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10 pages, 261 KiB

Open AccessArticle

Properties for Close-to-Convex and Quasi-Convex Functions Using q-Linear Operator

by Ekram E. Ali, Rabha M. El-Ashwah, Abeer M. Albalahi and Wael W. Mohammed

Mathematics 2025, 13(6), 900; https://doi.org/10.3390/math13060900 - 7 Mar 2025

Viewed by 516

In this work, we describe the

q

-analogue of a multiplier–Ruscheweyh operator of a specific family of linear operators

I_{q, ρ}^{s} (ν, τ)

, and we obtain findings related to geometric function theory (GFT) by utilizing approaches [...] Read more.

In this work, we describe the

q

-analogue of a multiplier–Ruscheweyh operator of a specific family of linear operators

I_{q, ρ}^{s} (ν, τ)

, and we obtain findings related to geometric function theory (GFT) by utilizing approaches established through subordination and knowledge of

q

-calculus operators. By using this operator, we develop generalized classes of quasi-convex and close-to-convex functions in this paper. Additionally, the classes

K_{q, ρ}^{s} (ν, τ) (φ)

,

Q_{q, ρ}^{s} (ν, τ) (φ)

are introduced. The invariance of these recently formed classes under the

q

-Bernardi integral operator is investigated, along with a number of intriguing inclusion relationships between them. Additionally, several unique situations and the beneficial outcomes of these studies are taken into account. Full article

(This article belongs to the Special Issue Advanced Research in Complex Analysis Operators and Special Classes of Analytic Functions)

10 pages, 314 KiB

Open AccessArticle

Applications of Some Subclasses of Meromorphic Functions Associated with the q-Derivatives of the q-Binomials

by Ekram E. Ali, Hari M. Srivastava, Abdel Moneim Y. Lashin and Abeer M. Albalahi

Mathematics 2023, 11(11), 2496; https://doi.org/10.3390/math11112496 - 29 May 2023

Cited by 11 | Viewed by 1366

Abstract

In this article, we make use of the q-binomial theorem to introduce and study two new subclasses

ℵ (α q, q)

and

ℵ (α, q)

of meromorphic functions in the open unit disk

U

, that [...] Read more.

In this article, we make use of the q-binomial theorem to introduce and study two new subclasses

ℵ (α q, q)

and

ℵ (α, q)

of meromorphic functions in the open unit disk

U

, that is, analytic functions in the punctured unit disk

U^{*} = U \ {0} = {z : z \in C and 0 < |z| < 1} .

We derive inclusion relations and investigate an integral operator that preserves functions which belong to these function classes. In addition, we establish a strict inequality involving a certain linear convolution operator which we introduce in this article. Several special cases and corollaries of our main results are also considered. Full article

(This article belongs to the Section E4: Mathematical Physics)

15 pages, 328 KiB

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 8 | Viewed by 1962

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)

13 pages, 306 KiB

Open AccessArticle

On a Certain Subclass of p-Valent Analytic Functions Involving q-Difference Operator

by Abdel Moneim Y. Lashin, Abeer O. Badghaish and Badriah Maeed Algethami

Symmetry 2023, 15(1), 93; https://doi.org/10.3390/sym15010093 - 29 Dec 2022

Cited by 4 | Viewed by 1684

Abstract

This paper introduces and studies a new class of analytic p-valent functions in the open symmetric unit disc involving the Sălăgean-type q-difference operator. Furthermore, we present several interesting subordination results, coefficient inequalities, fractional q-calculus applications, and distortion theorems. Full article

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

16 pages, 339 KiB

Open AccessArticle

Hankel and Symmetric Toeplitz Determinants for a New Subclass of q-Starlike Functions

by Isra Al-shbeil, Jianhua Gong, Shahid Khan, Nazar Khan, Ajmal Khan, Mohammad Faisal Khan and Anjali Goswami

Fractal Fract. 2022, 6(11), 658; https://doi.org/10.3390/fractalfract6110658 - 7 Nov 2022

Cited by 23 | Viewed by 1881

Abstract

This paper considers the basic concepts of q-calculus and the principle of subordination. We define a new subclass of q-starlike functions related to the Salagean q-differential operator. For this class, we investigate initial coefficient estimates, Hankel determinants, Toeplitz matrices, and [...] Read more.

This paper considers the basic concepts of q-calculus and the principle of subordination. We define a new subclass of q-starlike functions related to the Salagean q-differential operator. For this class, we investigate initial coefficient estimates, Hankel determinants, Toeplitz matrices, and Fekete-Szegö problem. Moreover, we consider the q-Bernardi integral operator to discuss some applications in the form of some results. Full article

(This article belongs to the Special Issue Fractional Operators and Their Applications)

11 pages, 286 KiB

Open AccessArticle

A Study on Certain Subclasses of Analytic Functions Involving the Jackson q-Difference Operator

by Abdel Moneim Y. Lashin, Abeer O. Badghaish and Badriah Maeed Algethami

Symmetry 2022, 14(7), 1471; https://doi.org/10.3390/sym14071471 - 19 Jul 2022

Cited by 5 | Viewed by 1918

Abstract

We introduce two new subclasses of analytic functions in the open symmetric unit disc using a linear operator associated with the q-binomial theorem. In addition, we discuss inclusion relations and properties preserving integral operators for functions in these classes. This paper generalizes [...] Read more.

We introduce two new subclasses of analytic functions in the open symmetric unit disc using a linear operator associated with the q-binomial theorem. In addition, we discuss inclusion relations and properties preserving integral operators for functions in these classes. This paper generalizes some known results, as well as provides some new ones. Full article

(This article belongs to the Special Issue Complex Analysis, in Particular Analytic and Univalent Functions)

12 pages, 294 KiB

Open AccessArticle

Majorization Results Based upon the Bernardi Integral Operator

by Isra Al-Shbeil, Hari Mohan Srivastava, Muhammad Arif, Mirajul Haq, Nazar Khan and Bilal Khan

Symmetry 2022, 14(7), 1404; https://doi.org/10.3390/sym14071404 - 8 Jul 2022

Cited by 11 | Viewed by 1960

Abstract

By making use of some families of integral and derivative operators, many distinct subclasses of analytic, starlike functions, and symmetric starlike functions have already been defined and investigated from numerous perspectives. In this article, with the help of the one-parameter Bernardi integral operator, [...] Read more.

By making use of some families of integral and derivative operators, many distinct subclasses of analytic, starlike functions, and symmetric starlike functions have already been defined and investigated from numerous perspectives. In this article, with the help of the one-parameter Bernardi integral operator, we investigate several majorization results for the class of normalized starlike functions, which are associated with the Janowski functions. We also give some particular cases of our main results. Finally, we direct the interested readers to the possibility of examining the fundamental or quantum (or

q

-) extensions of the findings provided in this work in the concluding section. However, the

(p, q)

-variations of the suggested

q

-results will provide relatively minor and inconsequential developments because the additional (rather forced-in) parameter

p

is obviously redundant. Full article

(This article belongs to the Section Mathematics)

14 pages, 319 KiB

Open AccessArticle

Certain New Subclass of Multivalent Q-Starlike Functions Associated with Q-Symmetric Calculus

by Mohammad Faisal Khan, Anjali Goswami and Shahid Khan

Fractal Fract. 2022, 6(7), 367; https://doi.org/10.3390/fractalfract6070367 - 30 Jun 2022

Cited by 14 | Viewed by 1850

Abstract

In our present investigation, we extend the idea of q-symmetric derivative operators to multivalent functions and then define a new subclass of multivalent q-starlike functions. For this newly defined function class, we discuss some useful properties of multivalent functions, such as [...] Read more.

In our present investigation, we extend the idea of q-symmetric derivative operators to multivalent functions and then define a new subclass of multivalent q-starlike functions. For this newly defined function class, we discuss some useful properties of multivalent functions, such as the Hankel determinant, symmetric Toeplitz matrices, the Fekete–Szego problem, and upper bounds of the functional

|a_{p + 1} - μ a_{p + 1}^{2}|

and investigate some new lemmas for our main results. In addition, we consider the q-Bernardi integral operator along with q-symmetric calculus and discuss some applications of our main results. Full article

(This article belongs to the Special Issue New Trends 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 2431

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)

23 pages, 384 KiB

Open AccessArticle

On Starlike Functions of Negative Order Defined by q-Fractional Derivative

by Sadia Riaz, Ubaid Ahmed Nisar, Qin Xin, Sarfraz Nawaz Malik and Abdul Raheem

Fractal Fract. 2022, 6(1), 30; https://doi.org/10.3390/fractalfract6010030 - 6 Jan 2022

Cited by 16 | Viewed by 2034

Abstract

In this paper, two new classes of q-starlike functions in an open unit disc are defined and studied by using the q-fractional derivative. The class

\tilde{S_{q}^{*}} (α)

,

α \in (- 3, 1]

[...] Read more.

In this paper, two new classes of q-starlike functions in an open unit disc are defined and studied by using the q-fractional derivative. The class

\tilde{S_{q}^{*}} (α)

,

α \in (- 3, 1]

,

q \in (0, 1)

generalizes the class

S_{q}^{*}

of q-starlike functions and the class

\tilde{T_{q}^{*}} (α)

,

α \in [- 1, 1]

,

q \in (0, 1)

comprises the q-starlike univalent functions with negative coefficients. Some basic properties and the behavior of the functions in these classes are examined. The order of starlikeness in the class of convex function is investigated. It provides some interesting connections of newly defined classes with known classes. The mapping property of these classes under the family of q-Bernardi integral operator and its radius of univalence are studied. Additionally, certain coefficient inequalities, the radius of q-convexity, growth and distortion theorem, the covering theorem and some applications of fractional q-calculus for these new classes are investigated, and some interesting special cases are also included. Full article

(This article belongs to the Special Issue Importance of Fractional Order Derivatives in Real-World Applications: New Aspects and Understanding the Natural Phenomena)

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 22 | Viewed by 2270

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)

14 pages, 312 KiB

Open AccessArticle

q-Generalized Linear Operator on Bounded Functions of Complex Order

by Rizwan Salim Badar and Khalida Inayat Noor

Mathematics 2020, 8(7), 1149; https://doi.org/10.3390/math8071149 - 14 Jul 2020

Cited by 1 | Viewed by 1930

Abstract

This article presents a q-generalized linear operator in Geometric Function Theory (GFT) and investigates its application to classes of analytic bounded functions of complex order

S_{q} (c; M)

and

C_{q} (c; M)

where [...] Read more.

This article presents a q-generalized linear operator in Geometric Function Theory (GFT) and investigates its application to classes of analytic bounded functions of complex order

S_{q} (c; M)

and

C_{q} (c; M)

where

0 < q < 1,

0 \neq c \in C

, and

M > \frac{1}{2} .

Integral inclusion of the classes related to the q-Bernardi operator is also proven. Full article

(This article belongs to the Special Issue Geometrical Theory of Analytic Functions)

18 pages, 331 KiB

Open AccessArticle

Fekete-Szegö Type Problems and Their Applications for a Subclass of q-Starlike Functions with Respect to Symmetrical Points

by Hari Mohan Srivastava, Nazar Khan, Maslina Darus, Shahid Khan, Qazi Zahoor Ahmad and Saqib Hussain

Mathematics 2020, 8(5), 842; https://doi.org/10.3390/math8050842 - 22 May 2020

Cited by 41 | Viewed by 3660

Abstract

In this article, by using the concept of the quantum (or q-) calculus and a general conic domain

Ω_{k, q}

, we study a new subclass of normalized analytic functions with respect to symmetrical points in an open unit disk. [...] Read more.

In this article, by using the concept of the quantum (or q-) calculus and a general conic domain

Ω_{k, q}

, we study a new subclass of normalized analytic functions with respect to symmetrical points in an open unit disk. We solve the Fekete-Szegö type problems for this newly-defined subclass of analytic functions. We also discuss some applications of the main results by using a q-Bernardi integral operator. Full article

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

12 pages, 282 KiB

Open AccessArticle

A Study of Multivalent q-starlike Functions Connected with Circular Domain

by Lei Shi, Qaiser Khan, Gautam Srivastava, Jin-Lin Liu and Muhammad Arif

Mathematics 2019, 7(8), 670; https://doi.org/10.3390/math7080670 - 27 Jul 2019

Cited by 50 | Viewed by 3028

Abstract

Starlike functions have gained popularity both in literature and in usage over the past decade. In this paper, our aim is to examine some useful problems dealing with q-starlike functions. These include the convolution problem, sufficiency criteria, coefficient estimates, and Fekete–Szegö type [...] Read more.

Starlike functions have gained popularity both in literature and in usage over the past decade. In this paper, our aim is to examine some useful problems dealing with q-starlike functions. These include the convolution problem, sufficiency criteria, coefficient estimates, and Fekete–Szegö type inequalities for a new subfamily of analytic and multivalent functions associated with circular domain. In addition, we also define and study a Bernardi integral operator in its q-extension for multivalent functions. Furthermore, we will show that the class defined in this paper, along with the obtained results, generalizes many known works available in the literature. Full article

(This article belongs to the Special Issue Mathematical Analysis and Analytic Number Theory 2019)

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