Mathematics

Research

22 pages, 341 KiB

Open AccessArticle

Variable Besov–Morrey Spaces Associated with Operators

by Khedoudj Saibi

Mathematics 2023, 11(9), 2038; https://doi.org/10.3390/math11092038 - 25 Apr 2023

Cited by 1 | Viewed by 980

Abstract

Let

(X, d, μ)

be a space of homogenous type and L be a non-negative self-adjoint operator on

L^{2} (X)

with heat kernels satisfying Gaussian upper bounds. In this paper, we introduce the variable Besov–Morrey space [...] Read more.

Let

(X, d, μ)

be a space of homogenous type and L be a non-negative self-adjoint operator on

L^{2} (X)

with heat kernels satisfying Gaussian upper bounds. In this paper, we introduce the variable Besov–Morrey space associated with the operator L and prove that this space can be characterized via the Peetre maximal functions. Then, we establish its atomic decomposition. Full article

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

10 pages, 284 KiB

Open AccessArticle

On Generalizations of the Close-to-Convex Functions Associated with

q

-Srivastava–Attiya Operator

by Daniel Breaz, Abdullah A. Alahmari, Luminiţa-Ioana Cotîrlă and Shujaat Ali Shah

Mathematics 2023, 11(9), 2022; https://doi.org/10.3390/math11092022 - 24 Apr 2023

Cited by 5 | Viewed by 1400

Abstract

The study of the

q

-analogue of the classical results of geometric function theory is currently of great interest to scholars. In this article, we define generalized classes of close-to-convex functions and quasi-convex functions with the help of the

q

-difference operator. Moreover, [...] Read more.

The study of the

q

-analogue of the classical results of geometric function theory is currently of great interest to scholars. In this article, we define generalized classes of close-to-convex functions and quasi-convex functions with the help of the

q

-difference operator. Moreover, by using the

q

-analogues of a certain family of linear operators, the classes

K_{q, b}^{s} (h)

,

{\tilde{K}}_{q, s}^{b} (h)

,

Q_{q, b}^{s} (h)

, and

{\tilde{Q}}_{q, s}^{b} (h)

are introduced. Several interesting inclusion relationships between these newly defined classes are discussed, and the invariance of these classes under the

q

-Bernadi integral operator was examined. Furthermore, some special cases and useful consequences of these investigations were taken into consideration. Full article

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

15 pages, 315 KiB

Open AccessArticle

New Applications of Faber Polynomial Expansion for Analytical Bi-Close-to-Convex Functions Defined by Using q-Calculus

by Ridong Wang, Manoj Singh, Shahid Khan, Huo Tang, Mohammad Faisal Khan and Mustafa Kamal

Mathematics 2023, 11(5), 1217; https://doi.org/10.3390/math11051217 - 1 Mar 2023

Cited by 3 | Viewed by 1385

Abstract

In this investigation, the q-difference operator and the Sălăgean q-differential operator are utilized to establish novel subclasses of analytical bi-close-to-convex functions. We determine the general Taylor–Maclaurin coefficient of the functions in this class using the Faber polynomial method. We demonstrate the [...] Read more.

In this investigation, the q-difference operator and the Sălăgean q-differential operator are utilized to establish novel subclasses of analytical bi-close-to-convex functions. We determine the general Taylor–Maclaurin coefficient of the functions in this class using the Faber polynomial method. We demonstrate the unpredictable behaviour of initial coefficients

|a_{2}|

,

|a_{3}|

and investigate the Fekete–Szegő problem

|a_{3} - a_{2}^{2}|

for the subclasses of bi-close-to-convex functions. To highlight the connections between existing knowledge and new research, certain known and unknown corollaries are also highlighted. Full article

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

12 pages, 309 KiB

Open AccessArticle

Third Hankel Determinant for a Subfamily of Holomorphic Functions Related with Lemniscate of Bernoulli

by Halit Orhan, Murat Çağlar and Luminiţa-Ioana Cotîrlă

Mathematics 2023, 11(5), 1147; https://doi.org/10.3390/math11051147 - 25 Feb 2023

Cited by 6 | Viewed by 1269

Abstract

The main goal of this investigation is to obtain sharp upper bounds for Fekete-Szegö functional and the third Hankel determinant for a certain subclass

{SL}^{*} (u, v, α)

of holomorphic functions defined by the Carlson-Shaffer operator in the unit disk. [...] Read more.

The main goal of this investigation is to obtain sharp upper bounds for Fekete-Szegö functional and the third Hankel determinant for a certain subclass

{SL}^{*} (u, v, α)

of holomorphic functions defined by the Carlson-Shaffer operator in the unit disk. Finally, for some special values of parameters, several corollaries were presented. Full article

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

17 pages, 308 KiB

Open AccessArticle

Some Applications of Analytic Functions Associated with q-Fractional Operator

by Nazar Khan, Shahid Khan, Qin Xin, Fairouz Tchier, Sarfraz Nawaz Malik and Umer Javed

Mathematics 2023, 11(4), 930; https://doi.org/10.3390/math11040930 - 12 Feb 2023

Cited by 6 | Viewed by 1606

Abstract

This paper introduces a new fractional operator by using the concepts of fractional q-calculus and q-Mittag-Leffler functions. With this fractional operator, Janowski functions are generalized and studied regarding their certain geometric characteristics. It also establishes the solution of the complex Briot–Bouquet [...] Read more.

This paper introduces a new fractional operator by using the concepts of fractional q-calculus and q-Mittag-Leffler functions. With this fractional operator, Janowski functions are generalized and studied regarding their certain geometric characteristics. It also establishes the solution of the complex Briot–Bouquet differential equation by using the newly defined operator. Full article

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

9 pages, 290 KiB

Open AccessArticle

On Certain Classes of Multivalent Analytic Functions Defined with Higher-Order Derivatives

by Abdel Moneim Y. Lashin and Fatma Z. El-Emam

Mathematics 2023, 11(1), 83; https://doi.org/10.3390/math11010083 - 26 Dec 2022

Cited by 3 | Viewed by 1270

Abstract

This paper examines two subclasses of multivalent analytic functions defined with higher-order derivatives. These classes of functions are generalizations of several known subclasses that have been studied in separate works. Moreover, we find several interesting results for functions in these classes, including subordination [...] Read more.

This paper examines two subclasses of multivalent analytic functions defined with higher-order derivatives. These classes of functions are generalizations of several known subclasses that have been studied in separate works. Moreover, we find several interesting results for functions in these classes, including subordination results, containment relations, and integral preserving properties. Full article

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

11 pages, 365 KiB

Open AccessArticle

Necessary and Sufficient Conditions for Normalized Wright Functions to Be in Certain Classes of Analytic Functions

by Tariq Al-Hawary, Ibtisam Aldawish, Basem Aref Frasin, Osama Alkam and Feras Yousef

Mathematics 2022, 10(24), 4693; https://doi.org/10.3390/math10244693 - 10 Dec 2022

Cited by 6 | Viewed by 1241

Abstract

In this paper, the function classes

{SP}_{p} (σ, ν)

and

UCSP (σ, ν)

are investigated for the normalized Wright functions. More precisely, several sufficient and necessary conditions are provided so that the aforementioned functions are in [...] Read more.

In this paper, the function classes

{SP}_{p} (σ, ν)

and

UCSP (σ, ν)

are investigated for the normalized Wright functions. More precisely, several sufficient and necessary conditions are provided so that the aforementioned functions are in these classes. Furthermore, several corollaries will follow from our results. Full article

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

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20 pages, 331 KiB

Open AccessArticle

Differential Subordination and Differential Superordination for Classes of Admissible Multivalent Functions Associated with a Linear Operator

by Ekram E. Ali, Hari M. Srivastava, Rabha M. El-Ashwah and Abeer M. Albalahi

Mathematics 2022, 10(24), 4690; https://doi.org/10.3390/math10244690 - 10 Dec 2022

Cited by 6 | Viewed by 1168

Abstract

In this paper, we first introduce a linear integral operator [...] Read more.

In this paper, we first introduce a linear integral operator

ℑ_{p} (a, c, μ) (μ > 0; a, c \in R; c > a > - μ p; p \in N^{+} : = {1, 2, 3, \dots}),

which is somewhat related to a rather specialized form of the Riemann–Liouville fractional integral operator and its varied form known as the Erdélyi–Kober fractional integral operator. We then derive some differential subordination and differential superordination results for analytic and multivalent functions in the open unit disk

U,

which are associated with the above-mentioned linear integral operator

ℑ_{p} (a, c, μ)

. The results presented here are obtained by investigating appropriate classes of admissible functions. We also obtain some Sandwich-type results. Full article

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

26 pages, 2506 KiB

Open AccessArticle

Radii of Starlikeness of Ratios of Analytic Functions with Fixed Second Coefficients

by Shalini Rana, Om P. Ahuja and Naveen Kumar Jain

Mathematics 2022, 10(23), 4428; https://doi.org/10.3390/math10234428 - 24 Nov 2022

Cited by 1 | Viewed by 3399

Abstract

We introduce three classes of analytic functions with fixed second coefficients that are defined using the class

P

of analytic functions with positive real parts. The objective of this paper is to determine the radii such that the three classes are contained in [...] Read more.

We introduce three classes of analytic functions with fixed second coefficients that are defined using the class

P

of analytic functions with positive real parts. The objective of this paper is to determine the radii such that the three classes are contained in various subclasses of starlike functions. The radii estimated in the present investigation are better than the radii obtained earlier. Furthermore, connections with previously known results are shown. Full article

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

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25 pages, 378 KiB

Open AccessArticle

Novel Formulae of Certain Generalized Jacobi Polynomials

by Waleed Mohamed Abd-Elhameed

Mathematics 2022, 10(22), 4237; https://doi.org/10.3390/math10224237 - 13 Nov 2022

Cited by 6 | Viewed by 1519

Abstract

The main goal of this article is to investigate theoretically a kind of orthogonal polynomials, namely, generalized Jacobi polynomials (

G J P s

). These polynomials can be expressed as certain combinations of Legendre polynomials. Some basic formulas of these polynomials such [...] Read more.

The main goal of this article is to investigate theoretically a kind of orthogonal polynomials, namely, generalized Jacobi polynomials (

G J P s

). These polynomials can be expressed as certain combinations of Legendre polynomials. Some basic formulas of these polynomials such as the power form representation and inversion formula of these polynomials are first introduced, and after that, some interesting formulas concerned with these polynomials are established. The formula of the derivatives of the moments of these polynomials is developed. As special cases of this formula, the moment and high-order derivative formulas of the

G J P s

are deduced. New expressions for the high-order derivatives of the

G J P s

, but in terms of different symmetric and non-symmetric polynomials, are also established. These expressions lead to some interesting connection formulas between the

G J P s

and some various polynomials. Full article

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

15 pages, 315 KiB

Open AccessArticle

Convexity, Starlikeness, and Prestarlikeness of Wright Functions

by Dong Liu, Muhey U Din, Mohsan Raza, Sarfraz Nawaz Malik and Huo Tang

Mathematics 2022, 10(20), 3858; https://doi.org/10.3390/math10203858 - 18 Oct 2022

Cited by 1 | Viewed by 1213

Abstract

This article deals with the normalized Wright function and its geometric properties. In particular, we find sufficiency criteria for close-to-convexity with respect to starlike function

\frac{ς}{1 - ς^{2}} .

We also find conditions such that the normalized Wright function is starlike. [...] Read more.

This article deals with the normalized Wright function and its geometric properties. In particular, we find sufficiency criteria for close-to-convexity with respect to starlike function

\frac{ς}{1 - ς^{2}} .

We also find conditions such that the normalized Wright function is starlike. The convexity along the imaginary axis and starlikeness of a certain order is also a part of our discussion. Moreover, we study the bounded turning of the partial sums and prestarlikeness of this function. We use positivity techniques to obtain these results. Full article

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

18 pages, 319 KiB

Open AccessArticle

Some Subclasses of Spirallike Multivalent Functions Associated with a Differential Operator

by Ekram Elsayed Ali, Mohamed Kamal Aouf, Rabha Mohamed El-Ashwah and Teodor Bulboacă

Mathematics 2022, 10(17), 3064; https://doi.org/10.3390/math10173064 - 25 Aug 2022

Viewed by 1230

Abstract

In this paper we study convolution properties of spirallike multivalent functions defined by using a differential operator and higher order derivatives. Using convolution product relations we determine necessary and sufficient conditions for multivalent functions to belong to these classes, and our results generalized [...] Read more.

In this paper we study convolution properties of spirallike multivalent functions defined by using a differential operator and higher order derivatives. Using convolution product relations we determine necessary and sufficient conditions for multivalent functions to belong to these classes, and our results generalized many previous results obtained by different authors. We obtain convolution and inclusion properties for new subclasses of multivalent functions defined by using the Dziok-Srivatava operator. Moreover, using a result connected with the Briot-Bouquet differential subordination, we obtain an inclusion relation between some of these classes of functions. Full article

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

16 pages, 693 KiB

Open AccessArticle

Subordinations and Norm Estimates for Functions Associated with Ma-Minda Subclasses

by Aaisha Farzana Habibullah, Muthusamy Palani Jeyaraman and Teodor Bulboacă

Mathematics 2022, 10(16), 2879; https://doi.org/10.3390/math10162879 - 11 Aug 2022

Viewed by 1285

Abstract

For a function p analytic in the open unit disc and satisfying

p (0) = 1

, we prove certain subordination implications of the first order differential subordination

1 + z p^{'} (z) ≺ 1 + M z

[...] Read more.

For a function p analytic in the open unit disc and satisfying

p (0) = 1

, we prove certain subordination implications of the first order differential subordination

1 + z p^{'} (z) ≺ 1 + M z

, which provides sufficient conditions for a function to belong to various subclasses of Ma-Minda starlike functions. The pre-Schwarzian norm estimate and inclusion criteria for certain subclasses of analytic function are also obtained. Additionally, using Gronwall’s inequality we give a sufficient condition for a normalized function to belong to a class of functions with bounded arguments that extends the class of strongly

α

-Bazilevič functions of order

γ

studied by Gao in 1996. Full article

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

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8 pages, 259 KiB

Open AccessEditor’s ChoiceArticle

On the Extremality of Harmonic Beltrami Coefficients

by Samuel L. Krushkal

Mathematics 2022, 10(14), 2460; https://doi.org/10.3390/math10142460 - 14 Jul 2022

Cited by 2 | Viewed by 1143

Abstract

We prove a general theorem, which provides a broad collection of univalent functions with equal Grunsky and Teichmüller norms and thereby the Fredholm eigenvalues and the reflection coefficients of associated quasicircles. It concerns an important problem to establish the exact or approximate values [...] Read more.

We prove a general theorem, which provides a broad collection of univalent functions with equal Grunsky and Teichmüller norms and thereby the Fredholm eigenvalues and the reflection coefficients of associated quasicircles. It concerns an important problem to establish the exact or approximate values of basic quasiinvariant functionals of Jordan curves, which is crucial in applications and in the numerical aspect of quasiconformal analysis. Full article

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

17 pages, 859 KiB

Open AccessArticle

Hölder Inequalities for a Generalized Subclass of Univalent Functions Involving Borel Distributions

by Gangadharan Murugusundaramoorthy and Luminiţa-Ioana Cotîrlǎ

Mathematics 2022, 10(14), 2430; https://doi.org/10.3390/math10142430 - 12 Jul 2022

Viewed by 1282

Abstract

In this article, by making use of the Borel distributions series, we introduce a new family of normalized holomorphic functions in the open unit disk and investigate necessary and sufficient conditions for functions f to be in this new class. Furthermore, results on [...] Read more.

In this article, by making use of the Borel distributions series, we introduce a new family of normalized holomorphic functions in the open unit disk and investigate necessary and sufficient conditions for functions f to be in this new class. Furthermore, results on the modified Hadamard product, Hölder inequalities, and closure properties under integral transforms and subordination results are discussed in detail. Full article

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

18 pages, 322 KiB

Open AccessArticle

On Certain Generalizations of Rational and Irrational Equivariant Functions

by Isra Al-Shbeil, Afis Saliu, Abbas Kareem Wanas and Adriana Cătaş

Mathematics 2022, 10(13), 2247; https://doi.org/10.3390/math10132247 - 27 Jun 2022

Viewed by 1547

Abstract

In this paper, we address the case of a particular class of function referred to as the rational equivariant functions. We investigate which elliptic zeta functions arising from integrals of power of ℘, where ℘ is the Weierstrass ℘-function attached to [...] Read more.

In this paper, we address the case of a particular class of function referred to as the rational equivariant functions. We investigate which elliptic zeta functions arising from integrals of power of ℘, where ℘ is the Weierstrass ℘-function attached to a rank two lattice of

C

, yield rational equivariant functions. Our concern in this survey is to provide certain examples of rational equivariant functions. In this sense, we establish a criterion in order to determine the rationality of equivariant functions derived from ratios of modular functions of low weight. Modular forms play an important role in number theory and many areas of mathematics and physics. Full article

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

9 pages, 270 KiB

Open AccessArticle

Hadamard Product of Certain Multivalent Analytic Functions with Positive Real Parts

by Abdel Moneim Y. Lashin and Mohamed K. Aouf

Mathematics 2022, 10(9), 1506; https://doi.org/10.3390/math10091506 - 1 May 2022

Cited by 1 | Viewed by 1476

Abstract

This paper aims to provide sufficient conditions for starlikeness and convexity of Hadamard product (convolution) of certain multivalent analytic functions with positive real parts. Moreover, the starlikeness conditions for a certain integral operator and other convolution results are also considered. Full article

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

14 pages, 352 KiB

Open AccessArticle

Fekete–Szegö Functional Problem for a Special Family of m-Fold Symmetric Bi-Univalent Functions

by Sondekola Rudra Swamy, Basem Aref Frasin and Ibtisam Aldawish

Mathematics 2022, 10(7), 1165; https://doi.org/10.3390/math10071165 - 3 Apr 2022

Cited by 13 | Viewed by 1810

Abstract

In the current work, we introduce a special family of the function family of analytic and m-fold symmetric bi-univalent functions and obtain estimates of the Taylor–Maclaurin coefficients

| d_{m + 1} |

and

| d_{2 m + 1} |

for functions [...] Read more.

In the current work, we introduce a special family of the function family of analytic and m-fold symmetric bi-univalent functions and obtain estimates of the Taylor–Maclaurin coefficients

| d_{m + 1} |

and

| d_{2 m + 1} |

for functions in the special family. For

δ

a real number, Fekete–Szegö functional

| d_{2 m + 1} - δ d_{m + 1}^{2} |

for functions in the special family is also estimated. We indicate several cases of the defined family and connections to existing results are also discussed. Full article

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

12 pages, 281 KiB

Open AccessArticle

A Subclass of q-Starlike Functions Defined by Using a Symmetric q-Derivative Operator and Related with Generalized Symmetric Conic Domains

by Shahid Khan, Saqib Hussain, Muhammad Naeem, Maslina Darus and Akhter Rasheed

Mathematics 2021, 9(9), 917; https://doi.org/10.3390/math9090917 - 21 Apr 2021

Cited by 20 | Viewed by 2150

Abstract

In this paper, the concepts of symmetric q-calculus and conic regions are used to define a new domain

\tilde{Ω_{k, q, α}},

which generalizes the symmetric conic domains. By using the domain [...] Read more.

In this paper, the concepts of symmetric q-calculus and conic regions are used to define a new domain

\tilde{Ω_{k, q, α}},

which generalizes the symmetric conic domains. By using the domain

\tilde{Ω_{k, q, α}}

, we define a new subclass of analytic and q-starlike functions in the open unit disk U and establish some new results for functions of this class. We also investigate a number of useful properties and characteristics of this subclass, such as coefficients estimates, structural formulas, distortion inequalities, necessary and sufficient conditions, closure and subordination results. The proposed approach is also compared with some existing methods to show the reliability and effectiveness of the proposed methods. Full article

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

5 pages, 215 KiB

Open AccessArticle

On the Ninth Coefficient of the Inverse of a Convex Function

by Toshiyuki Sugawa

Mathematics 2021, 9(7), 706; https://doi.org/10.3390/math9070706 - 25 Mar 2021

Cited by 1 | Viewed by 1714

Abstract

We consider the inverse function

z = g (w) = w + b_{2} w^{2} + \dots

of a normalized convex univalent function

w = f (z) = z + a_{2} z^{2} + \dots

on the [...] Read more.

We consider the inverse function

z = g (w) = w + b_{2} w^{2} + \dots

of a normalized convex univalent function

w = f (z) = z + a_{2} z^{2} + \dots

on the unit disk in the complex plane. So far, it is known that

| b_{n} | \leq 1

for

n = 2, 3, \dots, 8 .

On the other hand, the inequality

| b_{n} | \leq 1

is not valid for

n = 10 .

It is conjectured that

| b_{9} | \leq 1 .

The present paper offers the estimate

| b_{9} | < 1.617 .

Full article

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

11 pages, 291 KiB

Open AccessArticle

Differential Subordination and Superordination Results Associated with Mittag–Leffler Function

by Adel A. Attiya, Mohamed K. Aouf, Ekram E. Ali and Mansour F. Yassen

Mathematics 2021, 9(3), 226; https://doi.org/10.3390/math9030226 - 25 Jan 2021

Cited by 9 | Viewed by 1954

Abstract

In this paper, we derive a number of interesting results concerning subordination and superordination relations for certain analytic functions associated with an extension of the Mittag–Leffler function. Full article

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

8 pages, 244 KiB

Open AccessArticle

Subclasses of Multivalent Analytic Functions Associated with a q-Difference Operator

by Ekram Elsayed Ali and Teodor Bulboacă

Mathematics 2020, 8(12), 2184; https://doi.org/10.3390/math8122184 - 8 Dec 2020

Cited by 10 | Viewed by 1868

Abstract

In this article we introduced and studied some inclusion properties for new subclasses of multivalent analytic functions defined by using the q-derivative operator. With the aid of the Jackson q-derivative we defined two new operators that generalize many other previously studied [...] Read more.

In this article we introduced and studied some inclusion properties for new subclasses of multivalent analytic functions defined by using the q-derivative operator. With the aid of the Jackson q-derivative we defined two new operators that generalize many other previously studied operators, and help us to define two new subclasses of functions with several interesting properties studied in this paper. The methods used for the proof of our results are special tools of the differential subordination theory of one-variable functions. Full article

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

14 pages, 277 KiB

Open AccessArticle

Q-Extension of Starlike Functions Subordinated with a Trigonometric Sine Function

by Saeed Islam, Muhammad Ghaffar Khan, Bakhtiar Ahmad, Muhammad Arif and Ronnason Chinram

Mathematics 2020, 8(10), 1676; https://doi.org/10.3390/math8101676 - 1 Oct 2020

Cited by 14 | Viewed by 2353

Abstract

The main purpose of this article is to examine the q-analog of starlike functions connected with a trigonometric sine function. Further, we discuss some interesting geometric properties, such as the well-known problems of Fekete-Szegö, the necessary and sufficient condition, the growth and [...] Read more.

The main purpose of this article is to examine the q-analog of starlike functions connected with a trigonometric sine function. Further, we discuss some interesting geometric properties, such as the well-known problems of Fekete-Szegö, the necessary and sufficient condition, the growth and distortion bound, closure theorem, convolution results, radii of starlikeness, extreme point theorem and the problem with partial sums for this class. Full article

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

8 pages, 240 KiB

Open AccessArticle

Coefficient Estimates for a Subclass of Starlike Functions

by Dorina Răducanu

Mathematics 2020, 8(10), 1646; https://doi.org/10.3390/math8101646 - 24 Sep 2020

Cited by 3 | Viewed by 1925

Abstract

In this note, we consider a subclass

H_{3 / 2} (p)

of starlike functions f with

f^{″} (0) = p

for a prescribed

p \in [0, 2]

. Usually, in the study of univalent [...] Read more.

In this note, we consider a subclass

H_{3 / 2} (p)

of starlike functions f with

f^{″} (0) = p

for a prescribed

p \in [0, 2]

. Usually, in the study of univalent functions, estimates on the Taylor coefficients, Fekete–Szegö functional or Hankel determinats are given. Another coefficient problem which has attracted considerable attention is to estimate the moduli of successive coefficients

| a_{n + 1} | - | a_{n} |

. Recently, the related functional

| a_{n + 1} - a_{n} |

for the initial successive coefficients has been investigated for several classes of univalent functions. We continue this study and for functions

f (z) = z + \sum_{n = 2}^{\infty} a_{n} z^{n} \in H_{3 / 2} (p)

, we investigate upper bounds of initial coefficients and the difference of moduli of successive coefficients

| a_{3} - a_{2} |

and

| a_{4} - a_{3} |

. Estimates of the functionals

| a_{2} a_{4} - a_{3}^{2} |

and

| a_{4} - a_{2} a_{3} |

are also derived. The obtained results expand the scope of the theoretical results related with the functional

| a_{n + 1} - a_{n} |

for various subclasses of univalent functions. Full article

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

11 pages, 257 KiB

Open AccessArticle

On the Difference of Coefficients of Starlike and Convex Functions

by Young Jae Sim and Derek K. Thomas

Mathematics 2020, 8(9), 1521; https://doi.org/10.3390/math8091521 - 7 Sep 2020

Cited by 6 | Viewed by 1988

Abstract

Let f be analytic in the unit disk

D = {z \in C : | z | < 1}

, and

S

be the subclass of normalized univalent functions given by [...] Read more.

Let f be analytic in the unit disk

D = {z \in C : | z | < 1}

, and

S

be the subclass of normalized univalent functions given by

f (z) = z + \sum_{n = 2}^{\infty} a_{n} z^{n}

for

z \in D

. Let

S^{*} \subset S

be the subset of starlike functions in

D

and

C \subset S

the subset of convex functions in

D

. We give sharp upper and lower bounds for

| a_{3} | - | a_{2} |

for some important subclasses of

S^{*}

and

C

. Full article

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

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 7 | Viewed by 2477

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)

11 pages, 460 KiB

Open AccessFeature PaperArticle

Subordination Implications and Coefficient Estimates for Subclasses of Starlike Functions

by Nak Eun Cho, Ali Ebadian, Serap Bulut and Ebrahim Analouei Adegani

Mathematics 2020, 8(7), 1150; https://doi.org/10.3390/math8071150 - 14 Jul 2020

Cited by 5 | Viewed by 2347

Abstract

In the present paper, we consider various subclasses of star-like functions, which are defined by subordination and then we obtain several subordination implications related to these subclasses. Some coefficient bounds for functions belonging to some subclasses of star-like functions are also estimated. Moreover, [...] Read more.

In the present paper, we consider various subclasses of star-like functions, which are defined by subordination and then we obtain several subordination implications related to these subclasses. Some coefficient bounds for functions belonging to some subclasses of star-like functions are also estimated. Moreover, we give some related connections of the outcomes stated here with those obtained earlier. Full article

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

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14 pages, 479 KiB

Open AccessArticle

Hankel Determinants for New Subclasses of Analytic Functions Related to a Shell Shaped Region

by Gangadharan Murugusundaramoorthy and Teodor Bulboacă

Mathematics 2020, 8(6), 1041; https://doi.org/10.3390/math8061041 - 26 Jun 2020

Cited by 24 | Viewed by 3257

Abstract

Using the operator

L_{c}^{a}

defined by Carlson and Shaffer, we defined a new subclass of analytic functions

{ML}_{c}^{a} (λ; ψ)

defined by a subordination relation to the shell shaped function [...] Read more.

Using the operator

L_{c}^{a}

defined by Carlson and Shaffer, we defined a new subclass of analytic functions

{ML}_{c}^{a} (λ; ψ)

defined by a subordination relation to the shell shaped function

ψ (z) = z + \sqrt{1 + z^{2}}

. We determined estimate bounds of the four coefficients of the power series expansions, we gave upper bound for the Fekete–SzegőSzegő functional and for the Hankel determinant of order two for

f \in {ML}_{c}^{a} (λ; ψ)

. Full article

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

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