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16 pages, 639 KiB

Open AccessArticle

Some Results Related to Booth Lemniscate and Integral Operators

by Bilal Khan, Zahra Orouji and Ali Ebadian

Fractal Fract. 2025, 9(5), 271; https://doi.org/10.3390/fractalfract9050271 - 22 Apr 2025

Viewed by 195

In this work, we explore the impact of integral operators such as the Libera and Alexander operators on specific families of analytic functions introduced in the literature and find some of their remarkable results. Using techniques from differential subordination and convolution theory, we [...] Read more.

In this work, we explore the impact of integral operators such as the Libera and Alexander operators on specific families of analytic functions introduced in the literature and find some of their remarkable results. Using techniques from differential subordination and convolution theory, we establish results concerning the radius of convexity and convolution properties for these function classes. Additionally, we investigate how these integral operators influence the geometric properties of functions in

$BS (μ)$ and

$KS (μ)$ , leading to new insights into their structural behavior. Full article

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17 pages, 273 KiB

Open AccessArticle

A Class of Meromorphic Multivalent Functions with Negative Coefficients Defined by a Ruscheweyh-Type Operator

by Isabel Marrero

Axioms 2025, 14(4), 284; https://doi.org/10.3390/axioms14040284 - 9 Apr 2025

Viewed by 220

Abstract

We introduce and systematically study a new class

$k^{λ, p} (α, β)$ of meromorphic p-valent functions defined by means of the Ruscheweyh-type operator

$D_{*}^{λ, p}$ , where

$p \in N$ , [...] Read more.

We introduce and systematically study a new class

$k^{λ, p} (α, β)$ of meromorphic p-valent functions defined by means of the Ruscheweyh-type operator

$D_{*}^{λ, p}$ , where

$p \in N$ ,

$λ > - p$ ,

$0 \leq α < 1$ , and

$β > 0$ . Membership in this class is characterized through coefficient estimates. Also investigated are growth, distortion, stability under convex combinations, radii of starlikeness and convexity of order

$ρ$

$(0 \leq ρ < 1)$ , convolution, the action of an integral operator of Bernardi–Libera–Livingston type, and neighborhoods. Full article

(This article belongs to the Section Mathematical Analysis)

10 pages, 260 KiB

Open AccessArticle

On α-Pseudo Spiralike Functions Associated with Exponential Pareto Distribution (EPD) and Libera Integral Operator

by Jamiu Olusegun Hamzat, Matthew Olanrewaju Oluwayemi, Abiodun Tinuoye Oladipo and Alina Alb Lupas

Mathematics 2024, 12(9), 1305; https://doi.org/10.3390/math12091305 - 25 Apr 2024

Viewed by 788

Abstract

The present study aims at investigating some characterizations of a new subclass

$G_{α} (μ, τ)$ and obtaining the bounds on the first two Taylor–Maclaurin coefficients for functions belonging to the newly introduced subclass. In order to achieve this, a [...] Read more.

The present study aims at investigating some characterizations of a new subclass

$G_{α} (μ, τ)$ and obtaining the bounds on the first two Taylor–Maclaurin coefficients for functions belonging to the newly introduced subclass. In order to achieve this, a compound function

$L_{x, n}^{σ} (z)$ is derived from the convolution of the analytic function

$f (z)$ and a modified exponential Pareto distribution

$G (x)$ in conjunction with the famous Libera integral operator

$L (ζ)$ . With the aid of the derived function, the aforementioned subclass

$G_{α} (μ, τ)$ is introduced, while some properties of functions belonging to this subclass are considered in the open unit disk. Full article

(This article belongs to the Special Issue New Trends in Complex Analysis Research, 2nd Edition)

17 pages, 2931 KiB

Open AccessArticle

New Results on

$(r, k, μ)$ -Riemann–Liouville Fractional Operators in Complex Domain with Applications

by Adel Salim Tayyah and Waggas Galib Atshan

Fractal Fract. 2024, 8(3), 165; https://doi.org/10.3390/fractalfract8030165 - 13 Mar 2024

Cited by 8 | Viewed by 1393

Abstract

This paper introduces fractional operators in the complex domain as generalizations for the Srivastava–Owa operators. Some properties for the above operators are also provided. We discuss the convexity and starlikeness of the generalized Libera integral operator. A condition for the convexity and starlikeness [...] Read more.

This paper introduces fractional operators in the complex domain as generalizations for the Srivastava–Owa operators. Some properties for the above operators are also provided. We discuss the convexity and starlikeness of the generalized Libera integral operator. A condition for the convexity and starlikeness of the solutions of fractional differential equations is provided. Finally, a fractional differential equation is converted into an ordinary differential equation by wave transformation; illustrative examples are provided to clarify the solution within the complex domain. Full article

(This article belongs to the Section General Mathematics, Analysis)

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12 pages, 309 KiB

Open AccessArticle

Study on the Criteria for Starlikeness in Integral Operators Involving Bessel Functions

by Georgia Irina Oros, Gheorghe Oros and Daniela Andrada Bardac-Vlada

Symmetry 2023, 15(11), 1976; https://doi.org/10.3390/sym15111976 - 26 Oct 2023

Cited by 3 | Viewed by 1224

Abstract

The study presented in this paper follows a line of research familiar for Geometric Function Theory, which consists in defining new integral operators and conducting studies for revealing certain geometric properties of those integral operators such as univalence, starlikness, or convexity. The present [...] Read more.

The study presented in this paper follows a line of research familiar for Geometric Function Theory, which consists in defining new integral operators and conducting studies for revealing certain geometric properties of those integral operators such as univalence, starlikness, or convexity. The present research focuses on the Bessel function of the first kind and order

$ν$ unveiling the conditions for this function to be univalent and further using its univalent form in order to define a new integral operator on the space of holomorphic functions. For particular values of the parameters implicated in the definition of the new integral operator involving the Bessel function of the first kind, the well-known Alexander, Libera, and Bernardi integral operators can be obtained. In the first part of the study, necessary and sufficient conditions are obtained for the Bessel function of the first kind and order

$ν$ to be a starlike function or starlike of order

$α \in [0, 1)$ . The renowned prolific method of differential subordination due to Sanford S. Miller and Petru T. Mocanu is employed in the reasoning. In the second part of the study, the outcome of the first part is applied in order to introduce the new integral operator involving the form of the Bessel function of the first kind, which is starlike. Further investigations disclose the necessary and sufficient conditions for this new integral operator to be starlike or starlike of order

$\frac{1}{2}$ . Full article

(This article belongs to the Special Issue Selected Papers from International Symposium on Geometric Function Theory and Applications (GFTA))

8 pages, 238 KiB

Open AccessArticle

On the Strong Starlikeness of the Bernardi Transform

by Zahra Orouji, Ali Ebadian and Nak Eun Cho

Axioms 2023, 12(1), 91; https://doi.org/10.3390/axioms12010091 - 16 Jan 2023

Viewed by 1599

Abstract

Many papers concern both the starlikeness and the convexity of Bernardi integral operator. Using the Nunokawa’s Lemma, we want to determine conditions for the strong starlikeness of the Bernardi transform of normalized analytic functions g, such that [...] Read more.

Many papers concern both the starlikeness and the convexity of Bernardi integral operator. Using the Nunokawa’s Lemma, we want to determine conditions for the strong starlikeness of the Bernardi transform of normalized analytic functions g, such that

$| arg {g^{'} (z)} | < \frac{α π}{2}$ in the open unit disk

$Δ$ where

$0 < α < 2$ . Our results include the results of Mocanu, Nunokawa and others on the Libera transform. Full article

(This article belongs to the Special Issue Dedicated to Professor Hari Mohan Srivastava on the Occasion of His 82nd Birthday)

17 pages, 339 KiB

Open AccessArticle

Subordination Properties of Certain Operators Concerning Fractional Integral and Libera Integral Operator

by Georgia Irina Oros, Gheorghe Oros and Shigeyoshi Owa

Fractal Fract. 2023, 7(1), 42; https://doi.org/10.3390/fractalfract7010042 - 30 Dec 2022

Cited by 7 | Viewed by 1724

Abstract

The results contained in this paper are the result of a study regarding fractional calculus combined with the classical theory of differential subordination established for analytic complex valued functions. A new operator is introduced by applying the Libera integral operator and fractional integral [...] Read more.

The results contained in this paper are the result of a study regarding fractional calculus combined with the classical theory of differential subordination established for analytic complex valued functions. A new operator is introduced by applying the Libera integral operator and fractional integral of order

$λ$ for analytic functions. Many subordination properties are obtained for this newly defined operator by using famous lemmas proved by important scientists concerned with geometric function theory, such as Eenigenburg, Hallenbeck, Miller, Mocanu, Nunokawa, Reade, Ruscheweyh and Suffridge. Results regarding strong starlikeness and convexity of order

$α$ are also discussed, and an example shows how the outcome of the research can be applied. Full article

(This article belongs to the Special Issue Fractional Calculus and Hypergeometric Functions in Complex Analysis)

14 pages, 298 KiB

Open AccessArticle

Application of a Multiplier Transformation to Libera Integral Operator Associated with Generalized Distribution

by Jamiu Olusegun Hamzat, Abiodun Tinuoye Oladipo and Georgia Irina Oros

Symmetry 2022, 14(9), 1934; https://doi.org/10.3390/sym14091934 - 16 Sep 2022

Cited by 2 | Viewed by 1405

Abstract

The research presented in this paper deals with analytic p-valent functions related to the generalized probability distribution in the open unit disk U. Using the Hadamard product or convolution, function

$f_{s} (z)$ is defined as involving an analytic [...] Read more.

The research presented in this paper deals with analytic p-valent functions related to the generalized probability distribution in the open unit disk U. Using the Hadamard product or convolution, function

$f_{s} (z)$ is defined as involving an analytic p-valent function and generalized distribution expressed in terms of analytic p-valent functions. Neighborhood properties for functions

$f_{s} (z)$ are established. Further, by applying a previously introduced linear transformation to

$f_{s} (z)$ and using an extended Libera integral operator, a new generalized Libera-type operator is defined. Moreover, using the same linear transformation, subclasses of starlike, convex, close-to-convex and spiralike functions are defined and investigated in order to obtain geometrical properties that characterize the new generalized Libera-type operator. Symmetry properties are due to the involvement of the Libera integral operator and convolution transform. Full article

(This article belongs to the Special Issue Symmetry in Pure Mathematics and Real and Complex Analysis)

13 pages, 484 KiB

Open AccessArticle

Applications of Confluent Hypergeometric Function in Strong Superordination Theory

by Georgia Irina Oros, Gheorghe Oros and Ancuța Maria Rus

Axioms 2022, 11(5), 209; https://doi.org/10.3390/axioms11050209 - 29 Apr 2022

Cited by 5 | Viewed by 2592

Abstract

In the research presented in this paper, confluent hypergeometric function is embedded in the theory of strong differential superordinations. In order to proceed with the study, the form of the confluent hypergeometric function is adapted taking into consideration certain classes of analytic functions [...] Read more.

In the research presented in this paper, confluent hypergeometric function is embedded in the theory of strong differential superordinations. In order to proceed with the study, the form of the confluent hypergeometric function is adapted taking into consideration certain classes of analytic functions depending on an extra parameter previously introduced related to the theory of strong differential subordination and superordination. Operators previously defined using confluent hypergeometric function, namely Kummer–Bernardi and Kummer–Libera integral operators, are also adapted to those classes and strong differential superordinations are obtained for which they are the best subordinants. Similar results are obtained regarding the derivatives of the operators. The examples presented at the end of the study are proof of the applicability of the original results. Full article

(This article belongs to the Special Issue New Developments in Geometric Function Theory)

12 pages, 300 KiB

Open AccessArticle

Inclusion Relations for Dini Functions Involving Certain Conic Domains

by Bilal Khan, Shahid Khan, Jong-Suk Ro, Serkan Araci, Nazar Khan and Nasir Khan

Fractal Fract. 2022, 6(2), 118; https://doi.org/10.3390/fractalfract6020118 - 17 Feb 2022

Cited by 2 | Viewed by 2355

Abstract

In recent years, special functions such as Bessel functions have been widely used in many areas of mathematics and physics. We are essentially motivated by the recent development; in our present investigation, we make use of certain conic domains and define a new [...] Read more.

In recent years, special functions such as Bessel functions have been widely used in many areas of mathematics and physics. We are essentially motivated by the recent development; in our present investigation, we make use of certain conic domains and define a new class of analytic functions associated with the Dini functions. We derive inclusion relationships and certain integral preserving properties. By applying the Bernardi-Libera-Livingston integral operator, we obtain some remarkable applications of our main results. Finally, in the concluding section, we recall the attention of curious readers to studying the q-generalizations of the results presented in this paper. Furthermore, based on the suggested extension, the

$(p, q)$ -extension will be a relatively minor and unimportant change, as the new parameter

$p$ is redundant. Full article

(This article belongs to the Special Issue New Trends in Geometric Function Theory)

14 pages, 315 KiB

Open AccessArticle

A Class of k-Symmetric Harmonic Functions Involving a Certain q-Derivative Operator

by Hari M. Srivastava, Nazar Khan, Shahid Khan, Qazi Zahoor Ahmad and Bilal Khan

Mathematics 2021, 9(15), 1812; https://doi.org/10.3390/math9151812 - 30 Jul 2021

Cited by 21 | Viewed by 2118

Abstract

In this paper, we introduce a new class of harmonic univalent functions with respect to k-symmetric points by using a newly-defined q-analog of the derivative operator for complex harmonic functions. For this harmonic univalent function class, we derive a sufficient condition, [...] Read more.

In this paper, we introduce a new class of harmonic univalent functions with respect to k-symmetric points by using a newly-defined q-analog of the derivative operator for complex harmonic functions. For this harmonic univalent function class, we derive a sufficient condition, a representation theorem, and a distortion theorem. We also apply a generalized q-Bernardi–Libera–Livingston integral operator to examine the closure properties and coefficient bounds. Furthermore, we highlight some known consequences of our main results. In the concluding part of the article, we have finally reiterated the well-demonstrated fact that the results presented in this article can easily be rewritten as the so-called

$(p, q)$ -variations by making some straightforward simplifications, and it will be an inconsequential exercise, simply because the additional parameter p is obviously unnecessary. Full article

(This article belongs to the Special Issue Higher Transcendental Functions and Their Multi-Disciplinary Applications)

12 pages, 790 KiB

Open AccessArticle

New Applications of the Bernardi Integral Operator

by Shigeyoshi Owa and H. Özlem Güney

Mathematics 2020, 8(7), 1180; https://doi.org/10.3390/math8071180 - 17 Jul 2020

Cited by 8 | Viewed by 2738

Abstract

Let

$A (p, n)$ be the class of

$f (z)$ which are analytic p-valent functions in the closed unit disk

$\bar{U} = \{z \in C : |z| \leq 1\}$ . The expression [...] Read more.

Let

$A (p, n)$ be the class of

$f (z)$ which are analytic p-valent functions in the closed unit disk

$\bar{U} = \{z \in C : |z| \leq 1\}$ . The expression

$B_{- m - λ} f (z)$ is defined by using fractional integrals of order

$λ$ for

$f (z) \in A (p, n) .$ When

$m = 1$ and

$λ = 0,$

$B_{- 1} f (z)$ becomes Bernardi integral operator. Using the fractional integral

$B_{- m - λ} f (z),$ the subclass

$T_{p, n} (α_{s}, β, ρ; m, λ)$ of

$A (p, n)$ is introduced. In the present paper, we discuss some interesting properties for

$f (z)$ concerning with the class

$T_{p, n} (α_{s}, β, ρ; m, λ) .$ Also, some interesting examples for our results will be considered. Full article

(This article belongs to the Special Issue Complex Analysis and Geometric Function Theory)

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