Inequality Estimations of Subclass of Univalent Functions Involving Raducanu-Orhan Operator

Thirucheran Manickam; Saravanan Kumar; Navaneetha Krishnan Ramachandran; Stalin Thangamani; Boopathy Pandi

doi:10.22342/jims.v32i2.1822

Thirucheran Manickam ⁽¹⁾ , Saravanan Kumar ⁽²⁾ , Navaneetha Krishnan Ramachandran ⁽³⁾ , Stalin Thangamani ⁽⁴⁾ , Boopathy Pandi ⁽⁵⁾

(1) Department of Mathematics, L N Government College, India,

(2) Department of Mathematics, Dr Ambedkar Government Arts College, India,

(3) Department of Mathematics, Vel Tech Rangarajan Dr. Sagunthala R&D Institute of Science and Technology, India,

(4) Department of Mathematics, Vel Tech Rangarajan Dr. Sagunthala R&D Institute of Science and Technology, India,

(5) Department of Computer Science, Anna University Regional Campus, India

https://doi.org/10.22342/jims.v32i2.1822

Issue
Vol. 32 No. 2 (2026): JUNE

Submitted
October 2, 2024

Accepted
March 1, 2026

Published
April 27, 2026

Keywords:

univalent functions, differential operator, coefficient inequality, Fekete-Szego inequality, subordination

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Abstract

The vast number of new papers that have been written about the univalent function in recent years shows how fascinating it is. Univalent functions are injective analytic functions, which means that they do not take the same value at various places within their domain. Univalent functions are important in complex analysis and have many uses in other areas of mathematics, physics, and engineering. These days, there is a high need for operators of normalized analytic functions, particularly differential and integral operator. Operators are widely used in numerous mathematical and scientific domains. Differential equations can be solved using these operators, which are also used to describe a wide range of physical phenomena. Many researchers have reviewed and discussed a substantial amount of material for the operators. In this work, the new subclass of univalent functions is defined by the Raducanu-Orhan differential operator. In addition, the coefficient inequalities, extreme points, integral means of inequality, and Fekte-Szego inequality for the subclass have been obtained.

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References

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Authors

Thirucheran Manickam

Department of Mathematics, L N Government College

Saravanan Kumar

Department of Mathematics, Dr Ambedkar Government Arts College

Navaneetha Krishnan Ramachandran

Department of Mathematics, Vel Tech Rangarajan Dr. Sagunthala R&D Institute of Science and Technology

Stalin Thangamani

Department of Mathematics, Vel Tech Rangarajan Dr. Sagunthala R&D Institute of Science and Technology

https://orcid.org/0000-0002-8735-3567

drstalint@veltech.edu.in (Primary Contact)

Boopathy Pandi

Department of Computer Science, Anna University Regional Campus

Manickam, T., Kumar, S., Ramachandran, N. K., Thangamani, S., & Pandi, B. (2026). Inequality Estimations of Subclass of Univalent Functions Involving Raducanu-Orhan Operator. Journal of the Indonesian Mathematical Society, 32(2), 1822. https://doi.org/10.22342/jims.v32i2.1822

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Many researchers have reviewed and discussed a substantial amount of material for the operators. In this work, the new subclass of univalent functions is defined by the Raducanu-Orhan differential operator. In addition, the coefficient inequalities, extreme points, integral means of inequality, and Fekte-Szego inequality for the subclass have been obtained.

1. INTRODUCTION

A univalent analytic function, sometimes called a univalent function of a complex variable, is an analytic function that is also univalent, meaning it is a one-to-one. In many branches of mathematics, such as diferential equations and complex analysis, uniform functions play a significant role because of their unique characteristics and uses. For example, univalent functions can be used to express conformal mappings, which maintain angles locally; they are particularly relevant in complex analysis. In complex analysis, the study of the class of univalent functions is important because it has links to many other areas, including the theory of special functions, geometric function theory, and conformal mapping. To gain a better understanding of these functions’ behavior and applicability in other mathematical domains, researchers frequently examine aspects of these functions, such as growth requirements, distortion theorems, and coeficient bounds. In recent years, researchers have been interested in defining a new subclass of univalent analytic functions associated with diferential operators [1,2,3,4,5] because it has many applications in mathematics, physics, and engineering. The core challenges in the theory of univalent functions are comprehending the limit correspondence in conformal mapping, figuring out univalent requirements, and addressing numerous functional extreme theory problems. More precisely, defining limits on a range of values for various functions in a given class.

The first substantial results in the theory of univalent functions were obtained using the area principle. With the aid of the outer area theorem (1916), Bieberbach[6]obtained precise upper and lower bounds for \(| f ( z ) |\) and \(\left| f ^ { \prime } ( z ) \right|\) for \(f \in { \mathcal { S } }\), provided \(| a _ { 2 } | \le 2\) and conjectured that \(\left. | a _ { n } | \right. \leq n\) . He also found the exact value of the Koebe[7]constant. For a long time, mathematicians have been challenged by this conjecture. Louis De Branges[8]found a solution to the conjecture \(| a _ { n } | \leq n , ( n = 2 , 3 , \ldots )\) in 1984. Following Loewner[9]’s 1923 proof of \(| a _ { 3 } | \le 3\), Fekete-Szego[10]astounded mathematicians with the troublesome inequality \(| a _ { 3 } - \eta a _ { 2 } ^ { 2 } | \leq 1 + 2 e ^ { \left( \frac { - 2 \eta } { 1 - \eta } \right) } , \ 0 \leq \eta \leq 1\). In recent years, the study of geometric properties of holomorphic functions has gained significant attention due to its applications in geometric function theory and univalent function classes. One key inequality in this field is the Fekete-Szeg¨o inequality, which provides sharp bounds for coeficients in certain classes of analytic functions. This paper focuses on extending these inequalities to more generalized domains using the Raducanu-Orhan operator, which has shown promise in preserving univalent functions across complex order starlike and convex classes. Moreover, we explore the implications of these findings, a special class of conformal maps with unique geometric properties.

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 \(f ( z )\) defined in a domain \(D \subset C\) maintains the angles between the curves that pass through \(z\), then \(z \in D\) 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. Conformal mapping is a crucial method in complex analysis with a wide range of real-world applications.

If the function is harmonic, that is, it satisfies \(\nabla ^ { 2 } f = 0\) according to Laplace, then the conformal mapping transformation of such functions is likewise harmonic. Therefore, conformal mapping can be used to solve equations related to any field that can be represented by a potential function. Many problems arising from fluid mechanics, electrostatics, heat conduction, and many other physical situations can be formulated mathematically using Laplace’s equation \(\phi _ { x x } + \phi _ { y y } = 0\) in a certain region D of the \(z\)-plane. For example, it can be applied to scattering and difraction problems, brain surface mapping problems, and electrostatic potential problems in the shaded region of the \(z\)-plane[11]. It can also be used in stealth technology. Although the concept of conformal mapping is not directly used in stealth technology, the development of efective stealth technologies greatly benefits from an understanding of shape optimization, material science, and electromagnetic wave behaviour[12]. In addition, the univalent function helps to analyze the frequency analysis problem[13].

In recent years, new subclasses have been defined

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Inequality Estimations of Subclass of Univalent Functions Involving Raducanu-Orhan Operator

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