Communications on Applied Nonlinear Analysis 

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A New Subclass of Univalent Functions Defined by Raducanu-Orhan 

Linear Differential Operator 

 

Thirucheran M1, Saravanan K2, Stalin T3, and Brotto Manoj A4 
1Department of Mathematics,L N Government College, Ponneri, Chennai, 601 204, India. drthirucheran@gmail.com 

2Department of Mathematics, Dr Ambedkar Government Arts College, Chennai, 600 039, India. 

saravanandagac@gmail.com 

3Department of Mathematics, Vel Tech Rangarajan Dr Sagunthala R &DInstitute of Science and Technology, Chennai, 

600 062, India. drstalint@veltech.edu.in 

4Department of Advanced Computer Science and Engineering, Vignan’s Foundation for Science, Technology & 

Research, Guntur(dist)-522213, Andhra Pradesh, India. brittomanoj@gmail.com 

 

Article History: 

Received: 23-08-2024 

Revised: 06-10-2024 

Accepted: 25-10-2024 

Abstract:  

The univalent function is incredibly exciting, as evidenced by the large number of new 

studies that have been written about it recently. As injective analytic functions, univalent 

functions do not take the same value at different points inside their domain. Univalent 

functions are widely used in several branches of mathematics, physics, and engineering and 

are particularly significant in complicated analysis. Operators of normalized analytic 

functions are in considerable demand these days, especially differential and integral 

operators. There are many mathematical and scientific applications for operators. These 

operators, which are also employed to explain a wide range of physical processes, can be 

utilized to solve differential equations. A significant amount of material has been studied 

and debated by numerous researchers for the operators. In this study, the Raducanu-Orhan 

differential operator defines the new subclass of univalent functions. Furthermore, the 

subclass’s Fekte-Szego inequality, extreme points, integral means of inequalities, and 

coefficient inequality have been determined. 

Keywords: univalent functions, differential operator, subordination, coefficient inequality. 

1. Introduction 

A univalent analytic function in the complex plane is a one-to-one function that is also referred to as a 

univalent function. In many branches of mathematics, such as differential equations and complex 

analysis, uniform functions play a significant role because of their unique characteristics and uses. For 

ex ample, univalent functions can be used to express conformal mappings, which maintain angles 

locally; they are particularly relevant in complex analysis. The detailed study for the class of univalent 

functions is more important because it has links to many other areas, including the theory of special 

functions, geo metric function theory, and conformal mapping. To gain a better understanding of these 

functions’ behaviour and applicability in other mathematical domains, researchers frequently look at 

aspects of these functions such growth requirements, distortion theorems, and coefficient bounds. In 

recent years, researchers have become popular for defining a new subclass of univalent analytic 

functions linked with some differential operators [17, 9, 3, 21, 23, 24]. Because it has many 

applications in mathematics, physics, and engineering. The core challenges in the theory of univalent 

functions are comprehending limit correspondence in conformal mapping, figuring out univalent 

mailto:drthirucheran@gmail.com
mailto:drstalint@veltech.edu.in
mailto:brittomanoj@gmail.com


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requirements and addressing numerous functional theory extreme problems. More precisely defining 

limits on a range of values for various functions in class.  

The area principle was used to generate the first significant findings in the theory of univalent 

functions. Bieberbach [4] found exact upper and lower bounds for |𝑓(𝜁)| and |𝑓 ′(𝜁)| for 𝑓 ∈  𝑆, given 

|𝑎2| ≤  2 and hypothesized that |𝑎𝑛| ≤  𝑛 with the help of the outer area theorem (1916). Additionally, 

he determined the Koebe [11] constant’s precise value. For a long time, mathematicians have been 

challenged by this conjecture. Louis De Branges [6] found a solution to the conjecture |𝑎𝑛| ≤

 𝑛, (𝑛 =  2,3, … . ) in 1984. Following Loewner [12]’s 1923 proof of |𝑎3| ≤  3, Fekete-Szego [7] 

astounded mathematicians with the troublesome inequality |𝑎3  −  µ𝑎2
2| ≤ 1 + 2𝑒

(
−2µ

1−µ 
) 

, 0 ≤  µ ≤  1. 

By then, univalent function theory had acquired its own name. 

A univalent function that is analytic in a domain and is also referred to as a one-to-one or injective 

function in complex analysis is said to be conformal. Locally, angles are preserved by conformal 

mappings. Formally speaking, if a function 𝑓(𝜁) that is defined on a domain 𝐷 ⊂  𝐶 maintains angles 

between curves that pass through 𝜁, then 𝜁 ∈  𝐷 is conformal at that point. A function is conformal in 

a domain 𝐷 if it is both univalent (injective) and analytic in 𝐷. In line with conformal maps in complex 

analysis, this indicates that the mapping maintains the local structure of the domain locally by not 

distorting the angles between curves. Despite of many practical uses, conformal mapping is an essential 

technique in complex analysis.  

If the function is harmonic that is, it satisfies 𝛻2𝑓 =  0 according to Laplace then the conformal 

mapping transformation of such functions is likewise harmonic. As a result, any field whose equations 

can be represented by a potential function can be solved using conformal mapping. Laplace’s equation 

𝜙𝑥𝑥  +  𝜙𝑦𝑦  =  0 can be used to formulate various mathematical issues related to the motion of fluids, 

the field of electrostatic, heat transfer, and many other physical circumstances in a certain region 𝐷 of 

the complex plane. For exam ple, it can be used to apply in scattering and diffraction problems, brain 

surface mapping problem, and the electrostatic potential problems in the shaded region of the 𝜁 plane 

[31, 32]. It can also be used in stealth technology. Although the concept of conformal mapping is not 

directly used in stealth technology, the development of effective stealth technologies greatly benefits 

from an un derstanding of shape optimisation, material science, and electromagnetic wave behaviour 

[1]. Further, the univalent function help to analysis the frequency analysis problem [16]. 

Recent years, the new subclasses defined by using the linear differential operators. The differential 

operator was first introduced by Ruscheweyh [19] in 1975, which is cleared the path. Salagean [20] 

followed in 1983 with an additional variation of differential and integral operators. Many scholars have 

examined and debated a wide range of properties related to these two operators. Al-Oboudi [2] 

generalized the Salagean operator in 2004. In 2010, Raducanu and Orhan [15] generalized the Al-

Oboudi differential operator. In this study we define two new subclasses, which is defined by the 

Raducanu-Orhan differential operator. Also, we have discussed some properties of these subclasses.  

Let 𝐴 be the class of univalent functions consists of the form  



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𝑓(𝜁) = 𝜁 +  ∑ 𝑎𝑙

∞

 𝑙=2

𝜁𝑙 , 𝜁 ∈ 𝑈 ∶=  {𝜁 ∈  𝐶 ∶  |𝜁| <  1},                                                                         (1) 

which is analytic in the unit disk U.  

For 𝑓(𝜁)  ∈  𝐴, the Raducanu-Orhan [15] differential operator is defined as 𝑅𝜌,µ
𝑛 𝑓(𝜁). 

𝑅𝜌,µ
0  =  𝑓 = 𝜁 +  ∑ 𝑎𝑙

∞

 𝑙=2

𝜁𝑙 , 

𝑅𝜌,µ
1  = (1 − 𝜌 + 𝜇)𝑓(𝜁) + (𝜌 − 𝜇)𝑓 ′(𝜁) + (𝜌𝜇)𝜁2𝑓 ′′(𝜁) 

= 𝜁 +  ∑[1 + (𝑙 − 1)(𝑙𝜌𝜇 + 𝜌 − 𝜇)]𝑎𝑙

∞

 𝑙=2

𝜁𝑙 , 

𝑅𝜌,µ
2 = 𝑅𝜌,µ

1  (𝑅𝜌,µ
1  ) 

Similarly,  

𝑅𝜌,µ
𝑛  = 𝑅𝜌,µ

1  (𝑅𝜌,µ
𝑛−1 ) = 𝜁 +  ∑[1 + (𝑙 − 1)(𝑙𝜌𝜇 + 𝜌 − 𝜇)]𝑛𝑎𝑙

∞

 𝑙=2

𝜁𝑙 .                                                         (2) 

where 𝑛 ∈ 𝑁0 = 𝑁 ∪ 0, 𝑁 = {1,2,3, … }, 𝜇, 𝜌 ≥ 0, 𝜁 ∈ 𝑈. 

Remark. 𝑅𝜌,0
𝑛   = 𝐷𝑛 yields the Al-Oboudi differential operator [2], 𝑅1,0

𝑛   = 𝐷𝑛  gives Salagean 

differential operator [20]. 

2. The subclass 𝑺𝒃,𝝉,𝝆,µ
𝒎,𝒏  (𝜸)  

Definition 2.1 

Let 𝑆𝑏,𝜏,𝜌,µ
𝑚,𝑛  (𝛾) denote the subclass of 𝐴 consisting of function 𝑓 which satisfies the inequality  

𝑅𝑒 (1 +
1

𝑏
(

𝑅𝜌,µ
𝑚 𝑓(𝜁) 

𝑅𝜌,µ
𝑛 𝑓(𝜁)

− 1))   > 𝜏 |
𝑅𝜌,µ

𝑚 𝑓(𝜁) 

𝑅𝜌,µ
𝑛 𝑓(𝜁)

− 1| + 𝛾.                                                                                      (3)                                                                                 

For some 𝑏 ∈ 𝐶 − {0}, 𝑚 ∈ 𝑁, 𝑛 ∈ 𝑁0, 𝜏, 𝜌,µ ≥ 0,0 ≤ 𝛾 < 1 and all 𝜁 ∈ 𝑈. 

 

For suitable choices of the parameters of the of 𝑆𝑏,𝜏,𝜌,µ
𝑚,𝑛  (𝛾) provides several well-known subclasses. 

Remark 2.2  

𝑆𝑏,0,0,𝜌
𝑚,𝑛 (𝛾) = 𝑆(𝑚, 𝑛, 𝛾, 𝜏, 𝜌, 𝑏, 𝑙)  studied by Stalin and Thiruchran [30].  

𝑆1,0,0,1
𝑚,𝑛 (𝛾) = 𝐾𝑚,𝑛(𝛾) studied by Sumer Eker and Owa [26]. 

𝑆1,0,0,𝜌
𝑚,𝑛  (𝛾) = 𝑆𝑚,𝑛,𝜌(𝛾) studied by Sumer Eker and Ozlem Guney [27]. 

𝑆1,0,0,1
𝑛+1,𝑛 (𝛾) = 𝑆𝑛(𝛾) studied by Kadioglu [10]. 

Theorem 2.3  

Let 𝑓 ∈ 𝐴 satisfies 



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∑ 𝜙(𝛾, 𝜏)|𝑎𝑙| ≤ 2(1 − 𝛾)|𝑏|.                                                                                                                  (4) 

∞

𝑙=2

 

For some 𝑏 ∈ 𝐶 − {0}, 𝑚 ∈ 𝑁, 𝑛 ∈ 𝑁0, 𝜏, 𝜌,µ ≥ 0, 0 ≤ 𝛾 < 1 and all 𝜁 ∈ 𝑈, then 𝑓 ∈ 𝑆𝑏,𝜏,𝜌,µ
𝑚,𝑛  (𝛾), 

where 𝜙(𝛾, 𝜏) = |(1 + (𝑙 − 1)(𝑙𝜌µ + 𝜌 − µ))𝑚 − (1 + 𝛾𝑏)(1 + (𝑙 − 1)(𝑙𝜌µ + 𝜌 − µ))𝑛|  

+(1 + (𝑙 − 1)(𝑙𝜌µ + 𝜌 − µ))𝑚 + ((2 − 𝛾)𝑏 − 1)(1 + (𝑙 − 1)(𝑙𝜌µ + 𝜌 − µ))𝑛  

+2𝑏𝜏|(1 + (𝑙 − 1)(𝑙𝜌µ + 𝜌 − µ))𝑚 − (1 + (𝑙 − 1)(𝑙𝜌µ + 𝜌 − µ))𝑛|. 

Proof. Suppose that ∑ 𝜙(𝛾, 𝜏)|𝑎𝑙| ≤ 2(1 − 𝛾)|𝑏|∞
𝑙=2  is true. For some 𝑏 ∈ 𝐶 − {0}, 𝑚 ∈ 𝑁, 𝑛 ∈

𝑁0, 𝜏, 𝜌,µ ≥ 0, 0 ≤ 𝛾 < 1 and all 𝜁 ∈ 𝑈, then it is sufficient to prove that  |
𝐹(𝜁)−1

𝐹(𝜁)+1
| < 1. 

For 𝑓 ∈ 𝐴, then define the function 𝐹 by  

 

𝐹(𝜁) = 1 +
1

𝑏
(

𝑅𝜌,µ
𝑚 𝑓(𝜁) 

𝑅𝜌,µ
𝑛 𝑓(𝜁)

− 1) −  𝜏 |
𝑅𝜌,µ

𝑚 𝑓(𝜁)

𝑅𝜌,µ
𝑛 𝑓(𝜁)

− 1| − 𝛾 

𝐹(𝜁) − 1 =
𝑅𝜌,µ

𝑚 𝑓(𝜁) − (1 + 𝛾𝑏)𝑅𝜌,µ
𝑛 𝑓(𝜁) − 𝑏𝜏|𝑅𝜌,µ

𝑚 𝑓(𝜁) − 𝑅𝜌,µ
𝑛 𝑓(𝜁)|

𝑏𝑅𝜌,µ
𝑛 𝑓(𝜁)

  

and 

𝐹(𝜁) − 1 =
𝑅𝜌,µ

𝑚 𝑓(𝜁) − (1 + (𝛾 − 2)𝑏)𝑅𝜌,µ
𝑛 𝑓(𝜁) − 𝑏𝜏|𝑅𝜌,µ

𝑚 𝑓(𝜁) − 𝑅𝜌,µ
𝑛 𝑓(𝜁)|

𝑏𝑅𝜌,µ
𝑛 𝑓(𝜁)

  

Therefore,  

|
𝐹(𝜁) − 1

𝐹(𝜁) + 1
| = |

𝑅𝜌,µ
𝑚 𝑓(𝜁) − (1 + 𝛾𝑏)𝑅𝜌,µ

𝑛 𝑓(𝜁) − 𝑏𝜏|𝑅𝜌,µ
𝑚 𝑓(𝜁) − 𝑅𝜌,µ

𝑛 𝑓(𝜁)|

𝑅𝜌,µ
𝑚 𝑓(𝜁) − (1 + (𝛾 − 2)𝑏)𝑅𝜌,µ

𝑛 𝑓(𝜁) − 𝑏𝜏|𝑅𝜌,µ
𝑚 𝑓(𝜁) − 𝑅𝜌,µ

𝑛 𝑓(𝜁)|
| < 1. 

⟹ ∑ {

|(1 + (𝑙 − 1)(𝑙𝜌µ + 𝜌 − µ))𝑚 − (1 + 𝛾𝑏)(1 + (𝑙 − 1)(𝑙𝜌µ + 𝜌 − µ))𝑛|

+(1 + (𝑙 − 1)(𝑙𝜌µ + 𝜌 − µ))𝑚 + ((2 − 𝛾)𝑏 − 1)(1 + (𝑙 − 1)(𝑙𝜌µ + 𝜌 − µ))𝑛 

+2𝑏𝜏|(1 + (𝑙 − 1)(𝑙𝜌µ + 𝜌 − µ))𝑚 − (1 + (𝑙 − 1)(𝑙𝜌µ + 𝜌 − µ))𝑛|
}

∞

𝑙=2

|𝑎𝑙|

≤ 2(1 − 𝛾)|𝑏|. 

∴ ∑ 𝜙(𝛾, 𝜏)|𝑎𝑙| ≤ 2(1 − 𝛾)|𝑏|.

∞

𝑙=2

 

Hence, equation (4) holds. 

Put 𝜏 = 0 and µ = 0 in  𝑆𝑏,𝜏,𝜌,µ
𝑚,𝑛  (𝛾) then this class reduces as 𝑆(𝑚, 𝑛, 𝛾, 𝜏, 𝜌, 𝑏), which was studied by 

Thirucheran and Stalin [30]. 

Corollary 2.4 

Let 𝑓 ∈ 𝐴 satisfies  

∑ 𝜙(𝑚, 𝑛, 𝛾, 𝜏, 𝜌, 𝑏, 𝑙)
∞

𝑙=2

  |𝑎𝑙| ≤  2(1 −  𝛾)𝑏. 

For some (0 ≤ 𝛾 < 1), 𝜏 ≥ 0, 𝑚 ∈ 𝑁, 𝑛 ∈ 𝑁0, 𝜌(𝜌 ≥  0), and all 𝜁 ∈ 𝑈, then 𝑓 ∈ 𝑆(𝑚, 𝑛, 𝛾, 𝜏, 𝜌, 𝑏),  



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Where 𝜙(𝑚, 𝑛, 𝛾, 𝜏, 𝜌, 𝑏, 𝑙) = |(1 +  (𝑙 –  1)𝜌)𝑚 − (1 + 𝛾𝑏)(1 + (𝑙 − 1)𝜌)𝑛| 

+(1 + (𝑙 − 1)𝜌)𝑚 + ((2 − 𝛾)𝑏 − 1)(1 + (𝑙 − 1)𝜌)𝑛 

+2𝑏𝜏 |(1 + (𝑙 − 1)𝜌)𝑚 − (1 + (𝑙 − 1)𝜌)𝑛|. 

If 𝑏 =  1 and 𝜏 =  0, then the class 𝑆𝑏,𝜏,𝜌,µ
𝑚,𝑛  (𝛾)  reduces to 𝑅𝑒 (

𝑅𝜌
𝑚𝑓(𝜁)

𝑅𝜌
𝑚𝑓(𝜁)

)  >  𝛾 which analogues to the 

class 𝑆𝑚,𝑛,𝜌(𝛾) introduced by Sevtap Sumer Eker and Ozlem Guney[27]. 

Corollary 2.5 

Let 𝑓 ∈  𝐴 satisfies the inequality ∑ 𝜙(𝛾, 𝑚, 𝑛, 𝜌, 𝑙)∞
𝑙=2  |𝑎𝑙| ≤  2(1 –  𝛾), for some 𝑏 ∈ 𝐶 − {0}, 

𝑚 ∈ 𝑁, 𝑛 ∈ 𝑁0, 𝜌,µ ≥ 0,0 ≤ 𝛾 < 1, then 𝑓 ∈ 𝑆𝑚,𝑛,𝜌(𝛾), where  

𝜙(𝛾, 𝑚, 𝑛, 𝜌, 𝑙) = |(1 + (𝑙 − 1)𝜌)𝑚 − (1 + 𝛾)(1 + (𝑙 − 1)𝜌)𝑛| 

+(1 + (𝑙 − 1)𝜌)𝑚  + (1 − 𝛾)(1 + (𝑙 − 1)𝜌)𝑛 

If 𝑏 =  1, 𝜏 =  0 and 𝜌 =  1, then the class 𝑆𝑏,𝜏,𝜌,µ
𝑚,𝑛  (𝛾) given the class 𝑆𝑚,𝑛(𝛾), which is discussed by 

Sevtap Sumer Eker and Owa [26]. 

Corollary 2.6 

Let 𝑓 ∈  𝐴 satisfies ∑ 𝜙(𝛾, 𝑚, 𝑛, 𝑙)∞
𝑙=2 |𝑎𝑙|  ≤  2(1 −  𝛾), for some 𝛾(0 ≤ 𝛾 < 1), 𝑚 ∈ 𝑁, 𝑛 ∈ 𝑁0, then 

𝑓 ∈ 𝑆𝑚,𝑛(𝛾), where 𝜙(𝛾, 𝑚, 𝑛, 𝑙)  =  |(𝑙)𝑚  − (1 + 𝛾)(𝑙)𝑛|  + (𝑙)𝑚  + (1 − 𝛾)(𝑙)𝑛. 

 

We define the subclass 𝑆𝑏,𝜏,𝜌,µ
𝑚,𝑛  (𝛾)̃  ⊂  𝑆𝑏,𝜏,𝜌,µ

𝑚,𝑛  (𝛾), and determine the extreme points of the subclass for 

the subclass 𝑆𝑏,𝜏,𝜌,µ
𝑚,𝑛  (𝛾)̃ . 

Theorem 2.7 

Let 𝑓1(𝜁) = 𝜁, and 𝑓𝑙(𝜁) = 𝜁 + ∑ 𝜂𝑙
2(1−𝛾)𝑏

𝜙(𝛾,𝜏)
𝜁𝑙 ,   (𝑙 = 2,3,4, … ),∞

𝑙=2  then 𝑓 ∈ 𝑆𝑏,𝜏,𝜌,µ
𝑚,𝑛  (𝛾)̃  if it is able to 

represented as 𝑓 = ∑ 𝜂𝑙𝑓𝑙(𝜁), 𝜂𝑙 > 0 ∞
𝑙=1  and ∑ 𝜂𝑙 = 1.∞

𝑙=1  

Proof. Suppose that 

𝑓 = ∑ 𝜂𝑙𝑓𝑙(𝜁)

∞

𝑙=1

 

= 𝜁 + ∑ 𝜂𝑙

2(1 − 𝛾)𝑏

𝜙(𝛾, 𝜏)
𝜁𝑙

∞

𝑙=2

 

= 2(1 − 𝛾)𝑏 ∑ 𝜂𝑙

∞

𝑙=2

 

= 2(1 − 𝛾)𝑏(1 − 𝜂1) 



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< 2(1 − 𝛾)𝑏. 

Which shows that 𝑓 ∈ 𝑆𝑏,𝜏,𝜌,µ
𝑚,𝑛  (𝛾)̃ . 

Conversely, suppose that 𝑓 ∈ 𝑆𝑏,𝜏,𝜌,µ
𝑚,𝑛  (𝛾) 

Since, 𝑎𝑙 ≤
2(1−𝛾)𝑏

𝜙(𝛾,𝜏)
 

Let 

 𝜂𝑙 ≤
𝜙(𝛾, 𝜏)

2(1 − 𝛾)𝑏
𝑎𝑙 

 and  

𝜂1 = 1 − ∑ 𝜂𝑙 ,

∞

𝑙=2

 

then we obtain  

𝑓 = ∑ 𝜂𝑙𝑓𝑙(𝜁)∞
𝑙=1 . 

 

Let  µ = 0, then the class 𝑆𝑏,𝜏,𝜌,µ
𝑚,𝑛  (𝛾) reduces and analogues to the class, which is examined by 

Thirucheran and Stalin [30]. 

Corollary 2.8 

Let 𝑓1(𝜁) = 𝜁, and 𝑓𝑙(𝜁) = 𝜁 + ∑ 𝜂𝑙
2(1−𝛾)𝑏

𝜙(𝑚,𝑛,𝛾,𝜏,𝜌,𝑏,𝑙)
𝜁𝑙 ,   (𝑙 = 2,3,4, … ),∞

𝑙=2  then 𝑓 ∈ 𝑆𝑏,𝜏,𝜌
𝑚,𝑛  (𝛾)̃  if it is able 

to represented as 𝑓 = ∑ 𝜂𝑙𝑓𝑙(𝜁), 𝜂𝑙 > 0 ∞
𝑙=1  and ∑ 𝜂𝑙 = 1.∞

𝑙=1  

If 𝑏 = 1 and 𝜏 = 0, we get the result of the class 𝑆𝑚,𝑛,𝜌(𝛾) introduced by Sevtap Sumer Eker and 

Ozlem Guney [27]. 

Corollary 2.9 

Let 𝑓1(𝜁) = 𝜁, and 𝑓𝑙(𝜁) = 𝜁 + ∑ 𝜂𝑙
2(1−𝛾)𝑏

𝜙(𝛾,𝑚,𝑛,𝜌,𝑙)
𝜁𝑙 ,   (𝑙 = 2,3,4, … ),∞

𝑙=2  then 𝑓 ∈ �̃�𝑚,𝑛,𝜌 if it is able to 

represented as 𝑓 = ∑ 𝜂𝑙𝑓𝑙(𝜁), 𝜂𝑙 > 0 ∞
𝑙=1  and ∑ 𝜂𝑙 = 1.∞

𝑙=1  

Theorem 2.10 

Let 𝑓 ∈ 𝑆𝑏,𝜏,𝜌,µ
𝑚,𝑛  (𝛾) and suppose that 𝑓 is defined by  

𝜁 +
2(1 − 𝛾)𝑏𝜀𝑙

𝜙(𝛾, 𝜏)
𝜁𝑙 , (𝑙 = 2,3,4, … ), |𝜀𝑙| = 1.  

If an analytic function is present 𝑤(𝜁) given by 

{𝑤(𝜁)}𝑙−1 =
𝜙(𝛾, 𝜏)

2(1 − 𝛾)𝑏𝜀𝑙
∑ 𝑎𝑙𝜁

𝑙−1,   (𝜁 = 𝑟𝑒𝑖𝜃, 0 < 𝑟 < 1),

∞

𝑙=2

 



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then 

∫ |𝑓(𝑟𝑒𝑖𝜃)|
𝜇

𝑑𝜃 ≤ ∫ |1 +
2(1 − 𝛾)𝑏𝜀𝑙

𝜙(𝛾, 𝜏)
𝜁𝑙−1|

𝜇

𝑑𝜃.
2𝜋

0

2𝜋

0

 

Proof. We show that  

∫ |1 + ∑ 𝑎𝑙

∞

𝑙=2

𝜁𝑙−1|

𝜇

𝑑𝜃 ≤ ∫ |1 +
2(1 − 𝛾)𝑏𝜀𝑙

𝜙(𝛾, 𝜏)
𝜁𝑙−1|

𝜇

𝑑𝜃.
2𝜋

0

2𝜋

0

 

By the help of little wood subordination theorem [13], it is sufficient to show that 

1 + ∑ 𝑎𝑙

∞

𝑙=2

𝜁𝑙−1 ≺ 1 +
2(1 − 𝛾)𝑏𝜀𝑙

𝜙(𝛾, 𝜏)
𝜁𝑙−1. 

Let  

1 + ∑ 𝑎𝑙

∞

𝑙=2

𝜁𝑙−1 = 1 +
2(1 − 𝛾)𝑏𝜀𝑙

𝜙(𝛾, 𝜏)
(𝑤(𝜁))𝑙−1. 

∴ (𝑤(𝜁))𝑙−1 =
𝜙(𝛾, 𝜏)

2(1 − 𝛾)𝑏𝜀𝑙
∑ 𝑎𝑙𝜁

𝑙−1.

∞

𝑙=2

 

Which readily yields 𝑤(0) = 0. 

Further, we prove that the analytic function 𝑤(𝜁) satisfies |𝑤(𝜁) | < 1 using schwarz lemma. We 

know that  

|(𝑤(𝜁))𝑙−1| = |
𝜙(𝛾, 𝜏)

2(1 − 𝛾)𝑏𝜀𝑙
∑ 𝑎𝑙𝜁𝑙−1.

∞

𝑙=2

| ≤ |𝜁| < 1. 

 

Let  µ = 0, then the class 𝑆𝑏,𝜏,𝜌,µ
𝑚,𝑛  (𝛾) reduces and analogues to the class, which is examined by 

Thirucheran and Stalin [30]. 

Corollary 2.11 

Let 𝑓 ∈ 𝑆(𝑚, 𝑛, 𝛾, 𝜏, 𝜌, 𝑏) and suppose that 𝑓 is defined by  

𝜁 +
2(1 − 𝛾)𝑏𝜀𝑙

𝜙(𝑚, 𝑛, 𝛾, 𝜏, 𝜌, 𝑏, 𝑙)
𝜁𝑙 , (𝑙 = 2,3,4, … ), |𝜀𝑙| = 1.  

If an analytic function is present 𝑤(𝜁) given by 

{𝑤(𝜁)}𝑙−1 =
𝜙(𝑚, 𝑛, 𝛾, 𝜏, 𝜌, 𝑏, 𝑙)

2(1 − 𝛾)𝑏𝜀𝑙
∑ 𝑎𝑙𝜁

𝑙−1,   (𝜁 = 𝑟𝑒𝑖𝜃, 0 < 𝑟 < 1),

∞

𝑙=2

 

 

 



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then 

∫ |𝑓(𝑟𝑒𝑖𝜃)|
𝜇

𝑑𝜃 ≤ ∫ |𝑔(𝑟𝑒𝑖𝜃)|
𝜇

𝑑𝜃,   𝜇 > 0.
2𝜋

0

2𝜋

0

 

3.   The Fekete-Szego inequality for the subclass  𝑺𝒃,𝝉,𝝆,µ
𝒎,𝒏  (𝜸)  

In 1933, Fekete-Szego [7] obtained the maximum value of |𝑎3  −  µ𝑎2
2| as a function of the real 

parameter µ, for the function of class 𝐴. Since then, the various authors were investigated and obtained 

the Fekete-Szego inequalities for different subclasses of the class 𝐴 [5] [7],[8],[18],[22],[25], [28], 

[29]. In this article, we introduced two new subclasses of univalent functions which are defined by 

using Raducanu-Ohran differential operator in the open unit disc. For these subclasses, we obtain the 

Fekete-Szego inequality |𝑎3  −  µ𝑎2
2|.  

If replacing special values for the subclass, we obtained Several well-known subclasses. 

Remark  

𝑆1,0,0,1
1,0  (𝛾)  =  𝑆∗(𝛾) studied by Ma and Minda [14]. 

𝑆𝑏,0,0,1
1,0  (𝛾)  =  𝑆𝑏

∗(𝛾) studied by Ravichandran et.al. [18]. 

𝑆𝑏,0,0,𝜌
2,0  (𝛾)  =  𝑀𝑎,𝑏(𝜙) studied by Suchitra et.al. [25]. 

𝑆𝑏,0,0,𝜌
2,1  (𝛾)  =  𝑀𝛾(𝜙)studied by Shanmugam and Sivasubramanian [22]. 

Lemma 3.2  

If 𝑃(𝜁) = 1 + 𝑐1𝜁 + 𝑐2𝜁2 + 𝑐3𝜁3 + ⋯ is a function with positive real part in 𝑈 and µ is a complex 

number, then |𝑐2 − µ𝑐1
2| ≤ 2 𝑀𝑎𝑥{1, |2µ − 1|}. The result is sharp for the function is given by  

𝑃(𝜁) =
1+𝜁2

1−𝜁2
    and  𝑃(𝜁) =

1+𝜁 

1−𝜁
. 

Theorem 3.3  

Let 𝜙(𝜁) = 1 + 𝐵1𝜁 + 𝐵2𝜁2 + 𝐵3𝜁3 + ⋯ , with 𝐵1 = 0. If 𝑓 ∈ 𝑆𝑏,𝜏,𝜌,µ
𝑚,𝑛  (𝛾) satisfies the inequality  

𝑅𝑒 (1 +
1

𝑏
(

𝑅𝜌,µ
𝑚 𝑓(𝜁) 

𝑅𝜌,µ
𝑛 𝑓(𝜁)

− 1)) −  𝜏 |
𝑅𝜌,µ

𝑚 𝑓(𝜁) 

𝑅𝜌,µ
𝑛 𝑓(𝜁)

− 1|  ≺ 𝜙(𝜁). 

Then 

 |𝑎3 − µ𝑎2
2| ≤

𝐵1|𝑏

| 𝑌2−𝑏𝜏|𝑌2|
max {1, |

𝐵2

𝐵1
+ [

[𝑌3−𝑏𝜏|𝑌3|]−µ[[𝑌2]−𝑏𝜏|𝑌2|]

[𝑌1−𝑏𝜏[𝑌1]]2
] 𝑏𝐵1|},                                    (5) 

where  

𝑌1 = ((1 + (𝜌 − µ + 2𝜌µ)))𝑚 − ((1 + (𝜌 − µ + 2𝜌µ)))𝑛,  

𝑌2 = ((1 + 2(𝜌 − µ + 3𝜌µ)))𝑚 − ((1 + 2(𝜌 − µ + 3𝜌µ)))𝑛  

and  



Communications on Applied Nonlinear Analysis 

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𝑌3 = ((1 + (𝜌 − µ + 2𝜌µ)))𝑚+𝑛 − ((1 + (𝜌 − µ + 2𝜌µ)))2𝑛.  

Then the result is sharp. 

Proof. If 𝑓 ∈ 𝑆𝑏,𝜏,𝜌,µ
𝑚,𝑛  (𝛾), then there is a Schwarz function 𝑤(𝜁), analytic in 𝑈 with 𝑤(0) = 0 and 

|𝑤(𝜁)| < 1 in such that  

1 +
1

𝑏
(

𝑅𝜌,µ
𝑚 𝑓(𝜁) 

𝑅𝜌,µ
𝑛 𝑓(𝜁)

− 1) −  𝜏 |
𝑅𝜌,µ

𝑚 𝑓(𝜁)

𝑅𝜌,µ
𝑛 𝑓(𝜁)

− 1|  ≺ 𝜙(𝑤(𝜁)). 

Define 𝑃(𝜁) by 𝑃(𝜁) =
1+𝑤(𝜁) 

1−𝑤(𝜁)
 = 1 + 𝑐1𝜁 + 𝑐2𝜁2 + 𝑐3𝜁3 + ⋯  

Since 𝑤(𝜁) is a Schwarz function, it is clear that 𝑅𝑒𝑃(𝜁) > 0 and 𝑃(0) = 1. 

∴ 𝜙(𝜁) = 𝜙 (
𝑃(𝜁)−1

𝑃(𝜁)+1
) = 1 +

𝐵1𝑐1

2
𝜁 + [

𝐵1

2
(𝑐2 −

𝑐1
2

2
) +

𝐵2𝑐1
2

4
] 𝜁2+. ... 

Now,  

1 +
1

𝑏
(

𝑅𝜌,µ
𝑚 𝑓(𝜁) 

𝑅𝜌,µ
𝑛 𝑓(𝜁)

− 1) −  𝜏 |
𝑅𝜌,µ

𝑚 𝑓(𝜁)

𝑅𝜌,µ
𝑛 𝑓(𝜁)

− 1| =  1 +
𝐵1𝑐1
2

𝜁 + [
𝐵1

2
(𝑐2 −

𝑐1
2

2
) +

𝐵2𝑐1
2

4
] 𝜁2 + ⋯ 

  

∴ 𝑎2 =
𝑏𝐵1𝑐1

2[𝑌1 − 𝑏𝜏|𝑌1|]
  

And 

𝑎3 =
𝑏𝐵1𝑐2

2[𝑌2 − 𝑏𝜏|𝑌2|]
+

𝑏𝐵1𝑐1
2

4[𝑌2 − 𝑏𝜏|𝑌2|]
[
[𝑌1 − 𝑏𝜏|𝑌1|]𝑏𝐵1

[𝑌1 − 𝑏𝜏|𝑌1|]2
− (1 −

𝐵2

𝐵1

)] 

∴ 𝑎3 − µ𝑎2
2 =

𝑏𝐵1

2[𝑌2 − 𝑏𝜏|𝑌2|]
{𝑐2 − 𝑣𝑐1

2}, 

Where 𝑣 =
1

2
(1 −

𝐵2

𝐵1
+

𝜇𝑏𝐵1[𝑌2−𝑏𝜏|𝑌2|]

[𝑌1−𝑏𝜏|𝑌1|]2
−

𝑏𝐵1[𝑌3−𝑏𝜏|𝑌3|]

[𝑌1−𝑏𝜏|𝑌1|]2
). 

Hence,  

|𝑎3 − µ𝑎2
2| ≤

𝐵1|𝑏

| 𝑌2 − 𝑏𝜏|𝑌2|
max {1, |

𝐵2

𝐵1

+ [
[𝑌3 − 𝑏𝜏|𝑌3|] − µ[[𝑌2] − 𝑏𝜏|𝑌2|]

[𝑌1 − 𝑏𝜏[𝑌1]]2
] 𝑏𝐵1|} . 

Therefore, the result (22) is sharp for the function defined by 

1 +
1

𝑏
(

𝑅𝜌,µ
𝑚 𝑓(𝜁) 

𝑅𝜌,µ
𝑛 𝑓(𝜁)

− 1) −  𝜏 |
𝑅𝜌,µ

𝑚 𝑓(𝜁)

𝑅𝜌,µ
𝑛 𝑓(𝜁)

− 1| =  𝜙(𝜁2) 

and 

1 +
1

𝑏
(

𝑅𝜌,µ
𝑚 𝑓(𝜁) 

𝑅𝜌,µ
𝑛 𝑓(𝜁)

− 1) −  𝜏 |
𝑅𝜌,µ

𝑚 𝑓(𝜁)

𝑅𝜌,µ
𝑛 𝑓(𝜁)

− 1| =  𝜙(𝜁). 

 



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4. Conclusion  

This present work, we got some results that were obtained from the new sub class of normalized 

analytic univalent function. We also acquired and examined a few fundamental characteristics of 

univalent functions, comparing them to earlier findings. Also, sharp boundaries were obtained for the 

subclass. The researchers are motivated to improve the findings of this subclass in future by using bi-

univalent, multivalent, q-analoque, and meromorpihic functions with positive and negative 

coefficients. 

Acknowledgment 

The authors sincerely thank the reviewers for their helpful remarks, that helped in the success of this 

article. 

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