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17 pages, 332 KB

Open AccessArticle

Starlike Functions of the Miller–Ross-Type Poisson Distribution in the Janowski Domain

by Gangadharan Murugusundaramoorthy, Hatun Özlem Güney and Daniel Breaz

Mathematics 2024, 12(6), 795; https://doi.org/10.3390/math12060795 - 8 Mar 2024

Cited by 6 | Viewed by 1320

In this paper, considering the various important applications of Miller–Ross functions in the fields of applied sciences, we introduced a new class of analytic functions

f,

utilizing the concept of Miller–Ross functions in the region of the Janowski domain. Furthermore, we obtained [...] Read more.

In this paper, considering the various important applications of Miller–Ross functions in the fields of applied sciences, we introduced a new class of analytic functions

f,

utilizing the concept of Miller–Ross functions in the region of the Janowski domain. Furthermore, we obtained initial coefficients of Taylor series expansion of

f,

coefficient inequalities for

f^{- 1}

and the Fekete–Szegö problem. We also covered some key geometric properties for functions f in this newly formed class, such as the necessary and sufficient condition, convex combination, sequential subordination and partial sum findings. Full article

16 pages, 323 KB

Open AccessArticle

New Subclass of Close-to-Convex Functions Defined by Quantum Difference Operator and Related to Generalized Janowski Function

by Suha B. Al-Shaikh, Mohammad Faisal Khan, Mustafa Kamal and Naeem Ahmad

Symmetry 2023, 15(11), 1974; https://doi.org/10.3390/sym15111974 - 25 Oct 2023

Cited by 2 | Viewed by 1482

Abstract

This work begins with a discussion of the quantum calculus operator theory and proceeds to develop and investigate a new family of close-to-convex functions in an open unit disk. Considering the quantum difference operator, we define and study a new subclass of close-to-convex [...] Read more.

This work begins with a discussion of the quantum calculus operator theory and proceeds to develop and investigate a new family of close-to-convex functions in an open unit disk. Considering the quantum difference operator, we define and study a new subclass of close-to-convex functions connected with generalized Janowski functions. We prove the necessary and sufficient conditions for functions that belong to newly defined classes, including the inclusion relations and estimations of the coefficients. The Fekete–Szegő problem for a more general class is also discussed. The results of this investigation expand upon those of the previous study. Full article

(This article belongs to the Special Issue Symmetry in Geometric Theory of Analytic Functions)

20 pages, 336 KB

Open AccessArticle

New Criteria for Starlikness and Convexity of a Certain Family of Integral Operators

by Hari M. Srivastava, Rogayeh Alavi, Saeid Shams, Rasoul Aghalary and Santosh B. Joshi

Mathematics 2023, 11(18), 3919; https://doi.org/10.3390/math11183919 - 14 Sep 2023

Cited by 2 | Viewed by 1319

Abstract

In this paper, we first modify one of the most famous theorems on the principle of differential subordination to hold true for normalized analytic functions with a fixed initial Taylor-Maclaurin coefficient. By using this modified form, we generalize and improve several results, which [...] Read more.

In this paper, we first modify one of the most famous theorems on the principle of differential subordination to hold true for normalized analytic functions with a fixed initial Taylor-Maclaurin coefficient. By using this modified form, we generalize and improve several results, which appeared recently in the literature on the geometric function theory of complex analysis. We also prove some simple conditions for starlikeness, convexity, and the strong starlikeness of several one-parameter families of integral operators, including (for example) a certain

μ

-convex integral operator and the familiar Bernardi integral operator. Full article

(This article belongs to the Special Issue Current Topics in Geometric Function Theory)

11 pages, 295 KB

Open AccessArticle

Studying the Harmonic Functions Associated with Quantum Calculus

by Abdullah Alsoboh, Ala Amourah, Maslina Darus and Carla Amoi Rudder

Mathematics 2023, 11(10), 2220; https://doi.org/10.3390/math11102220 - 9 May 2023

Cited by 14 | Viewed by 2324

Abstract

Using the derivative operators’

q

-analogs values, a wide variety of holomorphic function subclasses,

q

-starlike, and

q

-convex functions have been researched and examined. With the aid of fundamental ideas from the theory of

q

-calculus operators, we describe new

q

-operators [...] Read more.

Using the derivative operators’

q

-analogs values, a wide variety of holomorphic function subclasses,

q

-starlike, and

q

-convex functions have been researched and examined. With the aid of fundamental ideas from the theory of

q

-calculus operators, we describe new

q

-operators of harmonic function

H_{ϱ, χ; q}^{γ} F (ϖ)

in this work. We also define a new harmonic function subclass related to the Janowski and

q

-analog of Le Roy-type functions Mittag–Leffler functions. Several important properties are assigned to the new class, including necessary and sufficient conditions, the covering Theorem, extreme points, distortion bounds, convolution, and convex combinations. Furthermore, we emphasize several established remarks for confirming our primary findings presented in this study, as well as some applications of this study in the form of specific outcomes and corollaries. Full article

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

13 pages, 299 KB

Open AccessArticle

Faber Polynomial Coefficient Estimates for Janowski Type bi-Close-to-Convex and bi-Quasi-Convex Functions

by Shahid Khan, Şahsene Altınkaya, Qin Xin, Fairouz Tchier, Sarfraz Nawaz Malik and Nazar Khan

Symmetry 2023, 15(3), 604; https://doi.org/10.3390/sym15030604 - 27 Feb 2023

Cited by 10 | Viewed by 2089

Abstract

Motivated by the recent work on symmetric analytic functions by using the concept of Faber polynomials, this article introduces and studies two new subclasses of bi-close-to-convex and quasi-close-to-convex functions associated with Janowski functions. By using the Faber polynomial expansion method, it determines the [...] Read more.

Motivated by the recent work on symmetric analytic functions by using the concept of Faber polynomials, this article introduces and studies two new subclasses of bi-close-to-convex and quasi-close-to-convex functions associated with Janowski functions. By using the Faber polynomial expansion method, it determines the general coefficient bounds for the functions belonging to these classes. It also finds initial coefficients of bi-close-to-convex and bi-quasi-convex functions by using Janowski functions. Some known consequences of the main results are also highlighted. Full article

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

15 pages, 328 KB

Open AccessArticle

A Class of Janowski-Type (p,q)-Convex Harmonic Functions Involving a Generalized q-Mittag–Leffler Function

by Sarem H. Hadi, Maslina Darus and Alina Alb Lupaş

Axioms 2023, 12(2), 190; https://doi.org/10.3390/axioms12020190 - 11 Feb 2023

Cited by 9 | Viewed by 2213

Abstract

This research aims to present a linear operator

L_{p, q}^{ρ, σ, μ} f

utilizing the q-Mittag–Leffler function. Then, we introduce the subclass of harmonic

(p, q)

-convex functions [...] Read more.

This research aims to present a linear operator

L_{p, q}^{ρ, σ, μ} f

utilizing the q-Mittag–Leffler function. Then, we introduce the subclass of harmonic

(p, q)

-convex functions

{HT}_{p, q} (ϑ, W, V)

related to the Janowski function. For the harmonic p-valent functions f class, we investigate the harmonic geometric properties, such as coefficient estimates, convex linear combination, extreme points, and Hadamard product. Finally, the closure property is derived using the subclass

{HT}_{p, q} (ϑ, W, V)

under the q-Bernardi integral operator. Full article

(This article belongs to the Special Issue Nonlinear Dynamical Systems with Applications)

16 pages, 337 KB

Open AccessArticle

Geometric Properties for a New Class of Analytic Functions Defined by a Certain Operator

by Daniel Breaz, Gangadharan Murugusundaramoorthy and Luminiţa-Ioana Cotîrlǎ

Symmetry 2022, 14(12), 2624; https://doi.org/10.3390/sym14122624 - 11 Dec 2022

Cited by 6 | Viewed by 1883

Abstract

The aim of this paper is to define and explore a certain class of analytic functions involving the

(p, q)

-Wanas operator related to the Janowski functions. We discuss geometric properties, growth and distortion bounds, necessary and sufficient conditions, the [...] Read more.

The aim of this paper is to define and explore a certain class of analytic functions involving the

(p, q)

-Wanas operator related to the Janowski functions. We discuss geometric properties, growth and distortion bounds, necessary and sufficient conditions, the Fekete–Szegö problem, partial sums, and convex combinations for the newly defined class. We solve the Fekete–Szegö problem related to the convolution product and discuss applications to probability distribution. Full article

(This article belongs to the Special Issue Symmetry in Quantum Calculus)

22 pages, 360 KB

Open AccessArticle

On q-Limaçon Functions

by Afis Saliu, Kanwal Jabeen, Isra Al-Shbeil, Najla Aloraini and Sarfraz Nawaz Malik

Symmetry 2022, 14(11), 2422; https://doi.org/10.3390/sym14112422 - 15 Nov 2022

Cited by 17 | Viewed by 2038

Abstract

Very recently, functions that map the open unit disc U onto a limaçon domain, which is symmetric with respect to the real axis in the right-half plane, were initiated in the literature. The analytic characterization, geometric properties, and Hankel determinants of these families [...] Read more.

Very recently, functions that map the open unit disc U onto a limaçon domain, which is symmetric with respect to the real axis in the right-half plane, were initiated in the literature. The analytic characterization, geometric properties, and Hankel determinants of these families of functions were also demonstrated. In this article, we present a q-analogue of these functions and use it to establish the classes of starlike and convex limaçon functions that are correlated with q-calculus. Furthermore, the coefficient bounds, as well as the third Hankel determinants, for these novel classes are established. Moreover, at some stages, the radius of the inclusion relationship for a particular case of these subclasses with the Janowski families of functions are obtained. It is worth noting that many of our results are sharp. Full article

(This article belongs to the Special Issue Symmetry in Quantum Calculus)

16 pages, 316 KB

Open AccessArticle

Applications of Symmetric Quantum Calculus to the Class of Harmonic Functions

by Mohammad Faisal Khan, Isra Al-Shbeil, Najla Aloraini, Nazar Khan and Shahid Khan

Symmetry 2022, 14(10), 2188; https://doi.org/10.3390/sym14102188 - 18 Oct 2022

Cited by 16 | Viewed by 2203

Abstract

In the past few years, many scholars gave much attention to the use of q-calculus in geometric functions theory, and they defined new subclasses of analytic and harmonic functions. While using the symmetric q-calculus in geometric function theory, very little work [...] Read more.

In the past few years, many scholars gave much attention to the use of q-calculus in geometric functions theory, and they defined new subclasses of analytic and harmonic functions. While using the symmetric q-calculus in geometric function theory, very little work has been published so far. In this research, with the help of fundamental concepts of symmetric q-calculus and the symmetric q-Salagean differential operator for harmonic functions, we define a new class of harmonic functions connected with Janowski functions

\tilde{S_{H}^{0}} (m, q, A, B)

. First, we illustrate the necessary and sufficient convolution condition for

\tilde{S_{H}^{0}} (m, q, A, B)

and then prove that this sufficient condition is a sense preserving and univalent, and it is necessary for its subclass

\tilde{{TS}_{H}^{0}} (m, q, A, B)

. Furthermore, by using this necessary and sufficient coefficient condition, we establish some novel results, particularly convexity, compactness, radii of q-starlike and q-convex functions of order

α

, and extreme points for this newly defined class of harmonic functions. Our results are the generalizations of some previous known results. Full article

(This article belongs to the Special Issue Symmetry in Quantum Calculus)

19 pages, 334 KB

Open AccessArticle

Applications of a q-Differential Operator to a Class of Harmonic Mappings Defined by q-Mittag–Leffler Functions

by Mohammad Faisal Khan, Isra Al-shbeil, Shahid Khan, Nazar Khan, Wasim Ul Haq and Jianhua Gong

Symmetry 2022, 14(9), 1905; https://doi.org/10.3390/sym14091905 - 12 Sep 2022

Cited by 11 | Viewed by 1970

Abstract

Many diverse subclasses of analytic functions, q-starlike functions, and symmetric q-starlike functions have been studied and analyzed by using q-analogous values of integral and derivative operators. In this paper, we define a q-analogous value of differential operators for harmonic [...] Read more.

Many diverse subclasses of analytic functions, q-starlike functions, and symmetric q-starlike functions have been studied and analyzed by using q-analogous values of integral and derivative operators. In this paper, we define a q-analogous value of differential operators for harmonic functions with the help of basic concepts of quantum (q-) calculus operator theory; and introduce a new subclass of harmonic functions associated with the Janowski and q-Mittag–Leffler functions. We obtain several useful properties of the new class, such as necessary and sufficient conditions, criteria for analyticity, compactness, convexity, extreme points, radii of starlikeness, radii of convexity, distortion bounds, and integral mean inequality. Furthermore, we discuss some applications of this study in the form of some results and examples and highlight some known corollaries for verifying our main results presented in this investigation. Finally, the conclusion section summarizes the fact about the

(p, q)

-variations. Full article

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

15 pages, 331 KB

Open AccessArticle

On Geometric Properties of Bessel–Struve Kernel Functions in Unit Disc

by Najla M. Alarifi and Saiful R. Mondal

Mathematics 2022, 10(14), 2516; https://doi.org/10.3390/math10142516 - 19 Jul 2022

Cited by 4 | Viewed by 2337

Abstract

The Bessel–Struve kernel function defined in the unit disc is used in this study. The Bessel–Struve kernel functions are generalized in this article, and differential equations are derived. We found conditions under which the generalized Bessel–Struve function is Lemniscate convex by using a [...] Read more.

The Bessel–Struve kernel function defined in the unit disc is used in this study. The Bessel–Struve kernel functions are generalized in this article, and differential equations are derived. We found conditions under which the generalized Bessel–Struve function is Lemniscate convex by using a subordination technique. The relation between the Janowski class and exponential class is also derived. Full article

(This article belongs to the Special Issue New Trends in Complex Analysis Researches)

13 pages, 274 KB

Open AccessArticle

Applications of Borel-Type Distributions Series to a Class of Janowski-Type Analytic Functions

by Bakhtiar Ahmad, Muhammad Ghaffar Khan and Luminiţa-Ioana Cotîrlă

Symmetry 2022, 14(2), 322; https://doi.org/10.3390/sym14020322 - 4 Feb 2022

Cited by 2 | Viewed by 2087

Abstract

The main purpose of this article is to introduce the new subclass of analytic functions whose coefficients are Borel distributions in the Janowski domain. Further, we investigate some useful number of properties such as Fekete–Szegő inequality, necessary and sufficient condition, growth and distortion [...] Read more.

The main purpose of this article is to introduce the new subclass of analytic functions whose coefficients are Borel distributions in the Janowski domain. Further, we investigate some useful number of properties such as Fekete–Szegő inequality, necessary and sufficient condition, growth and distortion approximations, convex linear combination, arithmetic mean, radii of close-to-convexity and starlikeness and partial sums, followed by some extremal functions for this defined class. The symmetry properties and other properties of the subclass of functions introduced in this paper can be studied as future research directions. Full article

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

16 pages, 297 KB

Open AccessArticle

A Certain Subclass of Multivalent Analytic Functions Defined by the q-Difference Operator Related to the Janowski Functions

by Bo Wang, Rekha Srivastava and Jin-Lin Liu

Mathematics 2021, 9(14), 1706; https://doi.org/10.3390/math9141706 - 20 Jul 2021

Cited by 14 | Viewed by 2241

Abstract

A class of p-valent analytic functions is introduced using the q-difference operator and the familiar Janowski functions. Several properties of functions in the class, such as the Fekete–Szegö inequality, coefficient estimates, necessary and sufficient conditions, distortion and growth theorems, radii of [...] Read more.

A class of p-valent analytic functions is introduced using the q-difference operator and the familiar Janowski functions. Several properties of functions in the class, such as the Fekete–Szegö inequality, coefficient estimates, necessary and sufficient conditions, distortion and growth theorems, radii of convexity and starlikeness, closure theorems and partial sums, are discussed in this paper. Full article

(This article belongs to the Special Issue Mathematical Analysis and Analytic Number Theory 2020)

14 pages, 303 KB

Open AccessArticle

A Subclass of Multivalent Janowski Type q-Starlike Functions and Its Consequences

by Qiuxia Hu, Hari M. Srivastava, Bakhtiar Ahmad, Nazar Khan, Muhammad Ghaffar Khan, Wali Khan Mashwani and Bilal Khan

Symmetry 2021, 13(7), 1275; https://doi.org/10.3390/sym13071275 - 16 Jul 2021

Cited by 29 | Viewed by 2762

Abstract

In this article, by utilizing the theory of quantum (or q-) calculus, we define a new subclass of analytic and multivalent (or p-valent) functions class

A_{p}

, where class

A_{p}

is invariant (or symmetric) under rotations. The well-known class [...] Read more.

In this article, by utilizing the theory of quantum (or q-) calculus, we define a new subclass of analytic and multivalent (or p-valent) functions class

A_{p}

, where class

A_{p}

is invariant (or symmetric) under rotations. The well-known class of Janowski functions are used with the help of the principle of subordination between analytic functions in order to define this subclass of analytic and p-valent functions. This function class generalizes various other subclasses of analytic functions, not only in classical Geometric Function Theory setting, but also some q-analogue of analytic multivalent function classes. We study and investigate some interesting properties such as sufficiency criteria, coefficient bounds, distortion problem, growth theorem, radii of starlikeness and convexity for this newly-defined class. Other properties such as those involving convex combination are also discussed for these functions. In the concluding part of the article, we have finally given the well-demonstrated fact that the results presented in this article can be obtained for the

(p, q)

-variations, by making some straightforward simplification and will be an inconsequential exercise simply because the additional parameter

p

is obviously unnecessary. Full article

(This article belongs to the Special Issue Integral Transformation, Operational Calculus and Their Applications)

14 pages, 318 KB

Open AccessArticle

Bohr Radius Problems for Some Classes of Analytic Functions Using Quantum Calculus Approach

by Om Ahuja, Swati Anand and Naveen Kumar Jain

Mathematics 2020, 8(4), 623; https://doi.org/10.3390/math8040623 - 18 Apr 2020

Cited by 6 | Viewed by 3489

Abstract

The main purpose of this investigation is to use quantum calculus approach and obtain the Bohr radius for the class of q-starlike (q-convex) functions of order

α

. The Bohr radius is also determined for a generalized class of q [...] Read more.

The main purpose of this investigation is to use quantum calculus approach and obtain the Bohr radius for the class of q-starlike (q-convex) functions of order

α

. The Bohr radius is also determined for a generalized class of q-Janowski starlike and q-Janowski convex functions with negative coefficients. Full article

(This article belongs to the Special Issue Complex Analysis and Its Applications)

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