MDPI - Publisher of Open Access Journals

11 pages, 279 KiB

Open AccessArticle

Convolution Results with Subclasses of p-Valent Meromorphic Function Connected with

q

-Difference Operator

by Ekram E. Ali, Rabha M. El-Ashwah, Abeer M. Albalahi, Rabab Sidaoui, Marwa Ennaceur and Miguel Vivas-Cortez

Mathematics 2024, 12(22), 3548; https://doi.org/10.3390/math12223548 - 13 Nov 2024

Viewed by 333

Abstract

Applying the operator of

q

-difference, we examine the convolution properties of the subclasses

{MS}_{ζ, q}^{r, p} (A, B)

and

M K_{ζ, q}^{r, p} (A, B)

of p-valent [...] Read more.

Applying the operator of

q

-difference, we examine the convolution properties of the subclasses

{MS}_{ζ, q}^{r, p} (A, B)

and

M K_{ζ, q}^{r, p} (A, B)

of p-valent meromorphic functions defined in the punctured open-unit disc. We derived specific inclusion features and coefficient estimates for functions that fall into these subclasses. Additionally, connections between the results presented here and those discovered in earlier papers are emphasized. Full article

(This article belongs to the Special Issue Complex Analysis and Geometric Function Theory, 2nd Edition)

13 pages, 355 KiB

Open AccessArticle

Bi-Starlike Function of Complex Order Involving Mathieu-Type Series in the Shell-Shaped Region

by Ibrahim S. Elshazly, Gangadharan Murugusundaramoorthy, Borhen Halouani, Alaa H. El-Qadeem and Kaliappan Vijaya

Axioms 2024, 13(11), 747; https://doi.org/10.3390/axioms13110747 - 30 Oct 2024

Viewed by 476

Abstract

For functions of the form

ϕ (ξ) = ξ + \sum_{n = 2}^{\infty} c_{n} ξ^{n}

, we identified two new subclasses of bi-starlike functions and bi-convex functions by using Mathieu-type series defined in the disc [...] Read more.

For functions of the form

ϕ (ξ) = ξ + \sum_{n = 2}^{\infty} c_{n} ξ^{n}

, we identified two new subclasses of bi-starlike functions and bi-convex functions by using Mathieu-type series defined in the disc

Δ = {ξ \in C : | ξ | < 1}

. We derived constraints for

| c_{2} |

and

| c_{3} |

, and the subclasses are connected to the shell-shaped area. The Fekete–Szegö functional properties for the aforementioned function subclasses were also investigated. Additionally, a number of related corollaries are shown. Full article

(This article belongs to the Special Issue New Developments in Geometric Function Theory, 3rd Edition)

► Show Figures

Figure 1

11 pages, 395 KiB

Open AccessArticle

On λ-Pseudo Bi-Starlike Functions Related to Second Einstein Function

by Alaa H. El-Qadeem, Gangadharan Murugusundaramoorthy, Borhen Halouani, Ibrahim S. Elshazly, Kaliappan Vijaya and Mohamed A. Mamon

Symmetry 2024, 16(11), 1429; https://doi.org/10.3390/sym16111429 - 27 Oct 2024

Viewed by 834

Abstract

A new class

B_{Σ}^{λ} (γ, κ)

of bi-starlike

λ

-pseudo functions related to the second Einstein function is presented in this paper.

c_{2}

and

c_{3}

indicate the initial Taylor coefficients of [...] Read more.

A new class

B_{Σ}^{λ} (γ, κ)

of bi-starlike

λ

-pseudo functions related to the second Einstein function is presented in this paper.

c_{2}

and

c_{3}

indicate the initial Taylor coefficients of

ϕ \in B_{Σ}^{λ} (γ, κ),

and the bounds for

| c_{2} |

and

| c_{3} |

are obtained. Additionally, for

ϕ \in B_{Σ}^{λ} (γ, κ)

, we calculate the Fekete–Szegö functional. Full article

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

► Show Figures

Figure 1

16 pages, 818 KiB

Open AccessArticle

Starlikeness and Convexity of Generalized Bessel-Maitland Function

by Muhammad Umar Nawaz, Daniel Breaz, Mohsan Raza and Luminiţa-Ioana Cotîrlă

Axioms 2024, 13(10), 691; https://doi.org/10.3390/axioms13100691 - 4 Oct 2024

Viewed by 450

Abstract

The main objective of this research is to examine a specific sufficiency criteria for the starlikeness and convexity of order

δ

, k-uniform starlikeness, k-uniform convexity, lemniscate starlikeness and convexity, exponential starlikeness and convexity, uniform convexity of the Generalized Bessel-Maitland function. Applications of [...] Read more.

The main objective of this research is to examine a specific sufficiency criteria for the starlikeness and convexity of order

δ

, k-uniform starlikeness, k-uniform convexity, lemniscate starlikeness and convexity, exponential starlikeness and convexity, uniform convexity of the Generalized Bessel-Maitland function. Applications of these conclusions to the concept of corollaries are also provided. Additionally, an illustrated representation of these outcomes will be presented. So functional inequalities involving gamma function will be the main research tools of this exploration. The outcomes from this study generalize a number of conclusions from earlier studies. Full article

► Show Figures

Figure 1

22 pages, 1032 KiB

Open AccessArticle

Properties and Applications of Complex Fractal–Fractional Operators in the Open Unit Disk

by Adel A. Attiya, Soheil Salahshour, Rabha W. Ibrahim and Mansour F. Yassen

Fractal Fract. 2024, 8(10), 584; https://doi.org/10.3390/fractalfract8100584 - 3 Oct 2024

Viewed by 541

Abstract

A fractal–fractional calculus is presented in term of a generalized gamma function (ℓ−gamma function:

Γ_{ℓ} (.)

). The suggested operators are given in the symmetric complex domain (the open unit disk). A novel arrangement of the operators shows [...] Read more.

A fractal–fractional calculus is presented in term of a generalized gamma function (ℓ−gamma function:

Γ_{ℓ} (.)

). The suggested operators are given in the symmetric complex domain (the open unit disk). A novel arrangement of the operators shows the normalization associated with every operator. We investigate a number of significant geometric features thanks to this. Additionally, some integrals, such the Alexander and Libra integral operators, are associated with these operators. Simple power functions are among the illustrations that are provided. Additionally, the formulation of the discrete

ℓ -

fractal–fractional operators is conducted. We demonstrate that well-known examples are involved in the extended operators. Full article

(This article belongs to the Special Issue Recent Advances in Fractional Differential Equations and Their Applications, 2nd Edition)

► Show Figures

Figure 1

13 pages, 286 KiB

Open AccessArticle

Hankel Determinants of Normalized Analytic Functions Associated with Hyperbolic Secant Function

by Sushil Kumar, Daniel Breaz, Luminita-Ioana Cotîrlă and Asena Çetinkaya

Symmetry 2024, 16(10), 1303; https://doi.org/10.3390/sym16101303 - 3 Oct 2024

Cited by 2 | Viewed by 901

Abstract

In this paper, we consider a subclass of normalized analytic functions associated with the hyperbolic secant function. We compute the sharp bounds on third- and fourth-order Hermitian–Toeplitz determinants for functions in this class. Moreover, we determine the bounds on second- and third-order Hankel [...] Read more.

In this paper, we consider a subclass of normalized analytic functions associated with the hyperbolic secant function. We compute the sharp bounds on third- and fourth-order Hermitian–Toeplitz determinants for functions in this class. Moreover, we determine the bounds on second- and third-order Hankel determinants, as well as on the generalized Zalcman conjecture. We examine a Briot–Bouquet-type differential subordination involving the Bernardi integral operator. Finally, we obtain a univalent solution to the Briot–Bouquet differential equation, and discuss the majorization property for such function classes. Full article

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

13 pages, 259 KiB

Open AccessArticle

Starlikeness, Convexity, Close-to-Convexity, and Quasi-Convexity for Functions with Fixed Initial Coefficients

by Mohanad Kadhim Ahmed Alkarafi, Ali Ebadian and Saeid Shams

Axioms 2024, 13(10), 683; https://doi.org/10.3390/axioms13100683 - 2 Oct 2024

Viewed by 453

Abstract

In this paper, we employ the theory of differential subordination to establish a theorem that delineates certain sufficient conditions for starlikeness, convexity, close-to-convexity, and quasi-convexity in relation to functions with fixed initial coefficients. Furthermore, we introduce some results derived from these conditions. Building [...] Read more.

In this paper, we employ the theory of differential subordination to establish a theorem that delineates certain sufficient conditions for starlikeness, convexity, close-to-convexity, and quasi-convexity in relation to functions with fixed initial coefficients. Furthermore, we introduce some results derived from these conditions. Building upon this framework, we derive an extension of Nunokawa’s lemma for analytic functions with fixed initial coefficients. Full article

(This article belongs to the Section Mathematical Analysis)

17 pages, 301 KiB

Open AccessArticle

Sharp Results for a New Class of Analytic Functions Associated with the q-Differential Operator and the Symmetric Balloon-Shaped Domain

by Adeel Ahmad, Jianhua Gong, Akhter Rasheed, Saqib Hussain, Asad Ali and Zeinebou Cheikh

Symmetry 2024, 16(9), 1134; https://doi.org/10.3390/sym16091134 - 2 Sep 2024

Cited by 2 | Viewed by 726

Abstract

In our current study, we apply differential subordination and quantum calculus to introduce and investigate a new class of analytic functions associated with the q-differential operator and the symmetric balloon-shaped domain. We obtain sharp results concerning the Maclaurin coefficients the second and third-order [...] Read more.

In our current study, we apply differential subordination and quantum calculus to introduce and investigate a new class of analytic functions associated with the q-differential operator and the symmetric balloon-shaped domain. We obtain sharp results concerning the Maclaurin coefficients the second and third-order Hankel determinants, the Zalcman conjecture, and its generalized conjecture for this newly defined class of q-starlike functions with respect to symmetric points. Full article

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

9 pages, 778 KiB

Open AccessCommunication

Applications of Mittag–Leffler Functions on a Subclass of Meromorphic Functions Influenced by the Definition of a Non-Newtonian Derivative

by Daniel Breaz, Kadhavoor R. Karthikeyan and Gangadharan Murugusundaramoorthy

Fractal Fract. 2024, 8(9), 509; https://doi.org/10.3390/fractalfract8090509 - 29 Aug 2024

Cited by 1 | Viewed by 648

Abstract

In this paper, we defined a new family of meromorphic functions whose analytic characterization was motivated by the definition of the multiplicative derivative. Replacing the ordinary derivative with a multiplicative derivative in the subclass of starlike meromorphic functions made the class redundant; thus, [...] Read more.

In this paper, we defined a new family of meromorphic functions whose analytic characterization was motivated by the definition of the multiplicative derivative. Replacing the ordinary derivative with a multiplicative derivative in the subclass of starlike meromorphic functions made the class redundant; thus, major deviation or adaptation was required in defining a class of meromorphic functions influenced by the multiplicative derivative. In addition, we redefined the subclass of meromorphic functions analogous to the class of the functions with respect to symmetric points. Initial coefficient estimates and Fekete–Szegö inequalities were obtained for the defined function classes. Some examples along with graphs have been used to establish the inclusion and closure properties. Full article

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

► Show Figures

Figure 1

17 pages, 293 KiB

Open AccessArticle

Starlike Functions in the Space of Meromorphic Harmonic Functions

by Jacek Dziok

Symmetry 2024, 16(9), 1112; https://doi.org/10.3390/sym16091112 - 27 Aug 2024

Viewed by 711

Abstract

The Geometric Theory of Analytic Functions was initially developed for the space of functions that are analytic in the unit disk. The convexity and starlikeness of functions are the first geometric ideas considered in this theory. We can notice a symmetry between the [...] Read more.

The Geometric Theory of Analytic Functions was initially developed for the space of functions that are analytic in the unit disk. The convexity and starlikeness of functions are the first geometric ideas considered in this theory. We can notice a symmetry between the subjects considered in the space of analytic functions and those in the space of harmonic functions. In the presented paper, we consider the starlikeness of functions in the space of meromorphic harmonic functions. Full article

13 pages, 284 KiB

Open AccessArticle

On the Fekete–Szegö Problem for Certain Classes of (γ,δ)-Starlike and (γ,δ)-Convex Functions Related to Quasi-Subordinations

by Norah Saud Almutairi, Awatef Shahen, Adriana Cătaş and Hanan Darwish

Symmetry 2024, 16(8), 1043; https://doi.org/10.3390/sym16081043 - 14 Aug 2024

Viewed by 823

Abstract

In the present paper, we propose new generalized classes of (p,q)-starlike and (p,q)-convex functions. These classes are introduced by making use of a (p,q)-derivative operator. There are established Fekete–Szegö estimates

| a_{3} - μ a_{2}^{2} |

for functions belonging to [...] Read more.

In the present paper, we propose new generalized classes of (p,q)-starlike and (p,q)-convex functions. These classes are introduced by making use of a (p,q)-derivative operator. There are established Fekete–Szegö estimates

| a_{3} - μ a_{2}^{2} |

for functions belonging to the newly introduced subclasses. Certain subclasses of analytic univalent functions associated with quasi-subordination are defined. Full article

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

8 pages, 242 KiB

Open AccessArticle

Toeplitz Matrices for a Class of Bazilevič Functions and the λ-Pseudo-Starlike Functions

by Abbas Kareem Wanas, Salam Abdulhussein Sehen and Ágnes Orsolya Páll-Szabó

Axioms 2024, 13(8), 521; https://doi.org/10.3390/axioms13080521 - 2 Aug 2024

Viewed by 501

Abstract

In the present paper, we define and study a new family of holomorphic functions which involve the Bazilevič functions and the

λ

-pseudo-starlike functions. We establish coefficient estimates for the first four determinants of the symmetric Toeplitz matrices

T_{2} (2)

[...] Read more.

In the present paper, we define and study a new family of holomorphic functions which involve the Bazilevič functions and the

λ

-pseudo-starlike functions. We establish coefficient estimates for the first four determinants of the symmetric Toeplitz matrices

T_{2} (2)

,

T_{2} (3)

,

T_{3} (1)

and

T_{3} (2)

for the functions in this family. Further, we investigate several special cases and consequences of our results. Full article

(This article belongs to the Special Issue Advances in Geometric Function Theory and Related Topics)

14 pages, 367 KiB

Open AccessArticle

Subclasses of Bi-Univalent Functions Connected with Caputo-Type Fractional Derivatives Based upon Lucas Polynomial

by Kholood M. Alsager, Gangadharan Murugusundaramoorthy, Daniel Breaz and Sheza M. El-Deeb

Fractal Fract. 2024, 8(8), 452; https://doi.org/10.3390/fractalfract8080452 - 31 Jul 2024

Viewed by 774

Abstract

In the current paper, we introduce new subclasses of analytic and bi-univalent functions involving Caputo-type fractional derivatives subordinating to the Lucas polynomial. Furthermore, we find non-sharp estimates on the first two Taylor–Maclaurin coefficients

|a_{2}|

and

|a_{3}|

for functions in these subclasses. [...] Read more.

In the current paper, we introduce new subclasses of analytic and bi-univalent functions involving Caputo-type fractional derivatives subordinating to the Lucas polynomial. Furthermore, we find non-sharp estimates on the first two Taylor–Maclaurin coefficients

|a_{2}|

and

|a_{3}|

for functions in these subclasses. Using the values of

|a_{2}|

and

|a_{3}|

, we determined Fekete–Szegő inequality for functions in these subclasses. Moreover, we pointed out some more subclasses by fixing the parameters involved in Lucas polynomial and stated the estimates and Fekete–Szegő inequality results without proof. Full article

(This article belongs to the Special Issue Numerical Solutions of Caputo-Type Fractional Differential Equations and Derivatives)

24 pages, 349 KiB

Open AccessArticle

Sharp Coefficient Estimates for Analytic Functions Associated with Lemniscate of Bernoulli

by Rubab Nawaz, Rabia Fayyaz, Daniel Breaz and Luminiţa-Ioana Cotîrlă

Mathematics 2024, 12(15), 2309; https://doi.org/10.3390/math12152309 - 23 Jul 2024

Viewed by 555

Abstract

The main purpose of this work is to study the third Hankel determinant for classes of Bernoulli lemniscate-related functions by introducing new subclasses of star-like functions represented by

S_{L λ}^{*}

and

R L_{λ}

. In many geometric and physical applications [...] Read more.

The main purpose of this work is to study the third Hankel determinant for classes of Bernoulli lemniscate-related functions by introducing new subclasses of star-like functions represented by

S_{L λ}^{*}

and

R L_{λ}

. In many geometric and physical applications of complex analysis, estimating sharp bounds for problems involving the coefficients of univalent functions is very important because these coefficients describe the fundamental properties of conformal maps. In the present study, we defined sharp bounds for function-coefficient problems belonging to the family of

S_{L λ}^{*}

and

R L_{λ}

. Most of the computed bounds are sharp. This study will encourage further research on the sharp bounds of analytical functions related to new image domains. Full article

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

14 pages, 322 KiB

Open AccessArticle

Coefficient Functionals of Sakaguchi-Type Starlike Functions Involving Caputo-Type Fractional Derivatives Subordinated to the Three-Leaf Function

by Kholood M. Alsager, Sheza M. El-Deeb, Gangadharan Murugusundaramoorthy and Daniel Breaz

Mathematics 2024, 12(14), 2273; https://doi.org/10.3390/math12142273 - 20 Jul 2024

Viewed by 606

Abstract

A challenging part of studying geometric function theory is figuring out the sharp boundaries for coefficient-related problems that crop up in the Taylor–Maclaurin series of univalent functions. Using Caputo-type fractional derivatives to define the families of Sakaguchi-type starlike functions with respect to symmetric [...] Read more.

A challenging part of studying geometric function theory is figuring out the sharp boundaries for coefficient-related problems that crop up in the Taylor–Maclaurin series of univalent functions. Using Caputo-type fractional derivatives to define the families of Sakaguchi-type starlike functions with respect to symmetric points, this article aims to investigate the first three initial coefficient estimates, the bounds for various problems such as Fekete–Szegő inequality, and the Zalcman inequalities, by subordinating to the function of the three leaves domain. Fekete–Szegő-type inequalities and initial coefficients for functions of the form

H^{- 1}

and

\frac{ζ}{H (ζ)}

and

\frac{1}{2} log (\frac{H (ζ)}{ζ})

connected to the three leaves functions are also discussed. Full article

(This article belongs to the Special Issue Geometric Function Theory and Applications―Festschrift for Grigore-Stefan Salagean's 75th Birthday)

Search Results (284)

Further Information

Guidelines

MDPI Initiatives

Follow MDPI

Saved Queries

Search Filter Reset All

Years

Feature Papers

Subjects

Journals

Article Types

Countries / Regions

Search Results (284)

Further Information

Guidelines

MDPI Initiatives

Follow MDPI