Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 31 No. 6s (2024) 

 

49 https://internationalpubls.com 

Second Hankel Inequality for Certain Common Subclass of Classes of 

Starlike and Convex Functions 

 

Gurmeet Singh1, Gourav Saini2, Paras Uchat3, Dipa Sharma4, Chatinder Kaur5 
1Department of Mathematics, GSSDGS Khalsa College Patiala-14700, meetgur111@gmail.com 

2 Research scholar, Department of Mathematics, Punjabi University, Patiala-147002, 

sainig953@gmail.com 
3Assistant Professor, IITE, Gandhinagar, Gujarat, parasu@iite.ac.in 

4Department of mathematics, Sri Dev Suman Uttarakhand University Pt.L.M.S.Campus Rishikesh, 

Dehradun, Uttarakhand, dipa2014sharma@gmail.com 
5 Research scholar, Department of Mathematics, Punjabi University, Patiala-147002, 

chatinderkaur@gmail.com 

 

Article History: 

Received: 25-05-2024 

Revised: 28-06-2024 

Accepted: 20-07-2024 

Abstract:  

In this paper we will define a new class S* Csin (r) which is subordinate to function 1+sin z, 

we will find the Fekete - Szegö inequality for this class along with Fekete - Szegö 

inequality of the functions of this class defined S* Csin (r,Θ) through Poisson distribution. 

Further we have solved the second Hankel determinant of this new class. 

Keywords: Geometric function, Analytic univalent functions, Poisson distribution, 

subordination, Fekete - Szegö inequality, Upper bounds, Hankel determinant, coefficient 

inequalities. 

 

1. Introduction 

The class of all the analytic functions in a unit disk 𝔻:= {𝑧 ∈ ℂ: |𝑧| < 1}, whose Taylor's series 

expansion is of the form 

ℎ(𝑧):= 𝑧 +∑  

∞

𝑛=2

 𝑎𝑛𝑧
𝑛 = 𝑧 + 𝑎2𝑧

2 + 𝑎3𝑧
3 + 𝑎4𝑧

4⋯ ∀𝑧 ∈ 𝔻 (1) 

and normalized by the conditions: ℎ(0) = 0, ℎ′(0) = 1, is denoted by 𝒜. 

Let 𝒮 denotes a subclass of 𝒜 of all univalent analytic functions In the unit disk 𝔻. The class of 

Analytic -Univalent functions, with Taylor's series expansion of the form 

𝑝(𝑧) = 1 +∑  

∞

𝑛=1

  𝑐𝑛𝑧
𝑛 (2) 

in 𝔻, such that ℜ𝑒(𝑃(𝑧)) > 0 is denoted by 𝒫 

In 1916 a German Mathematician Ludwig Bieberbach proposed a conjecture on the coefficients of 

analytic functions of from (1) in 𝒮, i.e 

|𝑎𝑛| ≤ 𝑛, 𝑛 ∈ ℕ 



Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 31 No. 6s (2024) 

 

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This conjecture was known by the name of Bieberbach conjecture (1916) [2]. Until 1985 this 

conjecture was considered as very challenging problem in Geometric function theory of Complex 

Analysis. 

after 69 years of this conjecture, a French-American Mathematician Louis de Branges de 

Bourcia(1985) solved this conjecture [3].Before de Branges's proof many scholars around the world 

tried to prove or disprove this conjecture, as a consequences of their these efforts they found multiple 

subfamilies of class 𝒮. 

The most common subfamilies of 𝒮 are convex, star-like and close-to-convex functions whose set 

builder form is given by 

𝐶:= {ℎ ∈ 𝒮:ℜ𝑒 (
(𝑧(ℎ′(𝑧))′

ℎ′(𝑧)
) > 0, ∀𝑧 ∈ 𝔻}

𝑆∗: = {ℎ ∈ 𝒮:ℜ𝑒 (
𝑧ℎ′(𝑧)

ℎ(𝑧)
) > 0, ∀𝑧 ∈ 𝔻}

𝑅:= {ℎ ∈ 𝒮:ℜ𝑒[ℎ′(𝑧)] > 0, ∀𝑧 ∈ 𝔻}

 

 Two functions ℎ and 𝑔 in 𝒜,ℎ is said to be subordinated to 𝑔, or written as ℎ(𝑧) ≺ 𝑔(𝑧), if we have 

a Schwarz functions 𝜔(𝑧) analytic over 𝔻 with 𝜔(0) = 0 and also |𝜔(𝑧)| < 1, such that 

ℎ(𝑧) = 𝑔(𝜔(𝑧))  ∀𝑧 ∈ 𝔻 

, but if function 𝑔(𝑧) is univalent in 𝔻, then 

ℎ(𝑧) ≺ 𝑔(𝑧) iff ℎ(0) = 𝑔(0) and ℎ(𝔻) ⊂ 𝑔(𝔻) 

Ma and Minda [4] introduced two classes of analytic functions which are. 

𝑆∗(𝜙):= {ℎ ∈ 𝒜:
𝑧ℎ′(𝑧)

ℎ(𝑧)
≺ 𝜙(𝑧), ∀𝑧 ∈ 𝔻} 

and 

𝐶(𝜙):= {ℎ ∈ 𝒜: 1 +
𝑧ℎ′′(𝑧)

ℎ′(𝑧)
≺ 𝜙(𝑧), ∀𝑧 ∈ 𝔻} 

The function 𝜙(𝑧) is an univalent analytic function with positive real part in the unit disk 𝔻 such that 

𝜙(0) = 1, 𝜙′(0) > 0 where 𝜙 maps the open unit disk onto a region starlike with respect to 1 and 

symmetric with respect to the real axis, several other classes can be formed by varying the function 

𝜙, some of the examples are as follows 

• When 𝜙 = 𝑒𝑧, this class is denoted by 𝑆𝑒
∗, check out [5, 6] for more Details. 

• When 𝜙 = 1 +
2

𝜋2
(log 

1+√𝑧

1−√𝑧
)
2

, we get a new class, for further details see [7] 

• When 𝜙 =
1+𝐶𝑧

1+𝐷𝑧
(−1 ≤ 𝐷 << 𝐶 ≤ 1), we get the class 𝑆∗(𝐶, 𝐷). See [8] for more Details. 

• When 𝜙 = cosh (z), this new class is denoted by 𝑆cosh 
∗  see.[9] 



Communications on Applied Nonlinear Analysis 

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• When 𝜙 = 1 + sin (𝑧), the class is denoted by 𝑆sin
∗ , For more details see [10, 11] 

C. Pommerenke (1966-67), [12, 13] stated the 𝑝th  Hankel determinant for 𝑝 ≥ 1 and 𝑛 ≥ 1 where 

𝑝, 𝑛 ∈ ℕ of functions ℎ of form ??is defined as 

ℋ(𝑝,𝑛)(ℎ) = |

𝑎𝑛 𝑎𝑛+1 ⋯ 𝑎𝑛+𝑝−1
𝑎𝑛+1 𝑎𝑛+2 ⋯ 𝑎𝑛+𝑝
⋮ ⋮ ⋮ ⋮
𝑎𝑛+𝑝−1 𝑎𝑛+𝑝 ⋯ 𝑎𝑛+2𝑝−2

| 

In Geometric function theory of Complex analysis finding upper bounds of Hankel determinant of 

various subfamilies of 𝒜 is a widely famous and an interesting problem. Noonan(1976) and 

Noor(1983) [14, 15] studied the growth rate of ℋ(𝑝,𝑛) for fixed values of 𝑝 and 𝑛, as 𝑛 → ∞ of 

different subfamilies of the univalent function of class 𝒮. 

where 

ℋ(2,2)(𝑓) = |
𝑎2 𝑎3
𝑎3 𝑎4

| = 𝑎2𝑎4 − 𝑎3
2 

From past many years, a huge collection of research papers have been dedicated for finding the 

upper bounds for various orders of Hankel determinant, some recent work on second, third and 

fourth - order Hankel determinants see [[16] - [21]], Recently, Cho et al. [10] introduced the 

following function class 𝑆sin 
∗  

𝑆sin
∗ : = {ℎ ∈ 𝒜: 1 +

𝑧ℎ′(𝑧)

ℎ(𝑧)
≺ 1 + sin 𝑧, ∀𝑧 ∈ 𝔻} 

Various researchers established Fekete - Szegö inequality for various classes afterwards ([25] - [31]) 

 

. Lets define a new subclass 𝑆∗𝐶sin (𝑟)of 𝒜. This subclass contains all those Analytic univalent 

functions in 𝒜 which satisfies, 

(
𝑧ℎ′(𝑧)

ℎ(𝑧)
)

𝑟

(
(𝑧ℎ′(𝑧))′

ℎ′(𝑧)
)

1−𝑟

, 𝑧 ∈ 𝔻

𝑆∗𝐶sin(𝑟):= {ℎ ∈ 𝒜: (
𝑧ℎ′(𝑧)

ℎ(𝑧)
)

𝑟

(
(𝑧ℎ′(𝑧))′

ℎ′(𝑧)
)

1−𝑟

≺ 1 + sin (𝑧)}

 

From above we have 

𝑆∗𝐶sin(0):= 𝐶sin = 𝐶sin: = {ℎ ∈ 𝒜: (
(𝑧ℎ′(𝑧))′

ℎ′(𝑧)
) ≺ 1 + sin (𝑧)} 

and 

𝑆∗𝐶sin(1):= 𝑆sin
∗ : = {ℎ ∈ 𝒜: (

𝑧ℎ′(𝑧)

ℎ(𝑧)
) ≺ 1 + sin (𝑧)} 

 



Communications on Applied Nonlinear Analysis 

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2. Preliminary Lemmas. 

Let 𝒫 denotes the class of analytic functions of the form 

𝑝(𝑧) = 1 + ∑  ∞
𝑛=1   𝑐𝑛𝑧

𝑛, 𝑧 ∈ 𝔻     (3) 

whereℜ(𝑝(𝑧)) > 0 in 𝔻. 

Lemma 2.1. (see [13]) If 𝑝(𝑧) ∈ 𝒫, and 𝑐𝑛 be the 𝑛th cofficients of 𝑃(𝑧), then 

|𝑐𝑛| ≤ 2  for all 𝑛 ∈ ℕ     (4)  

|𝑐2 − 𝛾𝑐1
2| ≤ 2𝑚𝑎𝑥{1, |2𝛾 − 1|} where 𝛾 ∈ ℂ    (5)  

|𝑐𝑛+𝑚 − 𝛾𝑐𝑛𝑐𝑚| ≤ 2, 𝛾 ∈ [0,1], Where 𝑛,𝑚 ∈ ℕ    (6) 

|𝑐2 − 𝛿𝑐1
2| ≤ {

−4𝛿 + 2,  if 𝛿 ≤ 0,
2,  if 0 ≤ 𝛿 ≤ 1,
4𝛿 − 2,  if 1 ≤ 𝛿,

     (7)  

Lemma 2.2.(see[17]) If 𝑃(𝑧) ∈ 𝒫 then there exists 𝑥, 𝑧 ∈ �̅� with |𝑥| ≤ 1, |𝑦| ≤ 1, such that 

 2𝑐2 = 𝑐1
2 + 𝑥(4 − 𝑐1

2)      (8) 

4𝑐3 = 𝑐1
3 + 2(4 − 𝑐1

2)𝑐1𝑥 − (4 − 𝑐1
2)𝑐1𝑥

2 + 2(4 − 𝑐1
2)(1 − |𝑥|2)𝑦  (9)  

3. Important Theorems 

. Theorem 3.1. If the function ℎ(𝑧) ∈ 𝑆∗𝐶sin (𝑟) and is of the form (1), then 

|𝑎2| ≤
1

(2 − 𝑟)

|𝑎3| ≤
1

2(3 − 2𝑟)
max {1, |

𝑟2 + 5𝑟 − 8

2(−2 + 𝑟)2
|}

 

and 

|𝑎3 − 𝛿𝑎2
2| ≤

1

2(3 − 2𝑟)
max {1, |

𝑟2 + 5𝑟 + 4𝛿(3 − 2𝑟) − 8

2(𝑟 − 2)2
|} 

Proof.  From definition of subordination we have 

𝑆∗𝐶sin(𝑟) = {ℎ ∈ 𝒜: (
𝑧ℎ′(𝑧)

ℎ(𝑧)
)

𝑟

(
(𝑧ℎ′(𝑧))′

ℎ′(𝑧)
)

1−𝑟

≺ 1 + sin (𝑧)} 

(
𝑧ℎ′(𝑧)

ℎ(𝑧)
)
𝑟

(
(𝑧ℎ′(𝑧))

′

ℎ′(𝑧)
)
1−𝑟

=
𝑧𝑟[ℎ′(𝑧)]

2𝑟−1
[(𝑧ℎ′(𝑧))

′
]
1−𝑟

[ℎ(𝑧)]𝑟
   (10)  

After expanding 
𝑧𝑟[ℎ′(𝑧)]

2𝑟−1
[(𝑧ℎ′(𝑧))

′
]
1−𝑟

[ℎ(𝑧)]𝑟
, in a series form we have 



Communications on Applied Nonlinear Analysis 

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=1 + 𝑎2(2 − 𝑟)𝑧 + 𝑧
2 (
1

2
𝑎2
2(𝑟2 + 5𝑟 − 8) + 2𝑎3(3 − 2𝑟))

+𝑧3 (𝑎3𝑎2(4𝑟
2 + 11𝑟 − 18) +

1

6
𝑎2
3(−𝑟3 − 21𝑟2 − 20𝑟 + 48)

+3𝑎4(4 − 3𝑟)) + 𝑧
4(𝑎4𝑎2(9𝑟

2 + 19𝑟 − 32) + 2𝑎3
2(4𝑟2 + 4𝑟 − 9)

−2𝑎3𝑎2
2(𝑟3 + 16𝑟2 + 5𝑟 − 24) +

1

24
𝑎2
4(𝑟4 + 46𝑟3 + 371𝑟2 − 58𝑟 − 384) + ⋯

 

(11)  

Using principal of subordination and expending 1 + sin (𝜔(𝑧)) in series form 

1 + sin (𝜔(𝑧)) =1 +
𝑐1𝑧

2
+ (

𝑐2

2
−
𝑐1
2

4
) 𝑧2 +

1

48
(5𝑐1

3 − 24𝑐2𝑐1 + 24𝑐3)𝑧
3

+
1

32
(−𝑐1

4 + 10𝑐2𝑐1
2 − 16𝑐3𝑐1 − 8𝑐2

2 + 16𝑐4)𝑧
4

   (12)  

Comparing (11) and (12), we will get 

𝑎2 =
−𝑐1

2(𝑟−2)
      (13) 

𝑎3 =
3𝑐1

2𝑟2−4𝑐2𝑟
2−3𝑐1

2𝑟+16𝑐2𝑟−16𝑐2

16(𝑟−2)2(2𝑟−3)
     (14) 

𝑎4 =
1

288(𝑟−2)3(6𝑟2−17𝑟+12)
[−52𝑐1

3𝑟4 + 144𝑐1𝑐2𝑟
4 − 96𝑐3𝑟

4 + 165𝑐1
3𝑟3 − 780𝑐1𝑐2𝑟

3 +

720𝑐3𝑟
3−205𝑐1

3𝑟2 + 1464𝑐1𝑐2𝑟
2 − 2016𝑐3𝑟

2 + 46𝑐1
3𝑟 − 1104𝑐1𝑐2𝑟 + 2496𝑐3𝑟 + 48𝑐1

3 +

288𝑐1𝑐2 − 1152𝑐3]        (15)  

From equation (13) and (14) 

|𝑎2| ≤
1

2 − 𝑟

𝑎3 =
1

4(3 − 2𝑟)
[𝑐2 −

𝑐1
2(3𝑟2 − 3𝑟)

4(−2 + 𝑟)2
]

|𝑎3| ≤
1

4(3 − 2𝑟)
|𝑐2 − 𝛿𝑐1

2|

 

Where 𝛿 =
3𝑟2−3𝑟

4(−2+𝑟)2
 on applying Lemma (3.1) 

|𝑎3| =
1

2(3 − 2𝑟)
max {1, |

𝑟2 + 5𝑟 − 8

2(−2 + 𝑟)2
|} 

Now 

𝑎3 − 𝛿𝑎2
2 =

1

4(3−2𝑟)
[𝑐2 −

𝑐1
2(4𝛿(3−2𝑟)+3𝑟2−3𝑟)

4(𝑟−2)2
]     (16)  

Again using lemma we will Have 

|𝑎3 − 𝛿𝑎2
2| ≤

1

2(3 − 2𝑟)
max {1, |

𝑟2 + 5𝑟 + 4𝛿(3 − 2𝑟) − 8

2(𝑟 − 2)2
|} 



Communications on Applied Nonlinear Analysis 

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Theorem 3.2 : If function ℎ ∈ 𝑆∗𝐶sin , then 

|𝑎3 − 𝛿𝑎2
2| ≤

{
  
 

  
 

−1

4(3 − 2𝑟)
[
𝑟2 + 5𝑟 − 8

(−2 + 𝑟)2
+
4𝛿(3 − 2𝑟)

(−2 + 𝑟)2
]  if 𝛿 ≤

−3𝑟2 + 3𝑟

4(3 − 2𝑟)

1

2(3 − 2𝑟)
 if 
−3𝑟2 + 3𝑟

4(3 − 2𝑟)
≤ 𝛿 ≤

𝑟2 − 13𝑟 + 16

4(3 − 2𝑟)

1

4(3 − 2𝑟)
[
𝑟2 + 5𝑟 − 8

(−2 + 𝑟)2
+
4𝛿(3 − 2𝑟)

(−2 + 𝑟)2
]  if 

(𝑟2 − 13𝑟 + 16)

4(3 − 2𝑟)
≤ 𝛿

 

Proof. From (16) we have 

𝑎3 − 𝛿𝑎2
2 =

1

4(3−2𝑟)
[𝑐2 −

𝑐1
2(4𝛿(3−2𝑟)+3𝑟2−3𝑟)

4(𝑟−2)2
]    (17)  

this can be written as 

𝑎3 − 𝛿𝑎2
2 =

1

4(3 − 2𝑟)
[𝑐2 − 𝛿𝑐1

2] 

where𝛿:=
𝑐1
2(4𝛿(3−2𝑟)+3𝑟2−3𝑟)

4(𝑟−2)2
 

Using lemma (3.1) in (17) we have 

|𝑎3 − 𝛿𝑎2
2| ≤

{
  
 

  
 

−1

4(3 − 2𝑟)
[
𝑟2 + 5𝑟 − 8

(−2 + 𝑟)2
+
4𝛿(3 − 2𝑟)

(−2 + 𝑟)2
]  if 𝛿 ≤

−3𝑟2 + 3𝑟

4(3 − 2𝑟)

1

2(3 − 2𝑟)
 if 
−3𝑟2 + 3𝑟

4(3 − 2𝑟)
≤ 𝛿 ≤

𝑟2 − 13𝑟 + 16

4(3 − 2𝑟)

1

4(3 − 2𝑟)
[
𝑟2 + 5𝑟 − 8

(−2 + 𝑟)2
+
4𝛿(3 − 2𝑟)

(−2 + 𝑟)2
]  if 

(𝑟2 − 13𝑟 + 16)

4(3 − 2𝑟)
≤ 𝛿

 

Function defined with Poisson Distribution 

A discrete random variable 𝜉 is said to be poisson distribution it it takes the values 0,1,2,3,4,5⋯ 

with probabilities𝑒−𝜂 , 𝜂
𝑒−𝜂

1!
, 𝜂2

𝑒−𝜂

2!
, 𝜂3

𝑒−𝜂

3!
, 𝜂4

𝑒−𝜂

4!
+ 𝜂5

𝑒−𝜂

5!
+⋯, respectively, where 𝜂 > 0 is called 

the parameter. Thus 

𝑃(𝜉 = 𝑘) =
𝜂𝑘𝑒−𝜂

𝑘!
, 𝑘 = 0,1,2,3,⋯ 

A Power series with coefficients from probabilities of Poisson distribution, was introduced by 

Prowal, [17] 

𝑃(𝜂, 𝑧):= 𝑧 +∑  

∞

𝑛=2

𝜂𝑛−1

(𝑛 − 1)!
𝑒−𝜂𝑧𝑛, 𝑧 ∈ ℂ 

Where 𝜂 > 0. By using ratio test we can easily establish that radius of convergence of above series is 

Infinity. Recent works by G. Murugusundaramoorthy, et al and S. Porwal et al [23, 24], let the linear 

Operator 



Communications on Applied Nonlinear Analysis 

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𝑃𝜂(𝑧):𝒜 → 𝒜 

be given by 

𝑃𝜂ℎ(𝑧) =𝑃(𝜂, 𝑧) ⋆ ℎ(𝑧)

= 𝑧 +∑  

∞

𝑛=2

 
𝜂𝑛−1

(𝑛 − 1)!
𝑒−𝜂𝑎𝑛𝑧

𝑛

= 𝑧 +∑  

∞

𝑛=2

 Θ𝑛(𝜂)𝑎𝑛𝑧
𝑛

 

Where Θ𝑛 = Θ(𝜂) =
𝜂𝑛−1

(𝑛−1)!
𝑒−𝜂 and ⋆ denote the convolution or the hadmard product of the two 

Series. 

Here we will define a new class 𝑆∗𝐶sin(𝑟, Θ) whose functions having power series whose 

coefficients are probabilities of poisson Distribution Functions defined by Poisson distribution 

𝑆∗𝐶sin(𝑟, Θ):= {ℎ ∈ 𝒜: [
𝑧[𝑃𝜂ℎ(𝑧)]′

𝑃𝜂ℎ(𝑧)
]

𝑟

[
[𝑧(𝑃𝜂ℎ(𝑧))′]′

(𝑃𝜂ℎ(𝑧))′
]

1−𝑟

≺ 1 + sin (𝑧)} 

Theorem 3.3. Let 0 ≤ 𝑟 ≤ 1, 𝛿 ∈ ℂ if ℎ ∈ 𝑆∗𝐶sin(𝑟, Θ), and 𝑃𝜂ℎ(𝑧) = 𝑧 + Θ2𝑎2𝑧
2 + Θ3𝑎3𝑧

3 +

Θ4𝑎4𝑧
4 + Θ5𝑎5𝑧

5 +⋯ then we have 

|𝑎3 − 𝛿𝑎2
2| ≤

1

2(3 − 2𝑟)Θ3
max {1, |

3𝑟2 − 3𝑟

2(𝑟 − 2)2
−
4𝛿(2𝑟 − 3)Θ3

2(𝑟 − 2)2Θ2
2 |} 

Proof.  We have ℎ ∈ 𝑆∗𝐶sin(𝑟, Θ) which is defined as 𝑃𝜂ℎ(𝑧) = 𝑧 + Θ2𝑎2𝑧
2 + Θ3𝑎3𝑧

3 + Θ4𝑎4𝑧
4 +

Θ5𝑎5𝑧
5 +⋯. From the definition we Have 

[
𝑧[𝑃𝜂ℎ(𝑧)]′

𝑃𝜂ℎ(𝑧)
]

𝑟

[
[𝑧(𝑃𝜂ℎ(𝑧))′]′

(𝑃𝜂ℎ(𝑧))′
]

1−𝑟

= 1 + sin (𝑤(𝑧)) 

expansion of [
𝑧[𝑃𝜂ℎ(𝑧)]′

𝑃𝜂ℎ(𝑧)
]
𝑟

[
[𝑧(𝑃𝜂ℎ(𝑧))′]

′

(𝑃𝜂ℎ(𝑧))′
]
1−𝑟

is 

1 + 𝑎2Θ2(2 − 𝑟)𝑧 + 𝑧
2 (
1

2
𝑎2
2Θ2

2(𝑟2 + 5𝑟 − 8) + 2𝑎3Θ3(3 − 2𝑟))

+𝑧3 (𝑎2𝑎3Θ3Θ2(4𝑟
2 + 11𝑟 − 18) +

1

6
𝑎2
3Θ2

3(−𝑟3 − 21𝑟2 − 20𝑟 + 48) + 3𝑎4Θ4(4 − 3𝑟))

+𝑧4(𝑎2𝑎4Θ4Θ2(9𝑟
2 + 19𝑟 − 32) + 2𝑎3

2Θ3
2(4𝑟2 + 4𝑟 − 9) + ⋯                                                                          

 

(18)  

comparing above with expansion of 1 + sin (𝑤(𝑧)) 



Communications on Applied Nonlinear Analysis 

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1 +
𝑐1𝑧

2
+ (

𝑐2
2
−
𝑐1
2

4
) 𝑧2 +

1

48
(5𝑐1

3 − 24𝑐2𝑐1 + 24𝑐3)𝑧
3

+
1

32
(−𝑐1

4 + 10𝑐2𝑐1
2 − 16𝑐3𝑐1 − 8𝑐2

2 + 16𝑐4)𝑧
4

+
(𝑐1
5 − 480𝑐2𝑐1

3 + 1200𝑐3𝑐1
2 + 1200𝑐2

2𝑐1 − 1920𝑐4𝑐1 − 1920𝑐2𝑐3 + 1920𝑐5)𝑧
5

3840
+⋯

 

We will get 

𝑎2 =
−𝑐1

2Θ2(𝑟 − 2)

𝑎3 =
3𝑐1

2𝑟2 − 4𝑐2𝑟
2 − 3𝑐1

2𝑟 + 16𝑐2𝑟 − 16𝑐2
16Θ3(𝑟 − 2)2(2𝑟 − 3)

 

𝑎4 =
1

288(𝑟 − 2)3Θ4(6𝑟
2 − 17𝑟 + 12)

[−52𝑐1
3𝑟4 + 144𝑐1𝑐2𝑟

4 − 96𝑐3𝑟
4 + 165𝑐1

3𝑟3𝑧

  −780𝑐1𝑐2𝑟
3 + 720𝑐3𝑟

3 − 250𝑐1
3𝑟2 + 1464𝑐1𝑐2𝑟

2 − 2016𝑐3𝑟
2+46𝑐1

3𝑟 − 1104𝑐1𝑐2𝑟 + 2496𝑐3𝑟 + 48𝑐1
3 + 288𝑐1𝑐2 − 1152𝑐3]

 

(19)  

From above we will have 

[
𝑧[𝑃𝜂ℎ(𝑧)]′

𝑃𝜂ℎ(𝑧)
]

𝑟

[
[𝑧(𝑃𝜂ℎ(𝑧))′]′

(𝑃𝜂ℎ(𝑧))′
]

1−𝑟

= 1 + sin (𝑤(𝑧)) 

𝑎3 − 𝛿𝑎2
2 =

1

4(3−2𝑟)Θ3
[c2 −

𝑐1
2(4𝛿(3−2𝑟)Θ3+(3𝑟

2−3𝑟)Θ2
2𝑟)

4(𝑟−2)2Θ2
2 ]   (20) 

From lemma (3.1) we have 

|𝑎3 − 𝛿𝑎2
2| ≤

1

2(3 − 2𝑟)Θ3
max {1, |

3𝑟2 − 3𝑟

2(𝑟 − 2)2
−
4𝛿(2𝑟 − 3)Θ3

2(𝑟 − 2)2Θ2
2 |} 

Theorem 3.4. Let 0 ≤ 𝑟 ≤ 1and 𝑃𝜂ℎ(𝑧) = 𝑧 + Θ2𝑎2𝑧
2 + Θ3𝑎3𝑧

3 + Θ4𝑎4𝑧
4 + Θ5𝑎5𝑧

5 +⋯, with 

𝛿 ∈ ℝ Then 

|𝑎3 − 𝛿𝑎2
2| ≤

{
 
 
 

 
 
 

−1

4(3 − 2𝑟)Θ3
[
𝑟2 + 5𝑟 − 8

(−2 + 𝑟)2
+
4𝛿(3 − 2𝑟)Θ3

(−2 + 𝑟)2Θ2
2 ]  if 𝛿 ≤

(−3𝑟2 + 3𝑟)Θ2
2

4(3 − 2𝑟)Θ3
1

2(3 − 2𝑟)Θ3
 if 
(−3𝑟2 + 3𝑟)Θ3

4(3 − 2𝑟)Θ2
2 ≤ 𝛿 ≤

(𝑟2 − 13𝑟 + 16)Θ2
2

4(3 − 2𝑟)Θ3
1

4(3 − 2𝑟)Θ3
[
𝑟2 + 5𝑟 − 8

(−2 + 𝑟)2
+
4𝛿(3 − 2𝑟)Θ3
(−2 + 𝑟)2

]  if 
(𝑟2 − 13𝑟 + 16)Θ2

2

4(3 − 2𝑟)Θ3
≤ 𝛿

 

Proof. From (20) and along with using lemma (3.1) we will get the desired Result. 

Second Hankel inequality for 𝒉 ∈ 𝑺∗𝑪𝐬𝐢𝐧 

Theorem 3.5. If the function ℎ ∈ 𝑆∗𝐶sin(𝑟) then 

|𝑎2𝑎4 − 𝑎3
2| ≤

1

4(3 − 2𝑟)2
 



Communications on Applied Nonlinear Analysis 

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57 https://internationalpubls.com 

Proof. From (15), we Have 

𝑎4 =
1

288(𝑟 − 2)3(6𝑟2 − 17𝑟 + 12)
[−52𝑐1

3𝑟4 + 144𝑐1𝑐2𝑟
4 − 96𝑐3𝑟

4 + 165𝑐1
3𝑟3 − 780𝑐1𝑐2𝑟

3 + 720𝑐3𝑟
3 

−205𝑐1
3𝑟2  +  1464𝑐1𝑐2𝑟

2  −  2016𝑐3𝑟
2  +  46𝑐1

3𝑟 −  1104𝑐1𝑐2 𝑟 +  2496𝑐3 𝑟 +  48𝑐1
3  +  288𝑐1𝑐2  −

 1152𝑐3          (21)  

Using (13), (14) and (15) we will have 

𝑎2𝑎4 − 𝑎3
2 =

1

2304
[−
24𝑐2𝑐1

2(21𝑟3 − 77𝑟2 + 90𝑟 − 36)

(3 − 2𝑟)2(𝑟 − 2)2(3𝑟 − 4)

+
𝑐1
4(173𝑟5 − 1134𝑟4 + 2729𝑟3 − 2504𝑟2 + 168𝑟 + 576)

(3 − 2𝑟)2(𝑟 − 2)4(3𝑟 − 4)
+

192𝑐3𝑐1
(𝑟 − 2)(3𝑟 − 4)

−
144𝑐2

2

(3 − 2𝑟)2
]

 

Without loss of generality we can say That 𝑐:= 𝑐1, where |𝑐1| ≤ 2 and substituting the values of 𝑐2 

and 𝑐3 we have 

(𝑎2𝑎4 − 𝑎3
2) =

1

2304
[
96(1 − |𝑥|2)(4 − 𝑐2)𝑐𝑦

(𝑟 − 2)(3𝑟 − 4)
− 𝑥2(4 − 𝑐2) (

48𝑐2

(𝑟 − 2)(3𝑟 − 4)
+
36(4 − 𝑐2)

(3 − 2𝑟)2
)

−
12𝑥(4 − 𝑐2)𝑐2(12 − 6𝑟 − 13𝑟27𝑟3)

(3 − 2𝑟)2(𝑟 − 2)2(3𝑟 − 4)
+
𝑐4(5𝑟5 + 78𝑟4 − 607𝑟3 + 1816𝑟2 − 2424𝑟 + 1152)

(3 − 2𝑟)2(𝑟 − 2)4(3𝑟 − 4)
]

 

Replacing |𝑥| by 𝑏 and using triangular inequality and the inequality |𝑦| ≤ 1 in above we will get 

|𝑎2𝑎4 − 𝑎3
2| ≤

1

2304
[
96(1 − 𝑏2)(4 − 𝑐2)𝑐

(𝑟 − 2)(3𝑟 − 4)
+ 𝑏2(4 − 𝑐2) (

48𝑐2

(𝑟 − 2)(3𝑟 − 4)
+
36(4 − 𝑐2)

(3 − 2𝑟)2
)

+
12𝑏(4 − 𝑐2)𝑐2(12 − 6𝑟 − 3𝑟2 + 7𝑟3)

(3 − 2𝑟)2(𝑟 − 2)2(3𝑟 − 4)
+
𝑐4(5𝑟5 + 78𝑟4 − 607𝑟3 + 1816𝑟2 − 2424𝑟 + 1152)

(3 − 2𝑟)2(𝑟 − 2)4(4 − 3𝑟)
]

 

Let us assume that above is 

𝐼(𝑐, 𝑏) =
1

2304
[
96(1 − 𝑏2)(4 − 𝑐2)𝑐

(𝑟 − 2)(3𝑟 − 4)
+ 𝑏2(4 − 𝑐2) (

48𝑐2

(𝑟 − 2)(3𝑟 − 4)
+
36(4 − 𝑐2)

(3 − 2𝑟)2
)

+
12𝑏(4 − 𝑐2)𝑐2(12 − 6𝑟 − 13𝑟2 + 7𝑟3)

(3 − 2𝑟)2(𝑟 − 2)2(3𝑟 − 4)
+
𝑐4(5𝑟5 + 78𝑟4 − 607𝑟3 + 1816𝑟2 − 2424𝑟 + 1152)

(3 − 2𝑟)2(𝑟 − 2)4(3𝑟 − 4)
]

 

We have to maximize the function 𝐼(𝑐, 𝑏)for (𝑐, 𝑏) ∈ [0,2] × [0,1], now differentiation the above 

functions partially with respect to 𝑏 we will have 

∂𝐼

∂𝑏
=

1

2304
[
−192(4 − 𝑐2)𝑐

(𝑟 − 2)(3𝑟 − 4)
+2𝑏(4 − 𝑐2) (

48𝑐2

(𝑟 − 2)(3𝑟 − 4)
+
36(4 − 𝑐2)

(3 − 2𝑟)2
)

+
12(4 − 𝑐2)𝑐2(12 − 6𝑟 − 13𝑟2 + 7𝑟3)

(3 − 2𝑟)2(𝑟 − 2)2(3𝑟 − 4)
]

 

For 0 ≤ 𝑏 ≤ 1 and for any 𝑐 ∈ [0,2], we can easily check that 
∂𝐼

∂𝑏
> 0, from this we can conclude 

that 𝐼(𝑐, 𝑏) is an increasing function of 𝑐, and it will attain maximum value at 𝑏 = 1. 

By putting 𝑏 = 1 in above we will have 



Communications on Applied Nonlinear Analysis 

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𝐼(𝑐, 1) = 𝐸(𝑐) =
1

2304
[(4 − 𝑐2) (

48𝑐2

(𝑟 − 2)(3𝑟 − 4)
+
36(4 − 𝑐2)

(3 − 2𝑟)2
)

+
12(4 − 𝑐2)𝑐2(12 − 6𝑟 − 13𝑟2 + 7𝑟3)

(3 − 2𝑟)2(𝑟 − 2)2(3𝑟 − 4)
+
𝑐4(5𝑟5 + 78𝑟4 − 607𝑟3 + 1816𝑟2 − 2424𝑟 + 1152)

(3 − 2𝑟)2(𝑟 − 2)4(3𝑟 − 4)
]

 

We have 

𝐸′(𝑐)

=
𝑐3(−5𝑟5 + 150𝑟4 − 953𝑟3 + 2120𝑟2 − 1896𝑟 + 576) − 24𝑐(𝑟 − 2)2(9𝑟3 − 29𝑟2 + 30𝑟 − 12)

576(3 − 2𝑟)2(𝑟 − 2)4(3𝑟 − 4)
 

From 𝐸′(𝑐) = 0 we will have 𝑐 = 0 

𝐸′′(0) = −
9𝑟3 − 29𝑟2 + 30𝑟 − 12

24(3 − 2𝑟)2(𝑟 − 2)2(3𝑟 − 4)
 

that we can easily conclude that 𝐸′′(0) is Negative, which implies that at 𝑐 = 0 the function will 

attain its maxima. 

|𝑎2𝑎4 − 𝑎3
2| ≤

1

4(3 − 2𝑟)2
 

Corollary 

When 𝑟 = 0, we have ℎ ∈ 𝑆∗𝐶sin (0):= 𝐶sin  and we get 

|𝑎2𝑎4 − 𝑎3
2| ≤

1

36
 

When 𝑟 = 1 we have ℎ ∈ 𝑆∗𝐶sin(1):= 𝑆sin
∗ , and we get 

|𝑎2𝑎4 − 𝑎3
2| ≤

1

4
 

4. Conclusions 

In this paper we have introduced a new class 𝑆∗𝐶sin(𝑟), where (0 ≤ 𝑟 ≤ 1), and have worked on the 

Fekete-Szegö inequality along with upper bound of second Hankel determinant. We have stated the 

Fekete-Szegö inequality related to Poisson distribution of this new class. As ℎ ∈ 𝑆∗𝐶sin(𝑟) by fixing 

parameter 𝑟 = 0function ℎ ∈ 𝐶sin, and when we fix 𝑟 = 1 then we have ℎ ∈ 𝑆sin
∗ . Our this research 

paper is an extended work of research article [32], readers and interested scholars can extend our 

work further by working on higher order of Hankel determinant and coefficients inequalities on this 

newly introduced class. 

5. Acknowledgment 

The authors are very much thankful to the anonymous Editor and Referees for their valuable 

suggestions to improve the paper. 

Funding: Not applicable 

Conflict of interest: Not applicable 



Communications on Applied Nonlinear Analysis 

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Vol 31 No. 6s (2024) 

 

59 https://internationalpubls.com 

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Communications on Applied Nonlinear Analysis 

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