Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 31 No. 2 (2024) 

 

 

22 
https://internationalpubls.com 

Some Applications of Fractional Derivative Operator 
 

B.D. Karande1, Arun B. Damkondwar2 

1Department of Mathematics, Udaygiri Mahavidyalaya, 

Udgir, Dis: Latur, M.S, India. 

e-mail  : bdkarande2@gmail.com 
2 Department of Mathematics, Government Polytechnic, 

Hingoli - 431 513, Maharashtra, India. 

e-mail  : arundigras@gmail.com 

Coresponding Author : Arun B. Damkondwar 

 

 

Article History: 

Received: 18-01-2024 

Revised: 30-03-2024 

Accepted: 22-04-2024 

Abstract:  

Introduction: The aim of this paper is to introduce a new subclass TS(ω, σ, ς, ϑ) of univalent 

functions with negative coefficients related to fractional derivative operator in the unit disk 𝕌 =

{z ∈ ℂ: |z| < 1}. We obtain basic properties like coefficient inequality, distortion and covering 

theorem, radii of starlikeness, convexity and close-to-convexity, extreme points, Hadamard 

product, and closure theorems for functions belonging to our class 

Keywords: Univalent , derivative opertator , Starlike, Extreme points, Hadamard product. 

 

1. Introduction 

Let 𝐴 signify the class of all functions 𝑢(𝑧) of the type 

                                              𝑢(𝑧) = 𝑧 + ∑ 𝑎𝑛
∞
𝑛=2 𝑧𝑛                                                                            (1.1)                  

in the open unit disc 𝕌 = {𝑧 ∈ ℂ: |𝑧| < 1}. Let 𝑆 be the subclass of 𝐴 consisting of univalent 

functions and satisfy the following usual normalization condition 𝑢(0) = 𝑢′(0) − 1 = 0. We denote 

by 𝑆 the subclass of 𝐴 consisting of functions 𝑢(𝑧) which are all univalent in 𝕌. A function 𝑢 ∈ 𝐴 is 

a starlike function of the order 𝑚, 0 ≤ 𝑚 < 1, if it satisfy 

                                     ℜ {
𝑧𝑢′(𝑧)

𝑢(𝑧)
} > 𝑚, 𝑧 ∈ 𝕌.                                                                             (1.2) 

We denote this class with 𝑆∗(𝑚) . 

A function 𝑢 ∈ 𝐴 is a convex function of the order 𝑚, 0 ≤ 𝑚 < 1, if it fulfil 

                     ℜ {1 +
𝑧𝑢″(𝑧)

𝑢′(𝑧)
} > 𝑚, 𝑧 ∈ 𝕌.                                                     (1.3) 

We denote this class with 𝐾(𝑚). 

Note that 𝑆∗(0) = 𝑆∗ and 𝐾(0) = 𝐾 are the usual classes of starlike and convex functions in 𝕌 

respectively. 

Let 𝑇 denote the class of functions analytic in 𝕌 that are of the form 



Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 31 No. 2 (2024) 

 

 

23 
https://internationalpubls.com 

           𝑢(𝑧) = 𝑧 − ∑ 𝑎𝑛

∞

𝑛=2

𝑧𝑛,    𝑎𝑛 ≥ 0 𝑧 ∈ 𝕌                                                (1.4) 

and let 𝑇∗(𝑚) = 𝑇 ∩ 𝑆∗(𝑚),  𝐶(𝑚) = 𝑇 ∩ 𝐾(𝑚). The class 𝑇∗(𝑚) and allied classes possess some 

interesting properties and have been extensively studied by Silverman [14]. 

Many basically equivalent definitions of fractional computation have been given in literature 

((cf.)e.g.,[13] and ([15], p. 45) ). We state the following definitions due to Owa and Srivastava [9] 

which have been used rather frequently in the theory of analytic functions (see also [4] ). 

Definition 1.1.  The fractional integral of order ϑ is defined, for a function u(z) , by  

Dz
−ϑu(z) =

1

ω(ϑ)
∫

f(ζ)

(z−ζ)1−ϑ

z

0
dζ,   (ϑ > 0)                                                                                           (1.5)  

and the fractional derivative of order ς is defined, for a function u(z) , by Dz
ϑu(z) =

1

ω(1−ϑ)

d

dz
∫

f(ζ)

(z−ζ)ϑ

z

0
dζ,   (0 ≤ ϑ < 1)                                                                        (1.6) 

where u(z) is an analytic function in a simply-connected region of the z-plane containing the origin, 

and the multiplicity of (z − ζ)ϑ−1 involved in (1.5) (and that of (z − ζ)−ϑ involved in (1.6) is 

removed by requiring log(z − ζ) to be real when (z − ζ) > 0. 

Definition 1.2.  Under the hypotheses of Definition1.1, the fractional derivative of order n + ϑ is 

defined by 

     Dz
n+ϑu(z) =

dn

dzn Dz
ϑu(z),   (0 ≤ ϑ < 1; n ∈ N0 = N ∪ {0}).                                                    (1.7)  

With the aid of the above definitions, Owa and Srivastava [9]  defined the fractional operator ℐz
ϑ by 

𝒥𝑧
𝜗𝑢(𝑧) = 𝜔(2 − 𝜗)𝑧𝜗𝐷𝑧

𝜗𝑢(𝑧),   (𝜗 ≠ 2,3,4, ⋯ )

   𝒥𝑧
𝜗𝑢(𝑧) = 𝑧 + ∑ 𝛩

∞

𝑛=2

(𝜗, 𝑛)𝑎𝑛𝑧𝑛                                                                             (1.8) 

where 𝛩(𝜗, 𝑛) =
𝜔(𝑛 + 1)𝜔(2 − 𝜗)

𝜔(𝑛 − 𝜗 + 1)

and 𝛩(𝜗, 2) =
2

(2 − 𝜗)
.

 

Now, by making use of the linear operator 𝒥𝑧
𝜗𝑢, we define a new subclass of functions belonging to 

the class 𝐴. 

Definition 1.3.  For 0 ≤ ω < 1,0 ≤ σ < 1,0 < ς < 1,  and  0 ≤ ϑ < 1, we let TS(ω, σ, ς, ϑ) be the 

subclass of u consisting of functions of the form (1.4) and its geometrical condition satisfy 

|
𝜔((𝒥𝑧

𝜗𝑢(𝑧))′−
𝒥𝑧

𝜗𝑢(𝑧)

𝑧
)

𝜎(𝒥𝑧
𝜗𝑢(𝑧))′+(1−𝜔)

𝒥𝑧
𝜗𝑢(𝑧)

𝑧

| < ς,   𝑧 ∈ 𝕌 where 𝒥𝑧
𝜗 , is given by (1.8). 



Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 31 No. 2 (2024) 

 

 

24 
https://internationalpubls.com 

2. Coefficient Inequality 

In the following theorem, we obtain a necessary and sufficient condition for function to be in the 

class 𝑇𝑆(𝜔, 𝜎, ς, 𝜗). 

Theorem 2.1.  Let the function u be defined by (1.4). Then u ∈ TS(ω, σ, ς, ϑ) if and only if  

 

∑ [𝜔(𝑛 − 1) + ς(𝑛𝜎 + 1 − 𝜔)]∞
𝑛=2 𝛩(𝑛, 𝜗)𝑎𝑛 ≤ ς(𝜎 + (1 − 𝜔)),                                        (2.1)  

 

where 0 < ς < 1,0 ≤ 𝜔 < 1,0 ≤ 𝜎 < 1,  and  0 ≤ 𝜗 < 1.  

 

The result (2.1) is sharp for the function 

 𝑢(𝑧) = 𝑧 −
ς(𝜎+(1−𝜔))

[𝜔(𝑛−1)+ς(𝑛𝜎+1−𝜔)]𝛩(𝑛,𝜗)
𝑧𝑛,   𝑛 ≥ 2. 

Proof. Suppose that the inequality (2.1) holds true and |𝑧| = 1. Then we obtain 

|𝜔 ((𝒥𝑧
𝜗𝑢(𝑧)) ′ −

𝒥𝑧
𝜗𝑢(𝑧)

𝑧
)| − ς |𝜎 (𝒥𝑧

𝜗𝑢(𝑧))′ + (1 − 𝜔)
𝒥𝑧

𝜗𝑢(𝑧)

𝑧
)|

= |−𝜔 ∑(𝑛 − 1)

∞

𝑛=2

𝛩(𝑛, 𝜗)𝑎𝑛𝑧𝑛−1|

      −ς |𝜎 + (1 − 𝜔) − ∑(𝑛𝜎 + 1 − 𝜔)

∞

𝑛=2

𝛩(𝑛, 𝜗)𝑎𝑛𝑧𝑛−1|

≤ ∑[𝜔(𝑛 − 1) + ς(𝑛𝜎 + 1 − 𝜔)]

∞

𝑛=2

𝛩(𝑛, 𝜗)𝑎𝑛 − ς(𝜎 + (1 − 𝜔))

≤ 0.

 

Hence, by maximum modulus principle,𝑢 ∈ 𝑇𝑆(𝜔, 𝜎, ς, 𝜗). Now assume that 𝑢 ∈ 𝑇𝑆(𝜔, 𝜎, ς, 𝜗) so 

that 

|

|
𝜔 ((𝒥𝑧

𝜗𝑢(𝑧)) ′ −
𝒥𝑧

𝜗𝑢(𝑧)
𝑧

)

𝜎(𝒥𝑧
𝜗𝑢(𝑧))′ + (1 − 𝜔)

𝒥𝑧
𝜗𝑢(𝑧)

𝑧
|

|
< ς,   𝑧 ∈ 𝕌 

Hence 

|𝜔 ((𝒥𝑧
𝜗𝑢(𝑧)) ′ −

𝒥𝑧
𝜗𝑢(𝑧)

𝑧
)| < ς |𝜎 (𝒥𝑧

𝜗𝑢(𝑧))′ + (1 − 𝜔)
𝒥𝑧

𝜗𝑢(𝑧)

𝑧
)|. 

Therefore, we get 



Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 31 No. 2 (2024) 

 

 

25 
https://internationalpubls.com 

|− ∑ 𝜔

∞

𝑛=2

(𝑛 − 1)𝛩(𝑛, 𝜗)𝑎𝑛𝑧𝑛−1|

<  ς |𝜎 + (1 − 𝜔) − ∑(𝑛𝜎 + 1 − 𝜔)

∞

𝑛=2

𝛩(𝑛, 𝜗)𝑎𝑛𝑧𝑛−1| .

 

Thus 

∑[𝜔(𝑛 − 1) + ς(𝑛𝜎 + 1 − 𝜔)]

∞

𝑛=2

𝛩(𝑛, 𝜗)𝑎𝑛 ≤ ς(𝜎 + (1 − 𝜔)) 

and this completes the proof.  

Corollary 2.1.  Let the function 𝑢 ∈ 𝑇𝑆(𝜔, 𝜎, ς, 𝜗).Then 

 

                             𝑎𝑛 ≤
ς(𝜎+(1−𝜔))

[𝜔(𝑛−1)+ς(𝑛𝜎+1−𝜔)]𝛩(𝑛,𝜗)
𝑧𝑛,   𝑛 ≥ 2. 

3. Distortion and Covering Theorem 

We introduce the growth and distortion theorems for the functions in the class 𝑇𝑆(𝜔, 𝜎, ς, 𝜗) 

Theorem  3.1.  Let the function  𝑢 ∈ 𝑇𝑆(𝜔, 𝜎, ς, 𝜗). Then 

|𝑧| −
ς(𝜎 + (1 − 𝜔))

𝛩(2, 𝜗)[𝜔 + ς(2𝜎 + 1 − 𝜔)]
|𝑧|2 ≤ |𝑢(𝑧)|

≤ |𝑧| +
ς(𝜎 + (1 − 𝜔))

𝛩(2, 𝜗)[𝜔 + ς(2𝜎 + 1 − 𝜔)]
|𝑧|2.

 

The result is sharp and attained 

 𝑢(𝑧) = 𝑧 −
ς(𝜎+(1−𝜔))

𝛩(2,𝜗)[𝜔+ς(2𝜎+1−𝜔)]
𝑧2. 

Proof. 

|𝑢(𝑧)| = |𝑧 − ∑ 𝑎𝑛

∞

𝑛=2

𝑧𝑛| ≤ |𝑧| + ∑ 𝑎𝑛

∞

𝑛=2

|𝑧|𝑛

≤ |𝑧| + |𝑧|2 ∑ 𝑎𝑛

∞

𝑛=2

.

 

By Theorem 2.1, we get 

∑ 𝑎𝑛

∞

𝑛=2

≤
ς(𝜎 + (1 − 𝜔))

[𝜔 + ς(2𝜎 + 1 − 𝜔)]𝛩(𝑛, 𝜗)
                                                   (3.1). 



Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 31 No. 2 (2024) 

 

 

26 
https://internationalpubls.com 

Thus 

|𝑢(𝑧)| ≤ |𝑧| +
ς(𝜎 + (1 − 𝜔))

𝛩(2, 𝜗)[𝜔 + ς(2𝜎 + 1 − 𝜔)]
|𝑧|2. 

Also 

|𝑢(𝑧)| ≥ |𝑧| − ∑ 𝑎𝑛

∞

𝑛=2

|𝑧|𝑛

≥ |𝑧| − |𝑧|2 ∑ 𝑎𝑛

∞

𝑛=2

≥ |𝑧| −
ς(𝜎 + (1 − 𝜔))

𝛩(2, 𝜗)[𝜔 + ς(2𝜎 + 1 − 𝜔)]
|𝑧|2.

 

  

Theorem 3.2.  Let 𝑢 ∈ 𝑇𝑆(𝜔, 𝜎, ς, 𝜗). Then 

1 −
2ς(𝜎 + (1 − 𝜔))

𝛩(2, 𝜗)[𝜔 + ς(2𝜎 + 1 − 𝜔)]
|𝑧| ≤ |𝑢′(𝑧)| ≤ 1 +

2ς(𝜎 + (1 − 𝜔))

𝛩(2, 𝜗)[𝜔 + ς(2𝜎 + 1 − 𝜔)]
|𝑧| 

with equality for 

                                  𝑢(𝑧) = 𝑧 −
2ς(𝜎+(1−𝜔))

𝛩(2,𝜗)[𝜔+ς(2𝜎+1−𝜔)]
𝑧2. 

Proof. Notice that 

𝛩(2, 𝜗)[𝜔 + ς(2𝜎 + 1 − 𝜔)] ∑ 𝑛

∞

𝑛=2

𝑎𝑛

≤ ∑ 𝑛

∞

𝑛=2

[𝜔(𝑛 − 1) + ς(𝑛𝜎 + 1 − 𝜔)]𝛩(𝑛, 𝜗)𝑎𝑛

≤ ς(𝜎 + (1 − 𝜔)),                                                                               (3.2)

 

from Theorem 2.1. Thus 

|𝑢′(𝑧)| = |1 − ∑ 𝑛

∞

𝑛=2

𝑎𝑛𝑧𝑛−1|

≤ 1 + ∑ 𝑛

∞

𝑛=2

𝑎𝑛|𝑧|𝑛−1

≤ 1 + |𝑧| ∑ 𝑛

∞

𝑛=2

𝑎𝑛

≤ 1 + |𝑧|
2ς(𝜎 + (1 − 𝜔))

𝛩(2, 𝜗)[𝜔 + ς(2𝜎 + 1 − 𝜔)]
.                                       (3.3)

 



Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 31 No. 2 (2024) 

 

 

27 
https://internationalpubls.com 

On the other hand 

|𝑢′(𝑧)| = |1 − ∑ 𝑛

∞

𝑛=2

𝑎𝑛𝑧𝑛−1|

≥ 1 − ∑ 𝑛

∞

𝑛=2

𝑎𝑛|𝑧|𝑛−1

≥ 1 − |𝑧| ∑ 𝑛

∞

𝑛=2

𝑎𝑛

≥ 1 − |𝑧|
2ς(𝜎 + (1 − 𝜔))

𝛩(2, 𝜗)[𝜔 + ς(2𝜎 + 1 − 𝜔)]
                        (3.4).

 

 

Combining  (3.3) and (3.4), we get the result.  

4. Radii of Starlikeness, Convexity and Close-to-Convexity 

In the following theorems, we obtain the radii of starlikeness, convexity and close-to-convexity for 

the class 𝑇𝑆(𝜔, 𝜎, ς, 𝜗). 

Theorem 4.1.  Let 𝑢 ∈ 𝑇𝑆(𝜔, 𝜎, ς, 𝜗). Then 𝑢 is starlike in |𝑧| < 𝑅1 of order 𝛿,   0 ≤ 𝛿 < 1, where 

𝑅1 = inf
𝑛

{
(1−𝛿)(𝜔(𝑛−1)+ς(𝑛𝜎+1−𝜔))𝛩(𝑛,𝜗)

(𝑛−𝛿)ς(𝜎+(1−𝜔))
}

1

𝑛−1
,   𝑛 ≥ 2                                                         (4.1). 

Proof. 𝑢 is starlike of order 𝛿, 0 ≤ 𝛿 < 1 if 

ℜ {
𝑧𝑢′(𝑧)

𝑢(𝑧)
} > 𝛿. 

Thus it is enough to show that 

|
𝑧𝑢′(𝑧)

𝑢(𝑧)
− 1| = |

− ∑ (𝑛 − 1)∞
𝑛=2 𝑎𝑛𝑧𝑛−1

1 − ∑ 𝑎𝑛
∞
𝑛=2 𝑧𝑛−1

| ≤
∑ (𝑛 − 1)∞

𝑛=2 𝑎𝑛|𝑧|𝑛−1

1 − ∑ 𝑎𝑛
∞
𝑛=2 |𝑧|𝑛−1

.                     (4.2) 

Thus 

|
𝑧𝑢′(𝑧)

𝑢(𝑧)
− 1| ≤ 1 − 𝛿  𝑖𝑓  ∑

(𝑛 − 𝛿)

(1 − 𝛿)

∞

𝑛=2

𝑎𝑛|𝑧|𝑛−1 ≤ 1. 

Hence by Theorem 2.1,  (4.2)   will be true if 

𝑛 − 𝛿

1 − 𝛿
|𝑧|𝑛−1 ≤

(𝜔(𝑛 − 1) + ς(𝑛𝜎 + 1 − 𝜔))𝛩(𝑛, 𝜗)

ς(𝜎 + (1 − 𝜔)
 



Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 31 No. 2 (2024) 

 

 

28 
https://internationalpubls.com 

or if 

|𝑧| ≤ [
(1 − 𝛿)(𝜔(𝑛 − 1) + ς(𝑛𝜎 + 1 − 𝜔))𝛩(𝑛, 𝜗)

(𝑛 − 𝛿)ς(𝜎 + (1 − 𝜔))
]

1
𝑛−1

, 𝑛 ≥ 2. 

The theorem is proved. 

Theorem 4.2.  Let 𝑢 ∈ 𝑇𝑆(𝜔, 𝜎, ς, 𝜗). Then u is convex in |z| < R2 of order δ, 0 ≤ δ < 1, where 

𝑅2 = inf
𝑛

{
(1−𝛿)(𝜔(𝑛−1)+ς(𝑛𝜎+1−𝜔))𝛩(𝑛,𝜗)

𝑛(𝑛−𝛿)ς(𝜎+(1−𝜔))
}

1

𝑛−1
,   𝑛 ≥ 2.                                                          (4.3) 

Proof. 𝑢 is convex of order 𝛿, 0 ≤ 𝛿 < 1 if 

ℜ {1 +
𝑧𝑢″(𝑧)

𝑢′(𝑧)
} > 𝛿. 

Thus it is enough to show that 

|
𝑧𝑢″(𝑧)

𝑢′(𝑧)
| = |

− ∑ 𝑛∞
𝑛=2 (𝑛 − 1)𝑎𝑛𝑧𝑛−1

1 − ∑ 𝑛∞
𝑛=2 𝑎𝑛𝑧𝑛−1

| ≤
∑ 𝑛∞

𝑛=2 (𝑛 − 1)𝑎𝑛|𝑧|𝑛−1

1 − ∑ 𝑛∞
𝑛=2 𝑎𝑛|𝑧|𝑛−1

.                 (4.4) 

Thus 

|
𝑧𝑢″(𝑧)

𝑢′(𝑧)
| ≤ 1 − 𝛿  𝑖𝑓  ∑

𝑛(𝑛 − 𝛿)

(1 − 𝛿)

∞

𝑛=2

𝑎𝑛|𝑧|𝑛−1 ≤ 1. 

Hence by Theorem 2.1,    (4.4)will be true if 

𝑛(𝑛 − 𝛿)

1 − 𝛿
|𝑧|𝑛−1 ≤

(𝜔(𝑛 − 1) + ς(𝑛𝜎 + 1 − 𝜔))𝛩(𝑛, 𝜗)

ς(𝜎 + (1 − 𝜔)
 

or if 

|𝑧| ≤ [
(1 − 𝛿)(𝜔(𝑛 − 1) + ς(𝑛𝜎 + 1 − 𝜔))𝛩(𝑛, 𝜗)

𝑛(𝑛 − 𝛿)ς(𝜎 + (1 − 𝜔))
]

1
𝑛−1

, 𝑛 ≥ 2. 

The theorem proved. 

Theorem 4.3.  Let 𝑢 ∈ 𝑇𝑆(𝜔, 𝜎, ς, 𝜗). Then u is close-to-convex in |z| < R3 of order δ,   0 ≤ δ < 1, 

where 𝑅3 = inf
𝑛

{
(1−𝛿)(𝜔(𝑛−1)+ς(𝑛𝜎+1−𝜔))𝛩(𝑛,𝜗)

𝑛ς(𝜎+(1−𝜔))
}

1

𝑛−1
,   𝑛 ≥ 2.                                            (4.5)  

Proof. 𝑢 is close-to-convex of order 𝛿, 0 ≤ 𝛿 < 1 if 

ℜ{𝑢′(𝑧)} > 𝛿. 

Thus it is enough to show that 



Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 31 No. 2 (2024) 

 

 

29 
https://internationalpubls.com 

|𝑢′(𝑧) − 1| = |− ∑ 𝑛

∞

𝑛=2

𝑎𝑛𝑧𝑛−1| ≤ ∑ 𝑛

∞

𝑛=2

𝑎𝑛|𝑧|𝑛−1. 

Thus 

|𝑢′(𝑧) − 1| ≤ 1 − 𝛿  𝑖𝑓  ∑
𝑛

(1 − 𝛿)

∞

𝑛=2

𝑎𝑛|𝑧|𝑛−1 ≤ 1.                            (4.6) 

Hence by Theorem 2.1, (4.6)   will be true if 

𝑛

1 − 𝛿
|𝑧|𝑛−1 ≤

(𝜔(𝑛 − 1) + ς(𝑛𝜎 + 1 − 𝜔))𝛩(𝑛, 𝜗)

ς(𝜎 + (1 − 𝜔)
 

or if 

|𝑧| ≤ [
(1 − 𝛿)(𝜔(𝑛 − 1) + ς(𝑛𝜎 + 1 − 𝜔))𝛩(𝑛, 𝜗)

𝑛ς(𝜎 + (1 − 𝜔))
]

1
𝑛−1

, 𝑛 ≥ 2. 

The theorem follows. 

5. Extreme Points 

In the following theorem, we obtain extreme points for the class 𝑇𝑆(𝜔, 𝜎, ς, 𝜗). 

Theorem 5.1.  Let 𝑢1(𝑧) = 𝑧 and 𝑢𝑛(𝑧) = 𝑧 −
𝜍(𝜎+(1−𝜔))

[𝜔(𝑛−1)+𝜍(𝑛𝜎+1−𝜔)]𝛩(𝑛,𝜗)
𝑧𝑛,   for  𝑛 = 2,3, ⋯. Then 

𝑢 ∈ 𝑇𝑆(𝜔, 𝜎, 𝜍, 𝜗) if and only if it can be expressed in the form 

𝑢(𝑧) = ∑ 𝜃𝑛
∞
𝑛=1 𝑢𝑛(𝑧),   𝑤ℎ𝑒𝑟𝑒  𝜃𝑛 ≥ 0  𝑎𝑛𝑑  ∑ 𝜃𝑛

∞
𝑛=1 = 1. 

Proof. Assume that 𝑢(𝑧) = ∑ 𝜃𝑛
∞
𝑛=1 𝑢𝑛(𝑧), hence we get 

𝑢(𝑧) = 𝑧 − ∑
ς(𝜎 + (1 − 𝜔))𝜃𝑛

[𝜔(𝑛 − 1) + ς(𝑛𝜎 + 1 − 𝜔)]𝛩(𝑛, 𝜗)

∞

𝑛=2

𝑧𝑛. 

Now, 𝑢 ∈ 𝑇𝑆(𝜔, 𝜎, ς, 𝜗), since 

∑
[𝜔(𝑛 − 1) + ς(𝑛𝜎 + 1 − 𝜔)]𝛩(𝑛, 𝜗)

ς(𝜎 + (1 − 𝜔))

∞

𝑛=2

    ×
ς(𝜎 + (1 − 𝜔))𝜃𝑛

[𝜔(𝑛 − 1) + ς(𝑛𝜎 + 1 − 𝜔)]𝛩(𝑛, 𝜗)

= ∑ 𝜃𝑛

∞

𝑛=2

= 1 − 𝜃1 ≤ 1.

 



Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 31 No. 2 (2024) 

 

 

30 
https://internationalpubls.com 

Conversely, suppose 𝑢 ∈ 𝑇𝑆(𝜔, 𝜎, ς, 𝜗). Then we show that 𝑢 can be written in the form 

∑ 𝜃𝑛
∞
𝑛=1 𝑢𝑛(𝑧). 

Now 𝑢 ∈ 𝑇𝑆(𝜔, 𝜎, ς, 𝜗) implies from Theorem 2.1, 

𝑎𝑛 ≤
ς(𝜎 + (1 − 𝜔))

[𝜔(𝑛 − 1) + ς(𝑛𝜎 + 1 − 𝜔)]𝛩(𝑛, 𝜗)
. 

Setting 𝜃𝑛 =
[𝜔(𝑛−1)+ς(𝑛𝜎+1−𝜔)]𝛩(𝑛,𝜗)

ς(𝜎+(1−𝜔))
𝑎𝑛, 𝑛 = 2,3, ⋯ 

and 𝜃1 = 1 − ∑ 𝜃𝑛
∞
𝑛=2 , we obtain 𝑢(𝑧) = ∑ 𝜃𝑛

∞
𝑛=1 𝑢𝑛(𝑧).  

6. Hadamard product 

In the following theorem, we obtain the convolution result for functions belongs to the class 

𝑇𝑆(𝜔, 𝜎, ς, 𝜗). 

Theorem 6.1.  Let 𝑢, 𝑔 ∈ 𝑇𝑆(𝜔, 𝜎, 𝜍, 𝜗). Then 𝑢 ∗ 𝑔 ∈ 𝑇𝑆(𝜔, 𝜎, 𝜁, 𝜗) for  

 

𝑢(𝑧) = 𝑧 − ∑ 𝑎𝑛
∞
𝑛=2 𝑧𝑛, 𝑔(𝑧) = 𝑧 − ∑ 𝑏𝑛

∞
𝑛=2 𝑧𝑛  and (𝑢 ∗ 𝑔)(𝑧) = 𝑧 − ∑ 𝑎𝑛

∞
𝑛=2 𝑏𝑛𝑧𝑛, where  

𝜁 ≥
𝜍2(𝜎 + (1 − 𝜔))𝜔(𝑛 − 1)

[𝜔(𝑛 − 1) + 𝜍(𝑛𝜎 + 1 − 𝜔)]2𝛩(𝑛, 𝜗) − 𝜍2(𝜎 + (1 − 𝜔))(𝑛𝜎 + 1 − 𝜔)
. 

Proof. 𝑢 ∈ 𝑇𝑆(𝜔, 𝜎, ς, 𝜗) and so 

∑
[𝜔(𝑛 − 1) + ς(𝑛𝜎 + 1 − 𝜔)]𝛩(𝑛, 𝜗)

ς(𝜎 + (1 − 𝜔))

∞

𝑛=2

𝑎𝑛 ≤ 1, 

and 

∑
[𝜔(𝑛 − 1) + ς(𝑛𝜎 + 1 − 𝜔)]𝛩(𝑛, 𝜗)

ς(𝜎 + (1 − 𝜔))

∞

𝑛=2

𝑏𝑛 ≤ 1. 

We have to find the smallest number 𝜁 such that 

∑
[𝜔(𝑛 − 1) + 𝜁(𝑛𝜎 + 1 − 𝜔)]𝛩(𝑛, 𝜗)

𝜁(𝜎 + (1 − 𝜔))

∞

𝑛=2

𝑎𝑛𝑏𝑛 ≤ 1. 

By Cauchy-Schwarz inequality 

∑
[𝜔(𝑛 − 1) + ς(𝑛𝜎 + 1 − 𝜔)]𝛩(𝑛, 𝜗)

ς(𝜎 + (1 − 𝜔))

∞

𝑛=2

√𝑎𝑛𝑏𝑛 ≤ 1.                       (6.1) 

Therefore it is enough to show that 



Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 31 No. 2 (2024) 

 

 

31 
https://internationalpubls.com 

[𝜔(𝑛 − 1) + 𝜁(𝑛𝜎 + 1 − 𝜔)]𝛩(𝑛, 𝜗)

𝜁(𝜎 + (1 − 𝜔))
𝑎𝑛𝑏𝑛

≤
[𝜔(𝑛 − 1) + ς(𝑛𝜎 + 1 − 𝜔)]𝛩(𝑛, 𝜗)

ς(𝜎 + (1 − 𝜔))
√𝑎𝑛𝑏𝑛.

 

That is 

√𝑎𝑛𝑏𝑛 ≤
[𝜔(𝑛 − 1) + ς(𝑛𝜎 + 1 − 𝜔)]𝜁

[𝜔(𝑛 − 1) + 𝜁(𝑛𝜎 + 1 − 𝜔)]ς
. 

From (6.1) 

√𝑎𝑛𝑏𝑛 ≤
ς(𝜎 + (1 − 𝜔))

[𝜔(𝑛 − 1) + ς(𝑛𝜎 + 1 − 𝜔)]𝛩(𝑛, 𝜗)
. 

Thus it is enough to show that 

ς(𝜎 + (1 − 𝜔))

[𝜔(𝑛 − 1) + ς(𝑛𝜎 + 1 − 𝜔)]𝛩(𝑛, 𝜗)
≤

[𝜔(𝑛 − 1) + ς(𝑛𝜎 + 1 − 𝜔)]𝜁

[𝜔(𝑛 − 1) + 𝜁(𝑛𝜎 + 1 − 𝜔)]ς
, 

which simplifies to 

𝜁 ≥
ς2(𝜎 + (1 − 𝜔))𝜔(𝑛 − 1)

[𝜔(𝑛 − 1) + ς(𝑛𝜎 + 1 − 𝜔)]2𝛩(𝑛, 𝜗) − ς2(𝜎 + (1 − 𝜔))(𝑛𝜎 + 1 − 𝜔)
. 

7. Closure Theorems 

We shall prove the following closure theorems for the class 𝑇𝑆(𝜔, 𝜎, ς, 𝜗). 

Theorem 7.1.  Let 𝑢𝑗  ∈ 𝑇𝑆(𝜔, 𝜎, ς, 𝜗).), j=1,2,… Then 𝑔(𝑧) = ∑ 𝑐𝑗
𝑠
𝑗=1 𝑢𝑗(𝑧) ∈ 𝑇𝑆(𝜔, 𝜎, ς, 𝜗). For  

 

𝑢𝑗(𝑧) = 𝑧 − ∑ 𝑎𝑛,𝑗
∞
𝑛=2 𝑧𝑛, where ∑ 𝑐𝑗

𝑠
𝑗=1 = 1. 

Proof. 

𝑔(𝑧) = ∑ 𝑐𝑗

𝑠

𝑗=1

𝑢𝑗(𝑧)

= 𝑧 − ∑ ∑ 𝑐𝑗

𝑠

𝑗=1

∞

𝑛=2

𝑎𝑛,𝑗𝑧𝑛

= 𝑧 − ∑ 𝑒𝑛

∞

𝑛=2

𝑧𝑛,

 

where 𝑒𝑛 = ∑ 𝑐𝑗
𝑠
𝑗=1 𝑎𝑛,𝑗. Thus 𝑔(𝑧) ∈ 𝑇𝑆(𝜔, 𝜎, ς, 𝜗) if 



Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 31 No. 2 (2024) 

 

 

32 
https://internationalpubls.com 

∑
[𝜔(𝑛 − 1) + ς(𝑛𝜎 + 1 − 𝜔)]𝛩(𝑛, 𝜗)

ς(𝜎 + (1 − 𝜔))

∞

𝑛=2

𝑒𝑛 ≤ 1, 

that is, if 

∑ ∑
[𝜔(𝑛 − 1) + ς(𝑛𝜎 + 1 − 𝜔)]𝛩(𝑛, 𝜗)

ς(𝜎 + (1 − 𝜔))

𝑠

𝑗=1

∞

𝑛=2

𝑐𝑗𝑎𝑛,𝑗

= ∑ 𝑐𝑗

𝑠

𝑗=1

∑
[𝜔(𝑛 − 1) + ς(𝑛𝜎 + 1 − 𝜔)]𝛩(𝑛, 𝜗)

ς(𝜎 + (1 − 𝜔))

∞

𝑛=2

𝑎𝑛,𝑗

≤ ∑ 𝑐𝑗

𝑠

𝑗=1

= 1.

 

  

Theorem 14.  Let 𝑢, 𝑔 ∈ 𝑇𝑆(𝜔, 𝜎, ς, 𝜗). Then ℎ(𝑧) = 𝑧 − ∑ (𝑎𝑛
2 + 𝑏𝑛

2)∞
𝑛=2 𝑧𝑛 ∈ 𝑇𝑆(𝜔, 𝜎, ς, 𝜗), 𝑤ℎ𝑒𝑟𝑒  

𝜁 ≥
2𝜔(𝑛 − 1)ς2(𝜎 + (1 − 𝜔))

[𝜔(𝑛 − 1) + ς(𝑛𝜎 + 1 − 𝜔)]2𝛩(𝑛, 𝜗) − 2ς2(𝜎 + (1 − 𝜔))(𝑛𝜎 + 1 − 𝜔)
. 

Proof. Since 𝑢, 𝑔 ∈ 𝑇𝑆(𝜔, 𝜎, ς, 𝜗), so Theorem 2.1, yields 

∑ [
(𝜔(𝑛 − 1) + ς(𝑛𝜎 + 1 − 𝜔))𝛩(𝑛, 𝜗)

ς(𝜎 + (1 − 𝜔))
𝑎𝑛]

2∞

𝑛=2

≤ 1 

and 

∑ [
(𝜔(𝑛 − 1) + ς(𝑛𝜎 + 1 − 𝜔))𝛩(𝑛, 𝜗)

ς(𝜎 + (1 − 𝜔))
𝑏𝑛]

2∞

𝑛=2

≤ 1. 

We obtain from the last two inequalities 

∑
1

2

∞

𝑛=2

[
(𝜔(𝑛 − 1) + ς(𝑛𝜎 + 1 − 𝜔))𝛩(𝑛, 𝜗)

ς(𝜎 + (1 − 𝜔))
]

2

(𝑎𝑛
2 + 𝑏𝑛

2) ≤ 1.                      (7.1) 

But ℎ(𝑧) ∈ 𝑇𝑆(𝜔, 𝜎, 𝜁, 𝑞, 𝑚), if and only if 

∑
[𝜔(𝑛 − 1) + 𝜁(𝑛𝜎 + 1 − 𝜔)]𝛩(𝑛, 𝜗)

𝜁(𝜎 + (1 − 𝜔))

∞

𝑛=2

(𝑎𝑛
2 + 𝑏𝑛

2) ≤ 1,                               (7.2) 

where 0 < 𝜁 < 1, however (7.1) implies  (7.2)  if 



Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 31 No. 2 (2024) 

 

 

33 
https://internationalpubls.com 

[𝜔(𝑛 − 1) + 𝜁(𝑛𝜎 + 1 − 𝜔)]𝛩(𝑛, 𝜗)

𝜁(𝜎 + (1 − 𝜔))

≤
1

2
[
(𝜔(𝑛 − 1) + ς(𝑛𝜎 + 1 − 𝜔))𝛩(𝑛, 𝜗)

ς(𝜎 + (1 − 𝜔))
]

2

.

 

Simplifying, we get 

𝜁 ≥
2𝜔(𝑛 − 1)ς2(𝜎 + (1 − 𝜔))

[𝜔(𝑛 − 1) + ς(𝑛𝜎 + 1 − 𝜔)]2𝛩(𝑛, 𝜗) − 2ς2(𝜎 + (1 − 𝜔))(𝑛𝜎 + 1 − 𝜔)
. 

Refrences 

[1] Cho, N. E., Woo, S. Y. and Owa, S., Uniform convexity properties for hypergeometric functions, Fract. Calc. Appl. 

Anal., 5(3) (2002), 303 - 313. 

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

[3] El-Deeb, S. M., Bulboaca, T. and Dziok, J., Pascal distribution series connected with certain subclasses of univalent 

functions, Kyungpook Math. J., 59 (2) (2019), 301 - 314. 

[4] Kim, Y.C. and Srivastava, H.M., Fractional integral and other linear operators asso ciated with the Gaussian 

hypergeometric function, Complex Variables Theory Appl. 34, (1997), 293 – 312. 

[5] Merkes, E. P. and Scott, W. T., Starlike hypergeometric functions, Proc. Amer. Math. Soc., 12 (1961), 885 - 888. 

[6] Mostafa, A. O., A study on starlike and convex properties for hypergeometric functions, JIPAM. J. Inequal. Pure 

Appl. Math., 10(3) (2009), Article 87, 8 pp. 

[7] Murugusundaramoorthy,G., Subclasses of starlike and convex functions involving Poisson distribution series, Afr. 

Mat. 28 no. 7-8, (2017), 1357 - 1366. 

[8] Murugusundaramoorthy,G., Vijaya,K., and Porwal,S., Some inclusion results of certain subclass of analytic 

functions associated with Poisson distribution series, Hacet. J. Math. Stat. 45 no. 4, (2016),  1101–1107. 

[9] Owa, S. and Srivastava, H.M., Univalent and starlike generalized hypergeometric functions, Can. J. Math. 39, , 

(1987), 1057-1077. 

[10] Silverman, H., Starlike and convexity properties for hypergeometric functions, J. Math. Anal. Appl. 172 (2)(1993), 

574 - 581. 

[11] Srivastava, H. M., Murugusundaramoorthy, G. and Sivasubramanian, S., Hypergeometric functions in the parabolic 

starlike and uniformly convex domains, Integral Transforms Spec. Funct. 18(7-8) (2007), 511 - 520. 

[12] Pommerenke, C., Univalent Functions, Vandenhoeck and Ruprecht, Gottingen, (1975). 

[13] Samko, S.G., Kilbas, A. A. and Marichev, O.I., Fractional Integral and Derivatives, Theory and Applications, 

Gordon and Breach, New York, Philadelphia, London, Paris, Montreux, Toronto and Melbourne, 1993. 

[14] Silverman, H., Univalent functions with negative coefficients, Proc. Amer. Math. Soc., 51,(1975), 109-116. 

[15] Srivastava, H.M. and Buschman, R.G., Theory and Applications of Convolution Integral Equations, Kluwer 

Academic Publishers, dordrecht, Boston and London, 1992.