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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

Viewed by 330

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)

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13 pages, 341 KiB

Open AccessArticle

Applications of Caputo-Type Fractional Derivatives for Subclasses of Bi-Univalent Functions with Bounded Boundary Rotation

by Kholood M. Alsager, Gangadharan Murugusundaramoorthy, Adriana Catas and Sheza M. El-Deeb

Fractal Fract. 2024, 8(9), 501; https://doi.org/10.3390/fractalfract8090501 - 26 Aug 2024

Viewed by 352

Abstract

In this article, for the first time by using Caputo-type fractional derivatives, we introduce three new subclasses of bi-univalent functions associated with bounded boundary rotation in an open unit disk to obtain non-sharp estimates of the first two Taylor–Maclaurin coefficients, [...] Read more.

In this article, for the first time by using Caputo-type fractional derivatives, we introduce three new subclasses of bi-univalent functions associated with bounded boundary rotation in an open unit disk to obtain non-sharp estimates of the first two Taylor–Maclaurin coefficients,

| a_{2} |

and

| a_{3} |

. Furthermore, the famous Fekete–Szegö inequality is obtained for the newly defined subclasses of bi-univalent functions. Several consequences of our results are pointed out which are new and not yet discussed in association with bounded boundary rotation. Some improved results when compared with those already available in the literature are also stated as corollaries. Full article

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

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 662

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)

13 pages, 266 KiB

Open AccessArticle

A General and Comprehensive Subclass of Univalent Functions Associated with Certain Geometric Functions

by Tariq Al-Hawary, Basem Frasin and Ibtisam Aldawish

Symmetry 2024, 16(8), 982; https://doi.org/10.3390/sym16080982 - 2 Aug 2024

Viewed by 703

Abstract

In this paper, taking into account the intriguing recent results of Rabotnov functions, Poisson functions, Bessel functions and Wright functions, we consider a new comprehensive subclass

O_{μ} (Δ_{1}, Δ_{2}, Δ_{3}, Δ_{4})

of univalent [...] Read more.

In this paper, taking into account the intriguing recent results of Rabotnov functions, Poisson functions, Bessel functions and Wright functions, we consider a new comprehensive subclass

O_{μ} (Δ_{1}, Δ_{2}, Δ_{3}, Δ_{4})

of univalent functions defined in the unit disk

Λ = {τ \in C : |τ| < 1}

. More specifically, we investigate some sufficient conditions for Rabotnov functions, Poisson functions, Bessel functions and Wright functions to be in this subclass. Some corollaries of our main results are given. The novelty and the advantage of this research could inspire researchers of further studies to find new sufficient conditions to be in the subclass

O_{μ} (Δ_{1}, Δ_{2}, Δ_{3}, Δ_{4})

not only for the aforementioned special functions but for different types of special functions, especially for hypergeometric functions, Dini functions, Sturve functions and others. 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 323

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 567

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)

16 pages, 277 KiB

Open AccessArticle

Initial Coefficient Bounds for Certain New Subclasses of Bi-Univalent Functions Involving Mittag–Leffler Function with Bounded Boundary Rotation

by Ibtisam Aldawish, Prathviraj Sharma, Sheza M. El-Deeb, Mariam R. Almutiri and Srikandan Sivasubramanian

Symmetry 2024, 16(8), 971; https://doi.org/10.3390/sym16080971 - 31 Jul 2024

Viewed by 690

Abstract

By using the Mittag–Leffler function associated with functions of bounded boundary rotation, the authors introduce a few new subclasses of bi-univalent functions involving the Mittag–Leffler function with bounded boundary rotation in the open unit disk

D .

For these new classes, the authors [...] Read more.

By using the Mittag–Leffler function associated with functions of bounded boundary rotation, the authors introduce a few new subclasses of bi-univalent functions involving the Mittag–Leffler function with bounded boundary rotation in the open unit disk

D .

For these new classes, the authors establish initial coefficient bounds of

| a_{2} |

and

| a_{3} |

. Furthermore, the famous Fekete–Szegö coefficient inequality is also obtained for these new classes of functions. Full article

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

14 pages, 280 KiB

Open AccessArticle

Coefficient Estimates for New Subclasses of Bi-Univalent Functions with Bounded Boundary Rotation by Using Faber Polynomial Technique

by Huo Tang, Prathviraj Sharma and Srikandan Sivasubramanian

Axioms 2024, 13(8), 509; https://doi.org/10.3390/axioms13080509 - 28 Jul 2024

Viewed by 371

Abstract

In this article, the authors use the Faber polynomial expansions to find the general coefficient estimates for a few new subclasses of bi-univalent functions with bounded boundary rotation and bounded radius rotation. Some of the results improve the existing coefficient bounds in the [...] Read more.

In this article, the authors use the Faber polynomial expansions to find the general coefficient estimates for a few new subclasses of bi-univalent functions with bounded boundary rotation and bounded radius rotation. Some of the results improve the existing coefficient bounds in the literature. Full article

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

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 414

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 480

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)

14 pages, 284 KiB

Open AccessArticle

Certain Geometric Study Involving the Barnes–Mittag-Leffler Function

by Abdulaziz Alenazi and Khaled Mehrez

Fractal Fract. 2024, 8(7), 400; https://doi.org/10.3390/fractalfract8070400 - 4 Jul 2024

Viewed by 708

Abstract

The main purpose of this paper is to study certain geometric properties of a class of analytic functions involving the Barnes–Mittag-Leffler function. The main mathematical tools are the monotonicity patterns of some class of functions associated with the gamma and digamma functions. Furthermore, [...] Read more.

The main purpose of this paper is to study certain geometric properties of a class of analytic functions involving the Barnes–Mittag-Leffler function. The main mathematical tools are the monotonicity patterns of some class of functions associated with the gamma and digamma functions. Furthermore, some consequences and examples are presented. Full article

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

10 pages, 263 KiB

Open AccessArticle

Convolution Properties of Meromorphic

P

-Valent Functions with Coefficients of Alternating Type Defined Using

q

-Difference Operator

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

Mathematics 2024, 12(13), 2104; https://doi.org/10.3390/math12132104 - 4 Jul 2024

Viewed by 466

Abstract

Certain characteristics of univalent functions with negative coefficients of the form

f (z) = z - \sum_{n = 1}^{\infty} a_{2 n} z^{2 n}, a_{2 n} > 0

have been studied by Silverman and Berman. Pokley, [...] Read more.

Certain characteristics of univalent functions with negative coefficients of the form

f (z) = z - \sum_{n = 1}^{\infty} a_{2 n} z^{2 n}, a_{2 n} > 0

have been studied by Silverman and Berman. Pokley, Patil and Shrigan have discovered some insights into the Hadamard product of

P

-valent functions with negative coefficients. S. M. Khairnar and Meena More have obtained coefficient limits and convolution results for univalent functions lacking a alternating type coefficient. In this paper, using the

q

-Difference operator, we developed the a subclass of meromorphically

P

-valent functions with alternating coefficients. Additionally, we obtained multivalent function convolution results and coefficient limits. Full article

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

0 pages, 558 KiB

Open AccessArticle

On Ozaki Close-to-Convex Functions with Bounded Boundary Rotation

by Prathviraj Sharma, Asma Alharbi, Srikandan Sivasubramanian and Sheza M. El-Deeb

Symmetry 2024, 16(7), 839; https://doi.org/10.3390/sym16070839 - 3 Jul 2024

Cited by 1 | Viewed by 871

Abstract

In the present investigation, we introduce a new subclass of univalent functions

F (u, λ)

and a subclass of bi-univalent function

F_{o, Σ} (u, λ)

with bounded boundary and bounded radius rotation. Some examples of [...] Read more.

In the present investigation, we introduce a new subclass of univalent functions

F (u, λ)

and a subclass of bi-univalent function

F_{o, Σ} (u, λ)

with bounded boundary and bounded radius rotation. Some examples of the functions belonging to the classes

F (u, λ)

are also derived. For these new classes, the authors derive many interesting relations between these classes and the existing familiar subclasses in the literature. Furthermore, the authors establish new coefficient estimates for these classes. Apart from the above, the first two initial coefficient bounds for the class

F_{o, Σ} (u, λ)

are established. Full article

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

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16 pages, 331 KiB

Open AccessArticle

Some Classes of Bazilevič-Type Close-to-Convex Functions Involving a New Derivative Operator

by Pishtiwan Othman Sabir, Alina Alb Lupas, Sipal Saeed Khalil, Pshtiwan Othman Mohammed and Mohamed Abdelwahed

Symmetry 2024, 16(7), 836; https://doi.org/10.3390/sym16070836 - 3 Jul 2024

Viewed by 768

Abstract

In the present paper, we are merging two interesting and well-known classes, namely those of Bazilevič and close-to-convex functions associated with a new derivative operator. We derive coefficient estimates for this broad category of analytic, univalent and bi-univalent functions and draw attention to [...] Read more.

In the present paper, we are merging two interesting and well-known classes, namely those of Bazilevič and close-to-convex functions associated with a new derivative operator. We derive coefficient estimates for this broad category of analytic, univalent and bi-univalent functions and draw attention to the Fekete–Szegö inequalities relevant to functions defined within the open unit disk. Additionally, we identify several specific special cases of our results by specializing the parameters. Full article

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

12 pages, 1495 KiB

Open AccessArticle

Geometric Features of the Hurwitz–Lerch Zeta Type Function Based on Differential Subordination Method

by Faten F. Abdulnabi, Hiba F. Al-Janaby, Firas Ghanim and Alina Alb Lupaș

Symmetry 2024, 16(7), 784; https://doi.org/10.3390/sym16070784 - 21 Jun 2024

Viewed by 878

Abstract

The interest in special complex functions and their wide-ranging implementations in geometric function theory (GFT) has developed tremendously. Recently, subordination theory has been instrumentally employed for special functions to explore their geometric properties. In this effort, by using a convolutional structure, we combine [...] Read more.

The interest in special complex functions and their wide-ranging implementations in geometric function theory (GFT) has developed tremendously. Recently, subordination theory has been instrumentally employed for special functions to explore their geometric properties. In this effort, by using a convolutional structure, we combine the geometric series, logarithm, and Hurwitz–Lerch zeta functions to formulate a new special function, namely, the logarithm-Hurwitz–Lerch zeta function (LHL-Z function). This investigation then contributes to the study of the LHL-Z function in terms of the geometric theory of holomorphic functions, based on the differential subordination methodology, to discuss and determine the univalence and convexity conditions of the LHL-Z function. Moreover, there are other subordination and superordination connections that may be visually represented using geometric methods. Functions often exhibit symmetry when subjected to conformal mappings. The investigation of the symmetries of these mappings may provide a clearer understanding of how subordination and superordination with the Hurwitz–Lerch zeta function behave under different transformations. Full article

(This article belongs to the Special Issue Symmetries of Difference Equations, Special Functions and Orthogonal Polynomials II)

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