ‖𝑎𝑛+1| − |𝑎𝑛‖ has a long history [3]. 

It is well - known [3], that ||𝑎𝑛+1| − |𝑎𝑛‖ ≤ 𝐶; however, finding exact values of the constant 𝐶 for 𝓐 

and its subclasses has proved difficult. It is clear from the definition that finding estimates for 𝑇𝑘(𝑛) 

is related to finding bounds for |𝑎𝑛+1 − 𝑎𝑛|. 

The pivotal moment in the exploration of univalent functions occurred in 1985, when Louis de Branges 

successfully proved the renowned Bieberbach conjecture, |𝑎𝑛| = 𝑛 for 𝑛 = 2 [22]. While this marked 

the conclusion of an era, numerous unresolved issues persist, including the notable Zalcman 

conjecture, which pertains to the coefficients 𝑎𝑛. One such is the Zalcman conjecture is 

|𝑎𝑛 − 𝑎2𝑛−1| ≤ (𝑛 − 1)2, (𝑛 ∈ ℕ, 𝑛 ≥ 2). (1.16) 

Formulated in the early 1970s, Krushkal [23]., made significant strides in this direction, employing the 

complex geometry of the universal Teichm ̈ üller space. In 1999, a broader perspective on the 

Generalized Zalcman conjecture was introduced by Ma [24]. The Generalized Zalcman conjecture is 

|𝑎𝑚𝑎𝑛 − 𝑎𝑚+𝑛−1| ≤ (𝑚 − 1)(𝑛 − 1), (𝑚, 𝑛 ∈ ℕ,𝑚 ≥ 2, 𝑛 ≥ 2). (1.17) 

Ma [23] successfully resolved the open problem within the realm of star-like functions and univalent 

functions with real coefficients. Ravichandran and Verma, as documented in [27], also tackled and 

closed the issue for star - like and convex functions of specified order, as well as for functions 

characterized by bounded turning. Ozaki and Nunokawa, as outlined in [25], established the univalence 

of functions within this class, deviating from the conventional characteristics observed in other 

univalent functions. Unlike the broad category of star - like functions, these exhibit unique patterns, 

adding intrigue to their study.  

The class 𝔻, being distinct, has garnered substantial interest over the previous decades. Chapter 12 of 

[28],  provides a comprehensive summary of the noteworthy findings in this field. We have 

|𝑎𝑛
𝑝 − 𝑎2

𝑝(𝑛−1)
| ≤ 2𝑝(𝑛−1) − 2𝑝, (𝑚, 𝑛 ∈ ℕ,𝑚 ≥ 2, 𝑛 ≥ 2). (1.18) 



Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 32 No. 2 (2025) 

 

387 
https://internationalpubls.com  

over the class 𝔻 for the cases 𝑛 = 4, 𝑝 = 1 and 𝑛 = 5, 𝑝 = 1. This inequality was introduced by 

Krushkal and proven for the whole class of univalent functions [23]. 

 

2.  Definitions and Preliminaries 

Lemma 2.1. [17] Let 𝑝 ∈ 𝒫, be given by (1.6), then 

|𝑝𝑛| ≤ 2, ∀𝑛 ∈ ℕ. (2.1) 

and 

|𝑝2 −
1

2
𝑝1
2| ≤ 2 −

1

2
|𝑝1|

2. (2.2) 

Lemma 2.2. [30], [16] 

Let 𝑝 ∈ 𝒫, be given by (1.6), then for some complex valued 𝑥 with |𝑥| ≤ 1, some complex valued 𝜚 

with |𝜚| ≤ 1 and some complex valued 𝜓 with |𝜓| ≤ 1. 

We have 

2𝑝2  = 𝑝1
2 + 𝑥(4 − 𝑝1

2) (2.3)

4𝑝3  = 𝑝1
3 + 2(4 − 𝑝1

2)𝑝1𝑥 − 𝑝1(4 − 𝑝1
2)𝑥2 + 2(4 − 𝑝1

2)(1 − |𝑥|2)𝜚 (2.4)

8𝑝4  = 𝑝1
4 + (4 − 𝑝1

2)𝑥[𝑝1
2(𝑥2 − 3𝑥 + 3) + 4𝑥] (2.4)

 −4(4 − 𝑝1
2)(1 − |𝑥|2)[𝑝(𝑥 − 1)𝜚 + 𝑥‾𝜚2 − 1 − |𝜚|2𝜓]. (2.5)

 

3. Coefficient estimates for Toeplitz determinant 𝓐(𝝀) 

In our first theorem we determinant a sharp bound for the coefficient body 𝑇2(2). 

Theorem 3.1. Let 𝑓 given by (1.1), be in the class 𝓐(𝝀); (0 ≤ 𝜆 ≤ 1). Then we have sharp bound 

|𝑇2(2)| = |𝑎3
2 − 𝑎2

2| ≤
4

(𝜆 + 2)2
max {1, |

−40𝜆3 − 60𝜆2 + 20

(𝜆 + 1)4
|}. 

Proof. First note that by equating the corresponding coefficients in the equation 

𝑧𝑓′(𝑧)

(1−𝜆)𝑓(𝑧)+𝜆𝑧
= 𝑝(𝑧) (3.1)

we get 



Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 32 No. 2 (2025) 

 

388 
https://internationalpubls.com  

𝑎2  =
𝑝1
𝜆 + 1

, (3.2)

𝑎3  =
𝑝1
2(1 − 𝜆)

(𝜆 + 1)(𝜆 + 2)
+

𝑝2
𝜆 + 2

, (3.3)

𝑎4  =
𝑝1
3(1 − 𝜆)2

(𝜆 + 1)(𝜆 + 2)(𝜆 + 3)
+
𝑝1𝑝2(1 − 𝜆)(3 + 2𝜆)

(𝜆 + 1)(𝜆 + 2)(𝜆 + 3)
+

𝑝3
𝜆 + 3

, (3.4)

𝑎5  =
𝑝1
4(1 − 𝜆)3

(𝜆 + 1)(𝜆 + 2)(𝜆 + 3)(𝜆 + 4)
+

𝑝1
2𝑝2(1 − 𝜆)

2(3 + 2𝜆)

(𝜆 + 1)(𝜆 + 2)(𝜆 + 3)(𝜆 + 4)

 +
𝑝1𝑝3(1 − 𝜆)

(𝜆 + 3)(𝜆 + 4)
+

𝑝1
2𝑝2(1 − 𝜆)

2

(𝜆 + 1)(𝜆 + 2)(𝜆 + 4)
+

𝑝2
2(1 − 𝜆)

(𝜆 + 2)(𝜆 + 4)
+

𝑝1𝑝3(1 − 𝜆)

(𝜆 + 1)(𝜆 + 4)

 +
𝑝4
𝜆 + 4

. (3.5)

 

In the view of (3.2) and (3.3), a simple computation leads to 

𝑎3
2 − 𝑎2

2 =
𝑝2
2

(𝜆 + 2)2
+

𝑝1
4(1 − 𝜆)2

(𝜆 + 2)2(𝜆 + 1)2
+

2𝑝2𝑝1
2(1 − 𝜆)

(𝜆 + 2)2(𝜆 + 1)
−

𝑝1
2

(𝜆 + 1)2
. (3.6) 

Note that, by Lemma (2.2), we may write 2𝑝2 = 𝑝1
2 + 𝑥(4 − 𝑝1

2), where without loss of generality, 

we let 0 ≤ 𝑝1 = 𝑝 ≤ 2. Substitute this into the above equation we obtain the 

following quadratic equation in terms of 𝑥. 

𝑎3
2 − 𝑎2

2 =
(4−𝑝2)

2

4(𝜆+2)2
𝑥2 +

𝑝2(4−𝑝2)(𝜆−3)

2(𝜆+2)2(𝜆+1)
𝑥 +

𝑝2[𝑝2(𝜆4−4𝜆3−2𝜆2+12𝜆+9)−4(𝜆+2)2(𝜆+1)2]

4(𝜆+2)2(𝜆+1)4
.      .                     (3.7) 

Using the triangular inequality, we gain 

|𝑎3
2 − 𝑎2

2| ≤
(4−𝑝2)

2

4(𝜆+2)2
+
𝑝2(4−𝑝2)(𝜆−3)

2(𝜆+2)2(𝜆+1 )
 +

𝑝2[𝑝2(𝜆4−4𝜆3−2𝜆2+12𝜆+9)+4(𝜆+2)2(𝜆+1)2]

4(𝜆+2)2(𝜆+1)4
  

                          = Υ(𝑝, 𝜆).                                                                                                                    (3.8) 

Differentiating Υ(𝑝, 𝜆) with respect to 𝑝, we obtain 

𝜕(Υ(𝑝, 𝜆))

𝜕𝑝
= 𝑝 [

16𝑝2 + (2𝜆2 − 8𝜆 − 8)

(𝜆 + 2)2(𝜆 + 1)2
]                                                                                              (3.9) 

Setting 
𝜕(Υ(𝑝,𝜆))

𝜕𝑝
= 0 yields either 𝑝 = 0 or 

𝑝2 = −
2𝜆2 − 8𝜆 − 8

16
(3.10) 

but −[2𝜆2 − 8𝜆 − 8] < 0 for 0 ≤ 𝜆 ≤ 1. 



Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 32 No. 2 (2025) 

 

389 
https://internationalpubls.com  

 

                               Figure 1. Graph of the bound −2𝜆2 + 8𝜆 + 8 in the range 𝜆 ∈ [0,1]. 

Therefore, the maximum value of |𝑎3
2 − 𝑎2

2| is attained at the end points 𝑝1 = 𝑝 ∈ [0,2]. 

For 𝑝1 = 0, 𝑝2 = 2𝑥. Then, we have (3.6). 

|𝑎3
2 − 𝑎2

2| =
4|𝑥|2

(𝜆 + 2)2
≤

4

(𝜆 + 2)2
(3.11) 

For 𝑝1 = 𝑝2 = 2, we get 

𝑎2 =
2

𝜆+2
(3.12)

𝑎3 =
4(1−𝜆)

(𝜆+2)(𝜆+1)
+

2

𝜆+2
(3.13)

which yields, 

|𝑎3
2 − 𝑎2

2| ≤ |
−40𝜆3 − 60𝜆2 + 20

(𝜆 + 1)4(𝜆 + 2)2
| (3.14) 

The result is sharp for the functions given by 

𝑧𝑓′(𝑧)

(1 − 𝜆)𝑓(𝑧) + 𝜆𝑧
=
1 + 𝑧

1 − 𝑧
(3.15) 

Remark 3.2. Theorem (3.1), for 𝜆 = 0 yields the bound |𝑎3
2 − 𝑎2

2| ≤ 5 for the class of star - like 

function 𝒮∗ conforming the bound obtained by Thomous and Halim [1] and for 𝜆 = 1 yields the 

bound |𝑎3
2 − 𝑎2

2| ≤
5

9
 for the class of functions with bounded boundary rotation ℛ̃ conforming the 

bound obtained by Radhika et al. [2]. 

In our next theorem, we determine an upper bound for the coefficient body 𝑇2(3). 

Theorem 3.3. Let 𝑓 given by (1.1) be in the class 𝓐(𝝀); (0 ≤ 𝜆 ≤ 1). Then we have sharp bound 

|𝑇2(3)| = |𝑎4
2 − 𝑎3

2| ≤ max {|64𝑅1(𝜆) − 16𝑅2(𝜆)|,
4

(𝜆 + 2)2
}

𝑅1(𝜆) =
𝜆4 − 14𝜆3 + 73𝜆2 − 168𝜆 + 144

16(𝜆 + 3)2(𝜆 + 2)2(𝜆 + 1)2

𝑅2(𝜆) =
𝜆2 − 6𝜆 + 9

4(𝜆 + 2)2(𝜆 + 1)2

 



Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 32 No. 2 (2025) 

 

390 
https://internationalpubls.com  

Proof. First note that by equating the corresponding coefficient in the equation (3.1). In the view of 

(3.3), (3.4) and applying Lemma (2.2), denoting 𝑋 = 4 − 𝑝2 and 𝑌 = (1 − |𝑥|2)𝜚, 

where 0 ≤ 𝑝 ≤ 2 and |𝜚| < 1, we get 

𝑎4
2 − 𝑎3

2 = [
𝜆4 − 14𝜆3 + 73𝜆2 − 168𝜆 + 144

16(𝜆 + 3)2(𝜆 + 2)2(𝜆 + 1)2
] 𝑝1

6 + [
−𝜆2 + 6𝜆 − 9

4(𝜆 + 2)2(𝜆 + 1)2
] 𝑝1

4 +
𝑋2𝑌2

4(𝜆 + 3)2
+
𝑝1𝑥

2𝑋2𝑌

4(𝜆 + 3)2

  + [
−𝜆2 + 2𝜆 + 5

2(𝜆 + 3)2(𝜆 + 2)(𝜆 + 1)
] 𝑝1𝑥𝑋

2𝑌

  + [
𝜆2 − 7𝜆 + 12

4(𝜆 + 3)2(𝜆 + 2)(𝜆 + 1)
] 𝑝1

3𝑋𝑌 +
𝑝1
2𝑥4𝑋2

16(𝜆 + 3)2

  + [
−𝜆2 + 2𝜆 + 5

4(𝜆 + 3)2(𝜆 + 2)(𝜆 + 1)
] 𝑝1

2𝑥3𝑋2

  + [
𝜆4 − 4𝜆3 − 6𝜆2 + 20𝜆 + 25

4(𝜆 + 3)2(𝜆 + 2)2(𝜆 + 1)2
] 𝑝1

2𝑥2𝑋2

  +
𝑥2𝑋2

4(𝜆 + 2)2
+ [

𝜆2 − 7𝜆 + 12

8(𝜆 + 3)2(𝜆 + 2)(𝜆 + 1)
] 𝑝1

4𝑥2𝑋

  + [
−𝜆4 + 9𝜆3 − 21𝜆2 − 11𝜆 + 60

4(𝜆 + 3)2(𝜆 + 2)2(𝜆 + 1)2
] 𝑝1

4𝑥𝑋

  + [
3 − 𝜆

2(𝜆 + 1)(𝜆 + 2)2
] 𝑝1

2𝑥𝑋.

 

As in the proof of theorem (3.1). Note that, by Lemma (2.2), where without loss of generality we let 

0 ≤ 𝑝1 = 𝑝 ≤ 2. Substitute this into the above equation, we get the following quadratic equation in 

terms of 𝑥. 

|𝑎4
2 − 𝑎3

2| ≤
(2 − 𝑝)2(4 − 𝑝2)2

16(𝜆 + 3)2
|𝑥|4 +

(−𝜆2 + 2𝜆 + 5)(4 − 𝑝2)(𝑝2 − 2𝑝)

4(𝜆 + 3)2(𝜆 + 2)(𝜆 + 1)
|𝑥|3

  + [
(𝜆2 − 7𝜆 + 12)(4 − 𝑝2)(𝑝 − 2)𝑝3

8(𝜆 + 3)2(𝜆 + 2)(𝜆 + 1)
+ [
𝜆4 − 4𝜆3 − 6𝜆2 + 20𝜆 + 25

4(𝜆 + 3)2(𝜆 + 2)2(𝜆 + 1)2
] 𝑝2(4 − 𝑝2)] |𝑥|2

  + [
𝑝(4 − 𝑝2)2

4(𝜆 + 3)2
− [

𝜆2 + 2𝜆 − 1

4(𝜆 + 3)2(𝜆 + 2)2
] (4 − 𝑝2)2] |𝑥|2

  + [
(3 − 𝜆)(4 − 𝑝2)𝑝2

2(1 + 𝜆)(2 + 𝜆)2
+
(−𝜆2 + 2𝜆 + 5)(4 − 𝑝2)𝑝

2(1 + 𝜆)(2 + 𝜆)(3 + 𝜆)2
+
−𝜆4 + 9𝜆3 − 21𝜆2 − 11𝜆 + 60

4(𝜆 + 3)2(𝜆 + 2)2(𝜆 + 1)2
𝑝4(4 − 𝑝2)] |𝑥|

  +|𝑅1(𝜆)𝑝
6 − 𝑅2(𝜆)𝑝

4| + [
𝜆2 − 7𝜆 + 12

4(𝜆 + 3)2(𝜆 + 2)(𝜆 + 1)
] 𝑝3(4 − 𝑝2) +

(4 − 𝑝2)2

4(𝜆 + 3)2
.

  = Θ(𝑝, |𝑥|)

 

Where, 

 𝑅1(𝜆) =
𝜆4 − 14𝜆3 + 73𝜆2 − 168𝜆 + 144

16(𝜆 + 3)2(𝜆 + 2)2(𝜆 + 1)2

 𝑅2(𝜆) =
𝜆2 − 6𝜆 + 9

4(𝜆 + 2)2(𝜆 + 1)2

 



Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 32 No. 2 (2025) 

 

391 
https://internationalpubls.com  

It is necessary to prove that the maximum value of Θ(𝑝, |𝑥|) on [0,2] × [0,1]. First, assume that there 

is a maximum at an interior point Θ(𝑝0, |𝑥0|) of [0,2] × [0,1]. Differentiating Θ(𝑝, |𝑥|) with respect 

to |𝑥| and equating it to 0 implies that 𝑝 = 𝑝0 = 2, which is contradiction. Thus, for the maximum of 

Θ(𝑝, |𝑥|), we need only to consider the end points of [0,2] × [0,1]. 

For 𝑝 = 0, we obtain 

Θ(0, |𝑥|) =
4|𝑥|4

(𝜆 + 3)2
−
4(𝜆2 + 2𝜆 − 1)|𝑥|

(𝜆 + 3)(𝜆 + 2)2
+

4

(𝜆 + 3)2
≤

4

(𝜆 + 2)2
(3.16) 

For 𝑝 = 2, we obtain 

Θ(2, |𝑥|) = |64𝑅1(𝜆) − 16𝑅2(𝜆)| (3.17) 

For |𝑥| = 0, we get 

Θ(𝑝, 0) = |𝑅1(𝜆)𝑝
6 − 𝑅2(𝜆)𝑝

4| + [
𝜆2 − 7𝜆 + 12

4(𝜆 + 3)2(𝜆 + 2)(𝜆 + 1)
] 𝑝1

3(4 − 𝑝2) +
(4 − 𝑝2)2

4(𝜆 + 3)2
(3.18) 

which has the maximum value |𝑅1(𝜆)𝑝
6 − 𝑅2(𝜆)𝑝

4| on [0,2]. 

 

For |𝑥| = 1, we gain 

Θ(𝑝, 1) =
(2 − 𝑝)2(4 − 𝑝2)2

16(𝜆 + 3)2
+
(−𝜆2 + 2𝜆 + 5)(4 − 𝑝2)(𝑝2 − 2𝑝)

4(𝜆 + 3)2(𝜆 + 2)(𝜆 + 1)

  + [
(𝜆2 − 7𝜆 + 12)(4 − 𝑝2)(𝑝 − 2)𝑝3

8(𝜆 + 3)2(𝜆 + 2)(𝜆 + 1)
+ [
𝜆4 − 4𝜆3 − 6𝜆2 + 20𝜆 + 25

4(𝜆 + 3)2(𝜆 + 2)2(𝜆 + 1)2
] 𝑝2(4 − 𝑝2)]

  + [
𝑝(4 − 𝑝2)2

4(𝜆 + 3)2
− [

𝜆2 + 2𝜆 − 1

4(𝜆 + 3)2(𝜆 + 2)2
] (4 − 𝑝2)2]

  + [
(3 − 𝜆)(4 − 𝑝2)𝑝2

2(1 + 𝜆)(2 + 𝜆)2
+
(−𝜆2 + 2𝜆 + 5)(4 − 𝑝2)𝑝

2(1 + 𝜆)(2 + 𝜆)(3 + 𝜆)2
+
−𝜆4 + 9𝜆3 − 21𝜆2 − 11𝜆 + 60

4(𝜆 + 3)2(𝜆 + 2)2(𝜆 + 1)2
𝑝4(4 − 𝑝2)]

  +|𝑅1(𝜆)𝑝
6 − 𝑅2(𝜆)𝑝

4| + [
𝜆2 − 7𝜆 + 12

4(𝜆 + 3)2(𝜆 + 2)(𝜆 + 1)
] 𝑝3(4 − 𝑝2) +

(4 − 𝑝2)2

4(𝜆 + 3)2
.

 

which has the maximum values |64𝑅1(𝜆) − 16𝑅2(𝜆)| for 𝑝 = 2 and 
4

(𝜆+2)2
 for 𝑝 = 0 

Remark 3.4. Theorem (3.3), for 𝜆 = 0 yields the bound |𝑎4
2 − 𝑎3

2| ≤ 7 for the class of star - like 

function 𝒮∗ conforming the bound obtained by Thomous and Halim [1]. and for 𝜆 = 1 yields the 

bound |𝑎4
2 − 𝑎3

2| ≤
4

9
 for the class of functions with bounded boundary rotation ℛ̃ conforming the 

bound obtained by Radhika et al. [2]. 

Theorem 3.5. Let 𝑓 given by (1.1), be in the class 𝓐(𝝀); (0 ≤ 𝜆 ≤ 1; 𝜆 ≠ 𝜆0). Then we have sharp 

bound 

|𝑇3(2)| = |(|

𝑎2 𝑎3 𝑎4
𝑎3 𝑎2 𝑎3
𝑎4 𝑎3 𝑎2

|)| ≤

{
 
 

 
 max {|𝑅(𝜆)𝑈(𝜆)|,

8|𝑅(𝜆)|

(𝜆 + 2)2
} ;  if 𝜆 ≠ 𝜆0

max {|𝑈(𝜆)𝐵(𝜆)|,
8|𝐵(𝜆)|

(𝜆 + 2)2
} ;  if 𝜆 = 𝜆0

 



Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 32 No. 2 (2025) 

 

392 
https://internationalpubls.com  

Where 𝜆0 ≈ 0.5 is the positive root of the polynomial 24𝑥 − 12 = 0, 

𝑅(𝜆)  =
24𝑥 − 12

(𝜆 + 3)(𝜆 + 2)(𝜆 + 1)
(3.19)

𝑈(𝜆)  =
4(8𝜆2 + 32𝜆 − 18)

(𝜆 + 3)(𝜆 + 2)2(𝜆 + 1)2
(3.20)

𝐵(𝜆)  =
4𝜆2 − 4𝜆 + 36

(𝜆 + 3)(𝜆 + 2)(𝜆 + 1)
(3.21)

 

Proof. Write 

|𝑇3(2)|  = |𝑎2
3 − 2𝑎2𝑎3

2 + 2𝑎3
2𝑎4 − 𝑎2𝑎4| (3.22)

 = |(𝑎2 − 𝑎4)(𝑎2
2 − 2𝑎3

2 + 𝑎2𝑎4)| (3.23)
 

Using the same techniques as the theorem (3.1), one can obtain with simple computations that 

|𝑎2 − 𝑎4| ≤ |𝑅(𝜆)| for 𝜆 ≠ 𝜆0. (3.24) 

We need to show that 

|𝑎2
3 − 2𝑎3

2 + 𝑎2𝑎4| ≤ |𝑈(∣ 𝜆)|. (3.25) 

In the view of (3.2),(3.3), (3.5) and Lemma (2.2), where we denote 𝑋 = 4 − 𝑝2 and 𝑌 =

(1 − |𝑥|2)𝜚, where 0 ≤ 𝑝 ≤ 2 and |𝜚| < 1, one may easily get 

|𝑎2
2 − 2𝑎3

2 + 𝑎2𝑎4| = |[
−𝜆3 + 𝜆2 + 16𝜆 − 30

4(𝜆 + 3)(𝜆 + 2)2(𝜆 + 1)2
] 𝑝1

4 +
𝑝1
2

(𝜆 + 1)2
−

𝑝1
2𝑋𝑥2

4(𝜆 + 3)(𝜆 + 1)
|

  + |
𝑋𝑌𝑝1

2(𝜆 + 3)(𝜆 + 1)
−

𝑋2𝑥2

2(𝜆 + 2)2
+ [

𝜆3 + 2𝜆2 − 9𝜆 − 8

2(𝜆 + 3)(𝜆 + 2)2(𝜆 + 1)2
] 𝑝1

2𝑋𝑥|

 

Applying the triangle inequality and assuming that 𝑝1 = 𝑝, where 0 ≤ 𝑝 ≤ 2, we obtain 

|𝑎2
2 − 2𝑎3

2 + 𝑎2𝑎4| ≤ [
𝑝2(4 − 𝑝2)

4(𝜆 + 3)(𝜆 + 1)
+

𝑝2(4 − 𝑝2)

2(𝜆 + 3)(𝜆 + 1)
+
(4 − 𝑝2)2

2(𝜆 + 2)2
] |𝑥|2

  + [[
𝜆3 + 2𝜆2 − 9𝜆 − 8

2(𝜆 + 3)(𝜆 + 2)(𝜆 + 1)2
] 𝑝(4 − 𝑝2)] |𝑥| +

𝑝(4 − 𝑝2)

2(𝜆 + 3)(𝜆 + 1)
+

𝑝2

(𝜆 + 1)2

  + [
−𝜆3 + 𝜆2 + 16𝜆 − 30

4(𝜆 + 3)(𝜆 + 2)2(𝜆 + 1)2
] 𝑝4

  = Ψ(𝑝, |𝑥|)

 

We have to prove that the maximum value of Ψ(𝑝, |𝑥|) on [0,2] × [0,1]. First, assume that there is a 

maximum at an interior point Ψ(𝑝0, |𝑥0|) of [0,2] × [0,1]. Differentiating Ψ(𝑝, |𝑥|) with respect to |𝑥| 

and equating it to 0 implies that 𝑝 = 𝑝0 = 2 which is contradiction. Thus, for the maximum of 

Ψ(𝑝, |𝑥|), we need only to consider the end points of [0,2] × [0,1]. 

For 𝑝 = 0, we obtain 

Ψ(0, |𝑥|) =
8|𝑥|2

(𝜆 + 2)2
≤

8

(𝜆 + 2)2
(3.26) 



Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 32 No. 2 (2025) 

 

393 
https://internationalpubls.com  

For 𝑝 = 2, we have 

Ψ(2, |𝑥|) =
4(8𝜆2 + 32𝜆 − 18)

(𝜆 + 1)(𝜆 + 2)2(3 + 𝜆)
= 𝑈(𝜆) (3.27) 

For 𝑥 = 0, we brought 

Ψ(𝑝, 0) = |
𝑝2

(𝜆 + 1)2
+

𝑝(4 − 𝑝2)

2(𝜆 + 3)(𝜆 + 1)
+

−𝜆3 + 𝜆2 + 16𝜆 − 30

4(𝜆 + 3)(𝜆 + 2)2(𝜆 + 1)2
𝑝4| (3.28) 

which has the maximum value Ψ(𝑝, 0) = 𝑈(𝜆) attained at the end point 𝑝 = 2. Hence, for |𝑥| = 1, 

we obtain 

Ψ(𝑝, 1) = [
𝑝2(4 − 𝑝2)

4(𝜆 + 3)(𝜆 + 1)
+

𝑝2(4 − 𝑝2)

2(𝜆 + 3)(𝜆 + 1)
+
(4 − 𝑝2)2

2(𝜆 + 2)2
]

  + [
𝜆3 + 2𝜆2 − 9𝜆 − 8

2(𝜆 + 3)(𝜆 + 2)2(𝜆 + 1)2
] 𝑝(4 − 𝑝2) +

𝑝(4 − 𝑝2)

2(𝜆 + 3)(𝜆 + 1)
+

𝑝2

(𝜆 + 1)2

  + [
−𝜆3 + 𝜆2 + 16𝜆 − 30

4(𝜆 + 3)(𝜆 + 2)2(𝜆 + 1)2
] 𝑝4.

 

which has maximum Ψ(𝑝, 1) =
8

(𝜆+2)2
 at 𝑝 = 0 and Ψ(𝑝, 1) = 𝑈(𝜆) at 𝑝 = 2. 

|𝑎2
2 − 2𝑎3

2 + 𝑎2𝑎4| ≤ max {|𝑈(𝜆)|,
8

(𝜆 + 2)2
} . (3.29) 

Thus 

|𝑇3(2)| = |(𝑎2 − 𝑎4)(𝑎2
2 − 2𝑎3

2 + 𝑎2𝑎4)| ≤ max {|𝑅(𝜆)𝑈(𝜆)|,
8|𝑅(𝜆)|

(𝜆 + 2)2
} . (3.30) 

For the case 𝜆 = 𝜆0, we compute |𝑎2 − 𝑎4| as follows 

|𝑎2 − 𝑎4| = |
𝑝1
𝜆 + 1

− [
𝑝1
3(1 − 𝜆)2

(𝜆 + 3)(𝜆 + 2)(𝜆 + 1)
+
𝑝1𝑝2(1 − 𝜆)(3 + 2𝜆)

(𝜆 + 3)(𝜆 + 2)(𝜆 + 1)
+

𝑝3
𝜆 + 3

]| (3.31) 

Since each |𝑝𝑛| ≤ 2, an application of triangle inequality shows that 

|𝑎2 − 𝑎4| ≤ |𝐵(𝜆)| =
4𝜆2 − 4𝜆 + 36

(𝜆 + 3)(𝜆 + 2)(𝜆 + 1)
. (3.32) 

Therefore, 

|𝑇3(2)| = |(𝑎2 − 𝑎4)(𝑎2
2 − 2𝑎3

2 + 𝑎2𝑎4)| ≤ max {|𝑈(𝜆)𝐵(𝜆)|,
8|𝐵(𝜆)|

(𝜆 + 2)2
} . (3.33) 

This completes the proof the Theorem (3.5). 

Remark 3.6. Theorem (3.5), for 𝜆 = 0 yields the bound |𝑇3(2)| ≤ 12 for the class of star - like 

function 𝒮∗ conforming the bound obtained by Thomous and Halim [1]. and for 𝜆 = 1 yields the bound 



Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 32 No. 2 (2025) 

 

394 
https://internationalpubls.com  

|𝑇3(2)| ≤
8

9
 for the class of functions with bounded boundary rotation ℛ̃ conforming the bound 

obtained by Radhika et al. [2]. 

Theorem 3.7. Let 𝑓 given by (1.1), be in the class 𝓐(𝝀); 0 ≤ 𝜆 ≤ 1. Then we have sharp bound 

|𝑇3(1)| = |(|
1 𝑎2 𝑎3
𝑎2 1 𝑎2
𝑎3 𝑎2 1

|)| ≤ max {1 +
1

4(𝜆 + 2)2
, |𝑁(𝜆)|} 

Where, 

𝑁(𝜆) =
𝜆5 + 7𝜆4 + 7𝜆3 − 11𝜆2 − 44𝜆 + 32

(𝜆 + 2)2(𝜆 + 1)3
 

Proof. Expanding the determinant by using equation (3.1), we get (3.2) and (3.3), we have 

𝑇3(1) = 1 + 2𝑎2
2(𝑎3 − 1) − 𝑎3

2

  = 1 +
2𝑝1

2

(𝜆 + 1)2
(

𝑝1
2(1 − 𝜆)

(𝜆 + 2)(𝜆 + 1)
+

𝑝2
𝜆 + 2

− 1)

  − [
𝑝1
4(1 − 𝜆)

(𝜆 + 2)2(𝜆 + 1)2
+

𝑝2
2

(𝜆 + 2)2
+

2(1 − 𝜆)𝑝1
2𝑝2

(𝜆 + 2)2(𝜆 + 1)
] .

  = 1 + [
−𝜆3 + 𝜆2 + 𝜆 + 15

4(𝜆 + 2)2(𝜆 + 1)3
] 𝑝1

4 + [
𝜆2 + 1

2(𝜆 + 2)2(𝜆 + 1)2
] 𝑝1

2𝑥𝑋

  −
2𝑝1

2

(𝜆 + 1)2
−

𝑥2𝑋2

4(𝜆 + 2)2
.

 

Note that, by Lemma (2.2), without loss of generality we let 0 ≤ 𝑝1 = 𝑝 ≤ 2. Substitute this into the 

above equation and applying the triangle inequality, we obtain the following quadratic equation in 

terms of 𝑥. 

𝑇3(1) ≤ [
(4 − 𝑝2)2

4(𝜆 + 2)2
] |𝑥|2 + [

𝑝2(4 − 𝑝2)(𝜆2 + 1)

2(𝜆 + 2)2(𝜆 + 1)2
] |𝑥|

  + [1 +
8(𝜆 + 2)2(𝜆 + 1) + (−𝜆3 + 𝜆2 + 𝜆 + 15)𝑝2

4(𝜆 + 2)2(𝜆 + 1)3
𝑝2] .

𝑇3(1) ≤ [
(4 − 𝑝2)

4(𝜆 + 2)2
] + [

𝑝2(4 − 𝑝2)(𝜆2 + 1)

2(𝜆 + 2)2(𝜆 + 1)2
]

  + [1 +
8(𝜆 + 2)2(𝜆 + 1) + (−𝜆3 + 𝜆2 + 𝜆 + 15)𝑝2

4(𝜆 + 2)2(𝜆 + 1)3
𝑝2] .

  = Ξ(𝑝, 𝜆).

 

Differentiating Ξ(𝑝, 𝜆) with respect to 𝑝, we obtain 

𝜕(Ξ(𝑝, 𝜆))

𝜕𝑝
= 𝑝 [

𝑝2(−2𝜆3 + 2𝜆2 + 2𝜆 + 14) + (4𝜆3 + 12𝜆2 + 24𝜆 + 16)

(𝜆 + 2)2(𝜆 + 1)3
] . (3.34) 

Setting 
𝜕(Ξ(𝑝,𝜆))

𝜕𝑝
= 0 yields either 𝑝 = 0 or 



Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 32 No. 2 (2025) 

 

395 
https://internationalpubls.com  

𝑝2 =
−4𝜆3 − 12𝜆2 − 24𝜆 − 16

−2𝜆3 + 2𝜆2 + 2𝜆 + 14
. (3.35) 

but −4𝜆3 − 12𝜆2 − 24𝜆 − 16 < 0 for 0 ≤ 𝜆 ≤ 1 and 

therefore the maximum value of 𝑇3(1) is attained at the end points 𝑝1 = 𝑝 ∈ [0,2]. 

 

Figure 2. Graph of the bound −4𝜆3 − 12𝜆2 − 24𝜆 − 16 in the range 𝜆 ∈ [0,1]. 

For 𝑝1 = 0 and 𝑝2 = 2𝑥. Then, we have 

|1 + 2𝑎2
2(𝑎3 − 1) − 𝑎3

2| = 1 −
4|𝑥|2

(𝜆 + 2)2
≤ 1 +

4

(𝜆 + 2)2
. (3.36) 

In the view of (3.2), (3.3) and 𝑝1 = 𝑝2 = 2, we get 

2𝑎2
2𝑎3  =

32(1 − 𝜆)

(𝜆 + 2)(𝜆 + 1)3
+

16

(𝜆 + 2)(𝜆 + 1)2
. (3.37)

−2𝑎2
2  =

−8

(𝜆 + 1)2
. (3.38)

−𝑎3
2  = −

16(1 − 𝜆)2

(𝜆 + 2)(𝜆 + 1)2
−

4

(2 + 𝜆)2
−

16(1 − 𝜆)

(𝜆 + 2)2(𝜆 + 1)
. (3.39)

 

Substitute the values of (3.37), (3.38) and (3.39) in (3.22), we may get 

|1 + 2𝑎2
2(𝑎3 − 1) − 𝑎3

2| ≤ |
𝜆5 + 7𝜆4 + 7𝜆3 − 11𝜆2 − 44𝜆 + 32

(𝜆 + 2)2(𝜆 + 1)3
| ≤ 𝑁(𝜆). (3.40) 

where, 

𝑁(𝜆) = |
𝜆5 + 7𝜆4 + 7𝜆3 − 11𝜆2 − 44𝜆 + 32

(𝜆 + 2)2(𝜆 + 1)3
| . (3.41) 

This completes the proof of the theorem (3.7). 

Remark 3.8. Theorem (3.7), for 𝜆 = 0 yields the bound |1 + 2𝑎2
2(𝑎3 − 1) − 𝑎3

2| ≤ 8 for the class of 

star - like function 𝒮∗ conforming the bound obtained by Thomous and Halim [1]. and for 

𝜆 = 1 yields the bound |1 + 2𝑎2
2(𝑎3 − 1) − 𝑎3

2| ≤
13

9
 for the class of functions with bounded boundary 

rotation ℛ̃ conforming the bound obtained by Radhika et al. [2]. 

 



Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 32 No. 2 (2025) 

 

396 
https://internationalpubls.com  

 

4. Zalcman Conjecture For the class 𝓐(𝝀) 

Theorem 4.1. Let 𝑓 given by (1.1), be in the class 𝓐(𝝀); (0 ≤ 𝜆 ≤ 1). Then we have sharp bound 

|𝑎2
2 − 𝑎3| ≤ max {

2

(𝜆 + 2)
,

𝒯(𝜆)

(𝜆 + 1)2(𝜆 + 2)
} (4.1) 

where 

𝒯(𝜆) = 2(𝜆2 + 1) (4.2) 

Proof. First note that by equating the corresponding coefficients in the equation (3.1). We get, in the 

view of (3.2) and (3.3), a simple computation leads to 

𝑎2
2 − 𝑎3 = [

𝑝1
𝜆 + 1

]
2

− [
𝑝1
2(1 − 𝜆)

(𝜆 + 1)(𝜆 + 2)
+

𝑝2
𝜆 + 2

] 

                                                   =
𝑝2

1

(𝜆 + 1)2
−

𝑝1
2(1 − 𝜆)

(𝜆 + 1)(𝜆 + 2)
−

𝑝2
𝜆 + 2 

    .                                              (4.3) 

Note that, by Lemma (2.2), we may write 2𝑝2 = 𝑝1
2 + 𝑥(4 − 𝑝1

2), we can easily get 

= [
𝜆2 + 1

2(𝜆 + 1)(𝜆 + 2)
] 𝑝1

2 −
𝑋𝑥

2(𝜆 + 2)
. (4.4) 

Without loss of generality, we let 0 ≤ 𝑝1 = 𝑝 ≤ 2. Substitute this into the above equation, we obtain 

the following quadratic equation in terms of 𝑥. 

|𝑎2
2 − 𝑎3|  =

4 − 𝑝2

2(𝜆 + 2)
|𝑥| + [

𝜆2 + 1

2(𝜆 + 1)2(𝜆 + 2)
] 𝑝1

2. (4.5)

 = ¥(𝑝, |𝑥|). (4.6)

 

We required to prove that the maximum value of ¥(𝑝, |𝑥|) on [0,2] × [0,1]. First, assume that there is 

a maximum at an interior point ¥(𝑝0, |𝑥0|) of [0,2] × [0,1]. Differentiating ¥(𝑝, |𝑥|) with respect to 

|𝑥| and equating it to 0 implies that 𝑝 = 𝑝0 = 2 which is contradiction.  

Thus, for the maximum ¥(𝑝, |𝑥|), we must consider the end points of [0,2] × [0,1]. For 𝑝 = 0, we 

obtain 

¥(0, |𝑥|) =
4

2(𝜆 + 2)
|𝑥|2 ≤

2

𝜆 + 2
. (4.7) 

For 𝑝 = 2, we owe 

¥(2, |𝑥|) = [
2(𝜆2 + 1)

(𝜆 + 1)2(𝜆 + 2)
] . (4.8) 

For |𝑥| = 0, we receive 

¥(𝑝, 0) = [
𝜆2 + 1

2(𝜆 + 1)2(𝜆 + 2)
] 𝑝2. (4.9) 



Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 32 No. 2 (2025) 

 

397 
https://internationalpubls.com  

which has maximum value 
|𝒯(𝜆)|

(𝜆+1)2(𝜆+2)
 attained at the end point 𝑝 = 2. 

For |𝑥| = 1, we obtained 

¥(𝑝, 1) = [
𝜆2 + 1

2(𝜆 + 1)2(𝜆 + 2)
] 𝑝2 +

4 − 𝑝2

2(𝜆 + 1)
. (4.10) 

which is maximum value of ¥(𝑝, 1) =
2

𝜆+1
 at 𝑝 = 0 and 

|𝒯(𝜆)|

(𝜆+1)2(𝜆+2)
 at 𝑝 = 2. Hence 

|𝑎2
2 − 𝑎3| ≤ max {

2

4(𝜆 + 2)
,

|𝒯(𝜆)|

(𝜆 + 1)2(𝜆 + 2)
} (4.11) 

where 

𝒯(𝜆) = 2(𝜆2 + 1) (4.12) 

Theorem 4.2. Let 𝑓 given by (1.1), be in the class 𝓐(𝝀); (0 ≤ 𝜆 ≤ 1). Then we have sharp bound 

|𝑎3
2 − 𝑎5| ≤ max {

(3 + 2𝜆)4

(𝜆 + 2)(𝜆 + 4)
+

6

𝜆 + 4
,

|𝒢(𝜆)|

(𝜆 + 1)2(𝜆 + 2)2(𝜆 + 3)(𝜆 + 4)
} 

where, 

𝒢(𝜆) = 2(𝜆5 − 7𝜆4 + 15𝜆3 + 12𝜆2 − 104𝜆 + 96) (4.13) 

Proof. First note that by equating the corresponding coefficients in the equation (3.1). We get, in the 

view of (3.3), (3.5) and Lemma (2.2), we may write 2𝑝2 = 𝑝1
2 + 𝑥(4 − 𝑝1

2), 𝑌 = (1 − |𝑥|2)𝜚, a simple 

computation leads to 

𝑎3
2 − 𝑎5 = [

𝜆5 − 7𝜆4 + 15𝜆3 + 12𝜆2 − 104𝜆 + 96

8(𝜆 + 1)2(𝜆 + 2)2(𝜆 + 3)(𝜆 + 4)
] 𝑝1

4

  + [
3 + 2𝜆

4(2 + 𝜆)(4 + 𝜆)
] 𝑥2𝑋2 + [

−𝜆2 + 8𝜆 + 17

8(1 + 𝜆)(3 + 𝜆)(4 + 𝜆)
] 𝑝1

2𝑥2𝑋

  + [
−7𝜆3 + 2𝜆2 + 35𝜆 + 58

8(1 + 𝜆)(2 + 𝜆)(3 + 𝜆)(4 + 𝜆)
] 𝑝1

2𝑥𝑋 + [
𝜆2 − 2𝜆 − 7

2(𝜆 + 1)(𝜆 + 3)(𝜆 + 4)
] 𝑝1𝑋𝑌 −

𝑝1
2𝑥3𝑋

8(𝜆 + 4)

  −
𝑥2𝑋

2(𝜆 + 4)
+

𝑥𝑋𝑌𝑝1
2(𝜆 + 4)

+
𝑋𝑌𝑥‾

2(𝜆 + 4)
.

 

Without loss of generality, we let 0 ≤ 𝑝1 = 𝑝 ≤ 2. Substitute this into the above equation, we obtain 

the following quadratic equation in terms of 𝑥. 



Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 32 No. 2 (2025) 

 

398 
https://internationalpubls.com  

|𝑎3
2 − 𝑎5| ≤ [

𝑝2(4 − 𝑝2)

8(𝜆 + 4)
−
𝑝(4 − 𝑝2)

2(𝜆 + 4)
] |𝑥|3

  + [
(3 + 2𝜆)(4 − 𝑝2)2

4(2 + 𝜆)(4 + 𝜆)
+
(−𝜆2 + 8𝜆 + 17)(4 − 𝑝2)𝑝2

8(1 + 𝜆)(𝜆 + 3)(𝜆 + 4)
−
(4 − 𝑝2)𝑥‾

2(𝜆 + 4)
] |𝑥|2

  + [
(4 − 𝑝2)

2(𝜆 + 4)
−
(𝜆2 − 2𝜆 − 7)𝑝(4 − 𝑝2)

2(𝜆 + 1)(𝜆 + 3)(𝜆 + 4)
] |𝑥|2

  + [[
−7𝜆3 + 2𝜆2 + 35𝜆 + 58

8(𝜆 + 1)(𝜆 + 2)(𝜆 + 3)(𝜆 + 4)
] 𝑝2(4 − 𝑝2) +

𝑝(4 − 𝑝2)

2(𝜆 + 4)
] |𝑥|

  + [
𝜆5 − 7𝜆4 + 15𝜆3 + 12𝜆2 − 104𝜆 + 96

8(𝜆 + 1)2(𝜆 + 2)2(𝜆 + 3)(𝜆 + 4)
] 𝑝4

  + [
𝜆2 − 2𝜆 − 7

2(𝜆 + 1)(𝜆 + 3)(𝜆 + 4)
] 𝑝(4 − 𝑝2) +

(4 − 𝑝2)𝑥‾

2(𝜆 + 4)
.

  = 𝐶(𝑝, |𝑥|).

 

We want to prove that the maximum value of 𝐶(𝑝, |𝑥|) on [0,2] × [0,1]. First, assume that there is a 

maximum at an interior point 𝐶(𝑝0, |𝑥0|) of [0,2] × [0,1]. Differentiating 𝐶(𝑝, |𝑥|) with respect to |𝑥| 

and equating it to 0 implies that 𝑝 = 𝑝0 = 2 which is contradiction. Thus for the maximum of 

𝐶(𝑝, |𝑥|), we need to consider the end points of [0,2] × [0,1]. 

For 𝑝 = 0, we obtain 

𝐶(0, |𝑥|) = [
(3 + 2𝜆)4

(𝜆 + 2)(𝜆 + 4)
−

2

(𝜆 + 4)
𝑥‾] |𝑥|2 + [

4

2(𝜆 + 4)
]| 𝑥|2 +

4

2(4 + 𝑥)
𝑥‾ 

                                   ≤
(3 + 2𝜆)4

(𝜆 + 2)(𝜆 + 4)
+

6

(𝜆 + 4)
                                                                            (4.14) 

For 𝑝 = 2, we acquire 

𝐶(2, |𝑥|) =
|𝒢(𝜆)|

(𝜆 + 1)2(𝜆 + 2)2(𝜆 + 3)(𝜆 + 4)
(4.15) 

where, 

𝒢(𝜆) = 2(𝜆5 − 7𝜆4 + 15𝜆3 + 12𝜆2 − 104𝜆 + 96) (4.16) 

For |𝑥| = 0, we attained 

𝐶(𝑝, 0) = [
𝜆5−7𝜆4+15𝜆3+12𝜆2−104𝜆+96

(𝜆+1)2(𝜆+2)2(𝜆+3)(𝜆+4)
] 𝑝4+[

−𝜆2−2𝜆−7

2(𝜆+1)(𝜆+3)(𝜆+4)
] 𝑝(4 − 𝑝2).                                         (4.17) 

For |𝑥| = 1, we have 



Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 32 No. 2 (2025) 

 

399 
https://internationalpubls.com  

𝐶(𝑝, 1) = [
𝑝2(4 − 𝑝2)

8(𝜆 + 4)
−
𝑝(4 − 𝑝2)

2(𝜆 + 4)
]

  + [
(3 + 2𝜆)(4 − 𝑝2)2

4(2 + 𝜆)(4 + 𝜆)
+
(−𝜆2 + 8𝜆 + 17)(4 − 𝑝2)𝑝2

8(1 + 𝜆)(𝜆 + 3)(𝜆 + 4)
−
(4 − 𝑝2)𝑥‾

2(𝜆 + 4)
]

  + [
(4 − 𝑝2)

2(𝜆 + 4)
−
(𝜆2 − 2𝜆 − 7)𝑝(4 − 𝑝2)

2(𝜆 + 1)(𝜆 + 3)(𝜆 + 4)
]

  + [
−7𝜆3 + 2𝜆2 + 35𝜆 + 58

8(𝜆 + 1)(𝜆 + 2)(𝜆 + 3)(𝜆 + 4)
𝑝2(4 − 𝑝2) +

𝑝(4 − 𝑝2)

2(𝜆 + 2)
]

  + [
𝜆5 − 7𝜆4 + 15𝜆3 + 12𝜆2 − 104𝜆 + 96

8(𝜆 + 1)2(𝜆 + 2)2(𝜆 + 3)(𝜆 + 4)
] 𝑝4 + [

−𝜆2 − 2𝜆 − 7

2(𝜆 + 1)(𝜆 + 3)(𝜆 + 4)
] 𝑝(4 − 𝑝2) +

4 − 𝑝2

2(𝜆 + 4)
.

 

   (4.18) 

which has maximum value 
|𝒢(𝜆)|

(𝜆+1)2(𝜆+2)2(𝜆+3)(𝜆+4)
 attained at the end point 𝑝 = 2 and                

(3+2𝜆)4

(𝜆+2)(𝜆+4)
+

6

(𝜆+4)
 at 𝑝 = 0. 

5. Generalized Zalcman Conjecture for the class 𝓐(𝝀) 

Theorem 5.1. Let 𝑓 given by (1.1), be in the class 𝓐(𝝀); (0 ≤ 𝜆 ≤ 1). Then we have sharp bound 

|𝑎2𝑎3 − 𝑎4| ≤ max {
4

(𝜆 + 3)
,

|ℋ(𝜆)|

(𝜆 + 1)2(𝜆 + 2)(𝜆 + 3)
} 

where 

ℋ(𝜆) = 2(−𝜆3 + 4𝜆2 − 5𝜆 + 6) (5.1) 

Proof. First note that by equating the corresponding coefficients in the equation (3.1), we bring, in 

the view of (3.2), (3.3) and (3.4), a simple computation leads to 

𝑎2𝑎3 − 𝑎4 = [
𝑝1
𝜆 + 1

] [
𝑝1
2(1 − 𝜆)

(𝜆 + 1)(𝜆 + 2)
+

𝑝2
𝜆 + 2

]. 

           − [
𝑝1
3(1 − 𝜆)2

(𝜆 + 1)(𝜆 + 2)(𝜆 + 3)
+
𝑝1𝑝2(1 − 𝜆)(3 + 2𝜆)

(𝜆 + 1)(𝜆 + 2)(𝜆 + 3)
+

𝑝3
𝜆 + 3

].                                    (5.2) 

Note that, by Lemma (2.2), we may write 

𝑎2𝑎3 − 𝑎4 =
𝑝1𝑋𝑥

2

4(𝜆+3)
+

𝑋𝑌

2(𝜆+3)
+ [

𝜆2−𝜆−2

2(𝜆+1)(𝜆+2)(𝜆+3)
] 𝑝1𝑥𝑋 + [

−𝜆3+4𝜆2−5𝜆+6

4(𝜆+1)2(𝜆+2)(𝜆+3)
] 𝑝1

3. 

Without loss of generality, we let 0 ≤ 𝑝1 = 𝑝 ≤ 2. Substitute this into the above equation, we obtain 

the following quadratic equation in terms of 𝑥. 

|𝑎2𝑎3 − 𝑎4| ≤ [
𝑝(4 − 𝑝2)

4(𝜆 + 3)
−
(4 − 𝑝2)

2(𝜆 + 3)
] |𝑥|2 + [

𝜆2 − 𝜆 − 2

2(𝜆 + 1)(𝜆 + 2)(𝜆 + 3)
] (4 − 𝑝2)𝑥𝑝

  + [
−𝜆3 + 4𝜆2 − 5𝜆 + 6

4(𝜆 + 1)2(𝜆 + 2)(𝜆 + 3)
] 𝑝3 +

4 − 𝑝2

2(𝜆 + 3)
 = £(𝑝, |𝑥|)                                          (5.3)

 



Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 32 No. 2 (2025) 

 

400 
https://internationalpubls.com  

We prove that the maximum value of £(𝑝, |𝑥|) on [0,2] × [0,1]. First, assume that there is a maximum 

at an interior point £(𝑝0, |𝑥0|) of [0,2] × [0,1]. Differentiating £(𝑝, |𝑥|) with respect to |𝑥| and 

equating it to 0 implies that 𝑝 = 𝑝0 = 2 which is contradiction. Thus, for the maximum of £(𝑝, |𝑥|), 

we need to consider the end points of [0,2] × [0,1]. For 𝑝 = 0, we obtain 

£(0, |𝑥|) =
−4

2(𝜆 + 3)
|𝑥|2 +

4

2(𝜆 + 3)
≤

4

𝜆 + 3
(5.4) 

For 𝑝 = 2, we obtain 

£(2, |𝑥|) =
2(−𝜆3 + 4𝜆2 − 5𝜆 + 6)

(𝜆 + 1)2(𝜆 + 2)(𝜆 + 3)
(5.5) 

For |𝑥| = 0, we get 

£(𝑝, 0) = [
−𝜆3 + 4𝜆2 − 5𝜆 + 6

4(𝜆 + 1)2(𝜆 + 2)(𝜆 + 3)
] 𝑝3 +

4 − 𝑝2

2(𝜆 + 3)
(5.6) 

which has maximum value 
|ℋ(𝜆)|

(𝜆+1)2(𝜆+2)(𝜆+3)
 attained at the end point 𝑝 = 2. For |𝑥| = 1, we obtain 

£(𝑝, 1) = [
𝑝(4 − 𝑝2)

4(𝜆 + 3)
−
(4 − 𝑝2)

2(𝜆 + 3)
] + [

𝜆2 − 𝜆 − 2

2(𝜆 + 3)(𝜆 + 1)
] (4 − 𝑝2)𝑝 

                                            + [
−𝜆3 + 4𝜆2 − 5𝜆 + 6 

4(𝜆 + 1)2(𝜆 + 𝜆)(𝜆 + 3)
] 𝑝3 +

4 − 𝑝2

2(𝜆 + 3)
   .                                          (5.7) 

which is maximum value of £(𝑝, 1) =
4

𝜆+1
 at 𝑝 = 0 and £(𝑝, 1) =

|ℋ(𝜆)|

(𝜆+1)2(𝜆+2)(𝜆+3)
 at 𝑝 = 2. 

Hence 

|𝑎2𝑎3 − 𝑎4| ≤ max {
4

(𝜆 + 3)
,

|ℋ(𝜆)|

(𝜆 + 1)2(𝜆 + 2)(𝜆 + 3)
} . (5.8)

ℋ(𝜆) = 2(−𝜆3 + 4𝜆2 − 5𝜆 + 6). (5.9)

 

Theorem 5.2. Let 𝑓 given by (1.1), be in the class 𝓐(𝝀); (0 ≤ 𝜆 ≤ 1). Then we have sharp bound 

|𝑎2𝑎4 − 𝑎5| ≤ max {
4

(𝜆 + 2)
,

|𝜕(𝜆)|

(𝜆 + 1)2(𝜆 + 2)(𝜆 + 3)(𝜆 + 4)
}. 

where, 

𝒥(𝜆) = 2(𝜆4 − 9𝜆3 + 29𝜆2 − 45𝜆 + 36). (5.10) 

Proof. 

 First note that by equating the corresponding coefficients in the equation (3.1). In view of (3.2), (3.4) 

and (3.5), we may write 2𝑝2 = 𝑝1
2 + 𝑥(4 − 𝑝1

2), 𝑌 = (1 − |𝑥|2)𝜚 and applying Lemma (2.2), a simple 

computation leads to 



Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 32 No. 2 (2025) 

 

401 
https://internationalpubls.com  

𝑎2𝑎4 − 𝑎5 = [
𝜆4−9𝜆3+29𝜆2−45𝜆+36

8(𝜆+1)2(𝜆+2)(𝜆+3)(𝜆+4)
] 𝑝1

4 

  + [
−3𝜆4+11𝜆3+9𝜆2−35𝜆−6

8(𝜆+1)2(𝜆+2)(𝜆+3)(𝜆+4)
] 𝑝1

2𝑥𝑋 + [
−𝜆2+6𝜆+9

8(𝜆+1)(𝜆+3)(𝜆+4)
] 𝑝1

2𝑥2𝑋

  + [
3𝜆2+7𝜆+3

2(𝜆+1)(𝜆+3)(𝜆+4)
] 𝑝1𝑋𝑌 + [

1−𝜆

4(𝜆+2)(𝜆+4)
] 𝑥2𝑋2 −

𝑝1
2𝑥3𝑋

8(𝜆+4)

    

       −
𝑥2𝑋

2(𝜆+4)
−

𝑥𝑋𝑝1

2(𝜆+4)
+

𝑋𝑌�̅�

2(𝜆+4)
                                                                                                       (5.11) 

Without loss of generality, we let 0 ≤ 𝑝1 = 𝑝 ≤ 2. Substitute this into the above equation, we obtain 

the following quadratic equation in terms of 𝑥. 

|𝑎2𝑎4 − 𝑎5| ≤ [
𝑝2(4 − 𝑝2)

8(𝜆 + 4)
−
𝑝(4 − 𝑝2)

2(𝜆 + 4)
] |𝑥|3

  + [
(−𝜆2 + 6𝜆 + 9)(4 − 𝑝2)𝑝2

8(𝜆 + 1)(𝜆 + 3)(𝜆 + 4)
+

4 − 𝑝2

2(𝜆 + 4)
] |𝑥|2

  + [
(4 − 𝑝2)2

4(2 + 𝜆)(𝜆 + 4)
−
(3𝜆2 + 7𝜆 + 3)(4 − 𝑝2)𝑝

2(𝜆 + 1)(𝜆 + 3)(𝜆 + 4)
−
(4 − 𝑝2)𝑥‾

2(𝜆 + 4)
] |𝑥|2

  + [
−3𝜆4 + 11𝜆3 + 9𝜆2 − 35𝜆 − 6

8(𝜆 + 1)2(𝜆 + 2)(𝜆 + 3)(𝜆 + 4)
𝑝2(4 − 𝑝2) +

𝑝(4 − 𝑝2)

2(𝜆 + 2)
] |𝑥|

  + [
3𝜆2 + 7𝜆 + 3

2(𝜆 + 1)(𝜆 + 3)(𝜆 + 4)
] 𝑝(4 − 𝑝2)

  +
(4 − 𝑝2)𝑥‾

2(𝜆 + 4)
+ [

𝜆4 − 9𝜆3 + 29𝜆2 − 45𝜆 + 36

8(𝜆 + 1)2(𝜆 + 2)(𝜆 + 3)(𝜆 + 4)
] 𝑝4.

 

                                     = Δ(𝑝, |𝑥|).                                                                                                         (5.12) 

We need to prove that the maximum value of Δ(𝑝, |𝑥|) on [0,2] × [0,1]. First, assume that there is a 

maximum at an interior point Δ(𝑝0, |𝑥0|) of [0,2] × [0,1]. Differentiating Δ(𝑝, |𝑥|) with respect to |𝑥| 

and equating it to 0 implies that 𝑝 = 𝑝0 = 2 which is contradiction. Thus, for the maximum of 

Δ(𝑝, |𝑥|), we should consider the end points of [0,2] × [0,1]. 

For 𝑝 = 0, we obtain 

Δ(0, |𝑥|) = [
2

(4 + 𝜆)
+

4

(𝜆 + 2)(4 + 𝜆)
−

2𝑥‾

(4 + 𝜆)
] |𝑥|2 +

2𝑥‾

(4 + 𝜆)
      .                                           (5.13) 

For 𝑝 = 2, we have 

Δ(2, |𝑥|) =
|𝒥(𝜆)|

(𝜆 + 1)2(𝜆 + 2)(𝜆 + 3)(𝜆 + 4)
. (5.14) 

For |𝑥| = 0, we gain 

Δ(𝑝, 0) = [
3𝜆3 + 7𝜆 + 3

2(𝜆 + 1)(𝜆 + 3)(𝜆 + 4)
] 𝑝(4 − 𝑝2) 

                                                                 + [
𝜆4 − 9𝜆3 + 29𝜆2 − 45𝜆 + 36

8(𝜆 + 1)2(𝜆 + 2)2(𝜆 + 3)(𝜆 + 4)
] 𝑝4 .                            (5.15) 



Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 32 No. 2 (2025) 

 

402 
https://internationalpubls.com  

For |𝑥| = 1, we obtain 

Δ(𝑝, 1) = [
𝑝2(4 − 𝑝2)

8(𝜆 + 4)
−
𝑝(4 − 𝑝2)

2(𝜆 + 4)
]

  + [
(−𝜆2 + 6𝜆 + 9)(4 − 𝑝2)𝑝2

8(𝜆 + 1)(𝜆 + 3)(𝜆 + 4)
+

4 − 𝑝2

2(𝜆 + 4)
]

  + [
(4 − 𝑝2)2

4(2 + 𝜆)(𝜆 + 4)
−
(3𝜆2 + 7𝜆 + 3)(4 − 𝑝2)𝑝

2(𝜆 + 1)(𝜆 + 3)(𝜆 + 4)
−
(4 − 𝑝2)

2(𝜆 + 4)
]

  + [
−3𝜆4 + 11𝜆3 + 9𝜆2 − 35𝜆 − 6

8(𝜆 + 1)2(𝜆 + 2)(𝜆 + 3)(𝜆 + 4)
𝑝2(4 − 𝑝2) +

𝑝(4 − 𝑝2)

2(𝜆 + 2)
]

  + [
3𝜆2 + 7𝜆 + 3

2(𝜆 + 1)(𝜆 + 3)(𝜆 + 4)
] 𝑝(4 − 𝑝2)

 

                                           +
4 − 𝑝2

2(𝜆 + 4)
+ [

𝜆4 − 9𝜆3 + 29𝜆2 − 45𝜆 + 36

8(𝜆 + 1)2(𝜆 + 2)2(𝜆 + 3)(𝜆 + 4)
] 𝑝4    .                       (5.16) 

which has maximum value 
|𝜕(𝜆)|

(𝜆+1)2(𝜆+2)(𝜆+3)(𝜆+4)
 attained at the end point 𝑝 = 2 and 

4

𝜆+2
 at 𝑝 = 0. 

Where, 

𝒥(𝜆) = 2(𝜆4 − 9𝜆3 + 29𝜆2 − 45𝜆 + 36). (5.17) 

6. Krushkal Inequality for the class 𝓐(𝝀) 

Theorem 6.1. Let 𝑓 given by (1.1), be in the class 𝓐(𝝀); (0 ≤ 𝜆 ≤ 1). Then we have sharp bound 

|𝑎4 − 𝑎2
3| = max {

4

𝜆 + 3
,

|ℒ(𝜆)|

(𝜆 + 1)3(𝜆 + 2)(𝜆 + 3)
}. 

where, 

ℒ(𝜆) = 𝜆4 − 5𝜆3 − 5𝜆2 − 3𝜆 − 12. (6.1) 

Proof. First note that by equating the corresponding coefficients in the equation (3.1). We get, in the 

view of (3.2) and (3.4), we may write 2𝑝2 = 𝑝1
2 + 𝑥(4 − 𝑝1

2), 𝑌 = (1 − |𝑥|2)𝜚 and applying Lemma 

(2.2), a simple computation leads to 

𝑎4 − 𝑎2
3 = [

𝑝1
3(1 − 𝜆)2

(𝜆 + 1)(𝜆 + 2)(𝜆 + 3)
+
𝑝1𝑝2(1 − 𝜆)(3 + 2𝜆)

(𝜆 + 1)(𝜆 + 2)(𝜆 + 3)
+

𝑝3
𝜆 + 3

] − [
𝑝1
𝜆 + 1

]
3

. (6.2) 

Note that, by Lemma (2.2), we have 

𝑎4 − 𝑎2
3 = [

(1 − 𝜆)2

(𝜆 + 1)(𝜆 + 2)(𝜆 + 3)
+

(1 − 𝜆)(3 + 2𝜆)

2(𝜆 + 1)(𝜆 + 2)(𝜆 + 3)
+

1

4(𝜆 + 3)
−

1

(𝜆 + 1)3
] 𝑝1

3 

                + [
(1 − 𝜆)(3 + 2𝜆)

2(𝜆 + 1)(𝜆 + 2)(𝜆 + 3)
+

1

2(𝜆 + 3)
] 𝑝1𝑋𝑥 −

𝑝1𝑋𝑥

4(𝜆 + 3)
+

𝑋𝑌

2(𝜆 + 3)
    .                       (6.3) 

Without loss of generality, we let 0 ≤ 𝑝1 = 𝑝 ≤ 2. Substitute this into the above equation, we obtain 

the following quadratic equation in terms of 𝑥. 



Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 32 No. 2 (2025) 

 

403 
https://internationalpubls.com  

|𝑎4 − 𝑎2
3| ≤ [

𝑝(4 − 𝑝2)

4(𝜆 + 3)
−

4 − 𝑝2

2(𝜆 + 3)
] |𝑥|2

  + [[
−𝜆2 + 2𝜆 + 5

2(1 + 𝜆)(2 + 𝜆)(3 + 𝜆)
] 𝑝(4 − 𝑝2)] |𝑥|

  + [
𝜆4 − 5𝜆3 − 5𝜆2 − 3𝜆 − 12

4(𝜆 + 1)3(𝜆 + 2)(𝜆 + 3)
] 𝑝3 +

4 − 𝑝2

2(𝜆 + 3)
.

 

                                                       = Γ(𝑝, |𝑥|)         .                                                                               (6.4) 

So, the maximum value of Γ(𝑝, |𝑥|) on [0,2] × [0,1]. First, assume that there is a maximum at an 

interior point Γ(𝑝0, |𝑥0|) of [0,2] × [0,1]. Differentiating Γ(𝑝, |𝑥|) with respect to |𝑥| and equating it 

to 0 implies that 𝑝 = 𝑝0 = 2 which is contradiction. Thus, for the maximum value of Γ(𝑝, |𝑥|), we 

need to consider the end points of [0,2] × [0,1]. 

For 𝑝 = 0, we obtain 

             Γ(0, |𝑥|) = [
−4

2(𝜆 + 3)
] |𝑥|2 +

4

2(𝜆 + 3)
≤

4

𝜆 + 3
   .                                                                (6.5) 

For 𝑝 = 2, we get 

Γ(2, |𝑥|) =
|ℒ(𝜆)|

(𝜆 + 1)3(𝜆 + 2)(𝜆 + 3)
. (6.6) 

For |𝑥| = 0, we gain 

Γ(𝑝, 0) = [
𝜆4 − 5𝜆3 − 5𝜆2 − 3𝜆 − 12

4(𝜆 + 1)3(𝜆 + 2)(𝜆 + 3)
] 𝑝3 +

4 − 𝑝2

2(𝜆 + 3)
. (6.7) 

For |𝑥| = 1, we have 

Γ(𝑝, 1) = [
𝑝(4 − 𝑝2)

4(𝜆 + 3)
−

4 − 𝑝2

2(𝜆 + 3)
]

  + [
−𝜆2 + 2𝜆 + 5

2(1 + 𝜆)(2 + 𝜆)(3 + 𝜆)
] 𝑝(4 − 𝑝2)

 

                                                           + [
𝜆4 − 5𝜆3 − 5𝜆2 − 3𝜆 − 12

4(𝜆 + 1)3(𝜆 + 2)(𝜆 + 3)
] 𝑝3 +

4 − 𝑝2

2(𝜆 + 3)
  .                        (6.8) 

which has maximum value 
|ℒ(𝜆)|

(𝜆+1)3(𝜆+2)(𝜆+3)
 attained at the end point 𝑝 = 2 and 

4

𝜆+3
 at 𝑝 = 0. 

Where, 

ℒ(𝜆) = 𝜆4 − 5𝜆3 − 5𝜆2 − 3𝜆 − 12. (6.9) 

Theorem 6.2. Let 𝑓 given by (1.1), be in the class 𝓐(𝝀); (0 ≤ 𝜆 ≤ 1). Then we have sharp bound 

|𝑎5 − 𝑎2
4| ≤ max {

4(1 + 𝜆)

(𝜆 + 2)(𝜆 + 4)
+

6

𝜆 + 4
,

|𝑄(𝜆)|

(𝜆 + 1)4(𝜆 + 2)(𝜆 + 3)(𝜆 + 4)
} . (6.10) 

where 



Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 32 No. 2 (2025) 

 

404 
https://internationalpubls.com  

𝒬(𝜆) = 2(𝜆6 + 21𝜆5 + 6𝜆4 − 54𝜆3 − 43𝜆2 − 87𝜆 − 132). (6.11) 

Proof. First note that by equating the corresponding coefficients in the equation (3.1) we get, in the 

view of (3.2) and (3.5), we may write 

𝑎5 − 𝑎2
4  =

𝑝1
4(1 − 𝜆)3

(𝜆 + 1)(𝜆 + 2)(𝜆 + 3)(𝜆 + 4)
+

𝑝1
2𝑝2(1 − 𝜆)

2(3 + 2𝜆)

(𝜆 + 1)(𝜆 + 2)(𝜆 + 3)(𝜆 + 4)
(6.12)

 +
𝑝1𝑝3(1 − 𝜆)

(𝜆 + 3)(𝜆 + 4)
+

𝑝1
2𝑝2(1 − 𝜆)

2

(𝜆 + 1)(𝜆 + 2)(𝜆 + 4)
+

𝑝2
2(1 − 𝜆)

(𝜆 + 2)(𝜆 + 4)
+

𝑝1𝑝3(1 − 𝜆)

(𝜆 + 1)(𝜆 + 4)

 +
𝑝4
𝜆 + 4

− (
𝑝1
𝜆 + 1

)
4

. (6.13)

 

Note that, by Lemma (2.2) and we have 2𝑝2 = 𝑝1
2 + 𝑥(4 − 𝑝1

2), 𝑌 = (1 − |𝑥|2)𝜚, a simple 

computation leads to 

𝑎5 − 𝑎2
4 = [

𝜆6+21𝜆5+6𝜆4−54𝜆3−43𝜆2−87𝜆−132

8(𝜆+1)4(𝜆+2)(𝜆+3)(𝜆+4)
] 𝑝1

4  + [
3𝜆3−14𝜆2−7𝜆+90

8(𝜆+1)(𝜆+2)(𝜆+3)(𝜆+4)
] 𝑝2𝑥𝑋

  + [
−𝜆2+2𝜆+7

2(𝜆+1)(𝜆+3)(𝜆+4)
] 𝑝1𝑋𝑌 − [

−𝜆2+8𝜆+17

8(𝜆+1)(𝜆+3)(𝜆+4
] 𝑝1

2𝑋𝑥2

  

  + [
(1−𝜆)

4(𝜆+2)(𝜆+4)
] 𝑋2𝑥2 +

𝑝1
2𝑋𝑥3

8(𝜆+4)
+

𝑋𝑥2

2(𝜆+4)
−

𝑋𝑌𝑝1𝑥

2(𝜆+4)
−

𝑋𝑌�̅�

2(𝜆+4)

    (6.14)          

without loss of generality, we let 0 ≤ 𝑝1 = 𝑝 ≤ 2. Substitute this into the above equation, we obtain 

the following quadratic equation in terms of 𝑥. 

|𝑎5 − 𝑎2
4| ≤ [

𝑝2(4 − 𝑝2)

8(𝜆 + 4)
−
𝑝(4 − 𝑝2)

2(𝜆 + 4)
] |𝑥|3

  + [
(−𝜆2 + 2𝜆 + 7)𝑝(4 − 𝑝2)

2(𝜆 + 1)(𝜆 + 3)(𝜆 + 4)
+
(−𝜆2 + 8𝜆 + 17)𝑝2(4 − 𝑝2)

8(𝜆 + 1)(𝜆 + 3)(𝜆 + 4)
] |𝑥|2

  + [
(4 − 𝑝2)2(1 − 𝜆)

4(𝜆 + 2)(𝜆 + 4)
+
(4 − 𝑝2)

2(𝜆 + 4)
−
(4 − 𝑝2)𝑥‾

2(𝜆 + 4)
] |𝑥|2

  + [[
(3𝜆3 − 14𝜆2 − 7𝜆 + 90)𝑝2

8(𝜆 + 1)(𝜆 + 3)(𝜆 + 4)(𝜆 + 2)
+

𝑝

2 + 𝜆
] (4 − 𝑝2)] |𝑥|

  + [[
−𝜆2 + 2𝜆 + 7

2(𝜆 + 1)(𝜆 + 3)(𝜆 + 4)
] 𝑝 +

𝑥‾

2(𝜆 + 4)
] (4 − 𝑝2)

  + [
𝜆6 + 21𝜆5 + 6𝜆4 − 54𝜆3 − 43𝜆2 − 87𝜆 − 132

8(𝜆 + 1)4(𝜆 + 2)(𝜆 + 3)(𝜆 + 4)
] 𝑝4

  = Λ(𝑝, |𝑥|)

 

We need to prove that the maximum value of Λ(𝑝, |𝑥|) on [0,2] × [0,1]. First assume, that there is a 

maximum at an interior point Λ(𝑝0, |𝑥0|) of [0,2] × [0,1]. Differentiating Λ(𝑝, |𝑥|) with respect to |𝑥| 

and equating it to 0 implies that 𝑝 = 𝑝0 = 2 which is contradiction. Thus, for the maximum of 

Λ(𝑝, |𝑥|), we have to consider the end points of [0,2] × [0,1]. For 𝑝 = 0 we obtain 



Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 32 No. 2 (2025) 

 

405 
https://internationalpubls.com  

Λ(0, |𝑥|)  = [
4(1 − 𝜆)

(𝜆 + 2)(𝜆 + 4)
+

2

(𝜆 + 4)
−

2𝑥‾

(𝜆 + 4)
] |𝑥|2 +

2𝑥‾

(𝜆 + 4)
. (6.15)

 ≤
4(1 − 𝜆)

(𝜆 + 2)(𝜆 + 4)
+

6

𝜆 + 4
. (6.16)

 

For 𝑝 = 2, we obtain 

Λ(2, |𝑥|) =
|𝒬(𝜆)|

(𝜆 + 1)4(𝜆 + 2)(𝜆 + 3)(𝜆 + 4)
. (6.17) 

For |𝑥| = 0, we have 

Λ(𝑝, 0) = [
𝜆6 + 21𝜆5 + 6𝜆4 − 54𝜆3 − 43𝜆2 − 87𝜆 − 132

8(𝜆 + 1)4(𝜆 + 2)(𝜆 + 3)(𝜆 + 4)
] 𝑝4 

+[
−𝜆2 + 2𝜆 + 7

2(𝜆 + 1)(𝜆 + 3)(𝜆 + 4)
] 𝑝(4 − 𝑝2)  . 

       (6.18)  

For |𝑥| = 1, we get 

Λ(𝑝, 1) = [
𝑝2(4 − 𝑝2)

8(𝜆 + 4)
−
𝑝(4 − 𝑝2)

2(𝜆 + 4)
]

  + [
(−𝜆2 + 2𝜆 + 7)𝑝(4 − 𝑝2)

2(𝜆 + 1)(𝜆 + 3)(𝜆 + 4)
+
(−𝜆2 + 8𝜆 + 17)𝑝2(4 − 𝑝2)

8(𝜆 + 1)(𝜆 + 3)(𝜆 + 4)
]

  + [
(4 − 𝑝2)2(1 − 𝜆)

4(𝜆 + 2)(𝜆 + 4)
+
(4 − 𝑝2)

2(𝜆 + 4)
−
(4 − 𝑝2)

2(𝜆 + 4)
]

  + [[
(3𝜆3 − 14𝜆2 − 7𝜆 + 90)𝑝2

8(𝜆 + 1)(𝜆 + 3)(𝜆 + 4)(𝜆 + 2)
+

𝑝

2 + 𝜆
] (4 − 𝑝2)]

  + [[
−𝜆2 + 2𝜆 + 7

2(𝜆 + 1)(𝜆 + 3)(𝜆 + 4
] 𝑝 +

1

2(𝜆 + 4)
] (4 − 𝑝2)

  + [
𝜆6 + 21𝜆5 + 6𝜆4 − 54𝜆3 − 43𝜆2 − 87𝜆 − 132

8(𝜆 + 1)4(𝜆 + 2)(𝜆 + 3)(𝜆 + 4)
] 𝑝4.

 

which has maximum value 
|𝑄(𝜆)|

(𝜆+1)4(𝜆+2)(𝜆+3)(𝜆+4)
 attained at the end point 𝑝 = 2 and  

4(1−𝜆)

(𝜆+2)(𝜆+4)
+

6

𝜆+4
 at 𝑝 = 0. 

Where, 

𝑄(𝜆) = 2(𝜆6 + 21𝜆5 + 6𝜆4 − 54𝜆3 − 43𝜆2 − 87𝜆 − 132). (6.19) 

Conclusion 

In this Present paper Toeplitz matrices characterized by coefficients from novel subclasses. The 

investigation establishes upper limits for the initial four determinants of these matrices, presenting 

innovative and unique findings. Notably, our results parallel recent works by Thomas and Halim [1], 



Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 32 No. 2 (2025) 

 

406 
https://internationalpubls.com  

specifically in the context of star-like and close - to - convex functions, as well as by Radhika et al. 

[2], which concentrate on functions with bounded boundary rotation. Additionally, we have 

determined the Zalcman and Generalized Zalcman conjecture, along with Krushkal inequalities for 

certain parameters. This contributes to the existing body of knowledge in the field and 

demonstrates the novelty of our results in comparison to recent literature. The result obtained perhaps 

give an opportunity for researchers to further investigate inequalities problems for functions of the 

class 𝒜 as well as other subclasses 𝒮. 

Data Availability: No data were used in this paper. 

Ethical Approval: This article does not contain any studies with human participants or animals 

performed by any of the authors. 

Conflicts of Interest: The authors confirm no competing interests. 

Funding Statement: The research did not receive any funding. 

Author's Contributions: The authors read and approved the final manuscript. 

References 

[1]  D. K. Thomas, S. A. Halim,: Toeplitz matrices whose elements are the coefficients of star-like and close - to - convex 

functions, Bull. Malays. Math. Sci. Soc., 2016. 

 https://dx.doi.org/10.1007/s40840-016-0385-4  , (published online). 

[2]  V. Radhika, S. Sivasubramanian, G. Murugusundaramoorthy, J. M. Jahangiri, : Toeplitz matrices whose elements are 

the co - efficient of functions with bounded boundary rotation,                         J. Complex Analysis, 2016 .                                                                     

https://dx.doi.org/10.1155/2016/4960704. 

[3]  P. L. Duren,: Univalent functions, Grundlehren der Mathematischen Wissenschaften, Springer, New York, 1983. 

[4]  Al - Oboudi F.M., 𝑛 - Bazilevič functions, Abstr. Appl. Anal., 2012, 1 - 10.     https://dx.doi.org/10.1155/2012/383592. 

[5]  A. A. Amer, M. Darus, Distortion theorem for certain class of Bazilevič functions,  Int. J. Math. Anal. 2012, Volume 

6(9 − 12), 591 − 597. 

[6]  Y. C. Kim, H. M. Srivastava, The Hardy space of a certain subclass of Bazilevič functions, Appl. Math. Comput, 

2006, Volume 183(2), 1201 – 1207 https://dx.doi.org/10.1016/j.amc.2006.06.044  .  

[7]  Y. C. Kim, T. Sugawa,: A note on Bazilevič functions, Taiwanese J. Math., 2009, Volume 13(5), 1489 – 1495                                                                               

https://dx.doi.org/10.11650/twjm/1500405555. 

[8]  R. Singh, On Bazilevič functions, Proc. Amer. Math. Soc. 1973, 38, 261 - 271. 

[9]  S. L. Krushkal,: Univalent functions and holomorphic motions, J. Anal. Math, 1995,  Volume 66(1), 253 − 275. 

[10]  R. J. Libera, E. J. Zloktiewicz,: Coefficients bounds for the inverse of a function with derivative in q, Proc. Amer. 

Math. Soc., 1983, Volume 87(2), 251 - 257.   https://dx.doi.org/10.1090/S0002-9939-1983-0681830-8. 

[11] D. A. Brannan,: On functions of bounded boundary rotation, Proceedings of the Edinburgh Mathematical Society, 

1968, Volume 16(2), 339 - 347.    https://dx.doi.org/10.2307/2039289 . 

[12] B. Pinchuk, A Variational method for functions of bounded boundary rotation, Transactions of the American 

Mathematical Society, 1969, Volume 138, 107 - 113.  https://dx.doi.org/10.1090/S0002-9947-1969-0237761-8. 

[13]  M. Obradović, N. Tuneski,: Zalcman and Generalized Zalcman conjecture for a subclass of univalent functions , Novi 

Sad J. Math, 2022, Volume 52(1), 185 - 190. https://dx.doi.org/10.30755/NSJOM.12436. 

[14]  K. Ye and L - H. Lim,: Every matrix is a product of Toeplitz matrices , Foundations of Computational Mathematics, 

2016, Volume 16(3), 577 - 598. https://dx.doi.org/10.48550/arXiv.1307.5132  .  

[15]  V. Radhika, J. M.Jahangiri, S. Sivasubramanian, G. Murugusundaramoorthy,: Toeplitz matrices whose elements are 

co - efficient of Bazilevič functions De Gruyter Open Access, 2018.    

  https://dx.doi.org/10.1515/math-2018-0093  . 

https://dx.doi.org/10.1090/S0002-9939-1983-0681830-8
https://dx.doi.org/10.2307/2039289
https://dx.doi.org/10.1090/S0002-9947-1969-0237761-8
https://dx.doi.org/10.30755/NSJOM
https://dx.doi.org/10.48550/arXiv.1307.5132
https://dx.doi.org/10.1515/math-2018-0093


Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 32 No. 2 (2025) 

 

407 
https://internationalpubls.com  

[16]  R. J. Libera, E. J. Zloktiewicz,: Coefficient bounds for the inverse of a function with derivative in 𝒫, Proceedings of 

the American mathematical Society . 

https://dx.doi.org/10.1090/S0002-9939-1983-0681830-8. 

[17]  M. Raza, S. N. Malik, Upper bound of the third Hankel determinant for a class of analytic functions related with 

Lemniscate of Bernoulli, J. Inequal. Appl, 2013.     https://dx.doi.org/10.1186/1029-242X-2013-412 . 

[18]  A. K. Sahoo and J. Patel,: Hankel determinant for a class of analytic functions related with Lemniscate of Bernoulli, 

Int. J. Anal. Appl., 2014, Volume 6(2), 170 - 177. 

[19]  J. Sokí, J. Stankiewicz, Radius of convexity of some subclasses of strongly star - like functions, Folia Scient. Univ. 

Tech. Resoviensis, Matematykaz.19, 1996, 101 - 105. 

[20]  C. Pommerenke, On the coefficients and Hankel determinants of univalent functions,  J. London Math. Soc, 1966, 

Volume 41, 111 - 122.                      https://dx.doi.org/10.1112/JLMS/S1-41.1.111. 

[21]  T.Panigrahi, J Sokól,: Coefficient inequalities For A Class of analytic functions associated with the Lemniscate af 

Bernoulli, Bol. Soc. Paran. Mat, 2019, Volume 37(4), 83 - 95.    https://dx.doi.org/10.5269/bspm.v37i4.32701 .  

[22]  L. De Branges,: A proof of the Bieberbach conjecture, Acta Math, 1985 , Volume 154(1-2),137 - 152.                                                                                                

https://dx.doi.org/10.1007/BF02392821  . 

[23] S. L. Krushkal,: A short geometric proof of the Zalcman and Bieberbach conjectures, 2014.    

https://dx.doi.org/10.48550/arXiv.1408.1948 . 

[24] W. Ma,: Generalized Zalcman conjecture for star - like and typically real functions,  J. Math. Anal. Appl, 1999, 

Volume 234(1), 328 - 339.   https://dx.doi.org/10.1006/jmaa.1999.6378    

[25]  M. Obradovic, N. Tuneski,: Some properties of the class U, Ann. Univ.Mariae Curie-Sk lodowska Sect. 2019, Volume 

73(1), 49 - 56.  https://dx.doi.org/10.48550/arXiv. 1812.08503. 

[26]  S. Ozaki, M. Nunokawa,: The Schwarzian derivative and univalent functions, Proc. Amer. Math. Soc, 1972, Volume 

33(2), 392 - 394.  https://dx.doi.org/10.2307/2038067.  

[27]  V. Ravichandran, S. Verma,: Generalized Zalcman conjecture for some classes of analytic functions, J. Math. Anal. 

Appl, 2017, Volume 450(1), 592 - 605. https://dx.doi.org/10.1016/j.jmaa.2017.01.053  . 

[28]  D. K. Thomas, N. Tuneski, A. Vasudevarao,: Univalent functions, A primer, De Gruyter Studies in Mathematics. De 

Gruyter, Berlin, 2018 , Volume 69.   https://dx.doi.org/10.1515/9783110560961. 

[29]  S.L. Krushkal,: Proof of the Zalcman conjecture for initial coefficients, Georgian Math. J., 2010, Volume 17, 663 - 

681.   https://dx.doi.org/10.1515/gmj-2012-0099. 

[30]  R. J. Libera, E. J. Zlotkiewicz,: Early coefficients of the inverse of a regular convex function, Proc. Amer. Math. Soc., 

1982, Volume 85(2), 225 - 230.   https://dx.doi.org/10.2307/2044286. 

[31]  W. Ma and D. Minda,: A unified treatment of some special classes of univalent functions. In Proceedings of the 

conference on Complex Analysis, Tianjin, China, 19-23, June, 1992, Conference on Proceedings Lecture Notes for 

Analysis. International Press; Cambridge, MA, USA, 1994,  157 - 169. 

 

 

https://dx.doi.org/10.1090/S0002-9939-1983-0681830-8
https://dx.doi.org/10.1186/1029-242X-2013-412
https://dx.doi.org/10.1112/JLMS/S1-41.1.111
https://dx.doi.org/10.5269/bspm.v37i4.32701
https://dx.doi.org/10.1007/BF02392821
https://dx.doi.org/10.48550/arXiv.1408.1948
https://dx.doi.org/10.1006/jmaa
https://dx.doi.org/10.48550/arXiv.%201812.08503
https://dx.doi.org/10.2307/2038067
https://dx.doi.org/10.1016/j.jmaa.2017.01.053
https://dx.doi.org/10.1515/9783110560961
https://dx.doi.org/10.2307/2044286