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A special issue of Mathematics (ISSN 2227-7390). This special issue belongs to the section "Mathematics and Computer Science".

Deadline for manuscript submissions: closed (31 October 2020) | Viewed by 29761

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Prof. Dr. Georgia Irina Oros

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Department of Mathematics and Computer Science, University of Oradea, Oradea 410 087, Romania
Interests: univalent functions; harmonic functions; differential subordination and superordination; geometric theory of analytic and non-analytic functions
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Special Issue Information

Dear Colleagues,

The Special Issue devoted to the Geometric Theory of Analytic Functions will bring together the newest research achievements of scholars studying the complex-valued functions of one variable. The issue will cover all aspects of this topic, starting with special classes of univalent functions, operator-related results, studies using the theory of differential subordination and superordination, or any other techniques which can be applied in the field of complex analysis and its applications.

The Editors of this Special Issue are pleased to invite the authors to submit their original results related to analytic functions and even studies related to non-analytic functions, if any research was done comparing the two types of complex-valued functions. We await the latest results related to classic differential subordination and superordination, strong differential subordination and superordination, and fuzzy differential subordination and superordination. We believe researchers are eager to see how differential, integral, and differential–integral operators are used on the special classes of univalent functions already known or newly introduced and what their importance in the field is. This Special Issue will also publish contributions related exclusively to complex analysis and we hope to find among the results new approaches for Geometric Function Theory which could inspire further achievements in the field.

Prof. Dr. Georgia Irina Oros
Guest Editor

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Keywords

analytic function
univalent function
harmonic function
differential subordination
differential superordination
strong differential subordination
strong differential superordination
fuzzy differential subordination
fuzzy differential superordination
differential operator
integral operator
differential–integral operator
linear operator

Published Papers (16 papers)

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Editorial

Jump to: Research

4 pages, 210 KiB

Open AccessEditorial

Geometrical Theory of Analytic Functions

by Georgia Irina Oros

Mathematics 2022, 10(18), 3267; https://0-doi-org.brum.beds.ac.uk/10.3390/math10183267 - 08 Sep 2022

Cited by 1 | Viewed by 878

Abstract

This Special Issue, devoted to the topic of the “Geometric Theory of Analytic Functions”, aims to bring together the newest research achievements of scholars studying the complex-valued functions of one variable [...] Full article

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

Research

Jump to: Editorial

10 pages, 275 KiB

Open AccessArticle

Systems of Simultaneous Differential Inclusions Implying Function Containment

by José A. Antonino and Sanford S. Miller

Mathematics 2021, 9(11), 1252; https://0-doi-org.brum.beds.ac.uk/10.3390/math9111252 - 30 May 2021

Cited by 3 | Viewed by 1568

Abstract

An important problem in complex analysis is to determine properties of the image of an analytic function p defined on the unit disc U from an inclusion or containment relation involving several of the derivatives of p. Results dealing with differential inclusions have led to the development of the field of Differential Subordinations, while results dealing with differential containments have led to the development of the field of Differential Superordinations. In this article, the authors consider a mixed problem consisting of special differential inclusions implying a corresponding containment of the form

D [p] (U) \subset Ω \Rightarrow Δ \subset p (U)

, where

Ω

and

Δ

are sets in

C

, and D is a differential operator such that D[p] is an analytic function defined on U. We carry out this research by considering the more general case involving a system of two simultaneous differential operators in two unknown functions. Full article

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

10 pages, 272 KiB

Open AccessArticle

Subordination Properties of Meromorphic Kummer Function Correlated with Hurwitz–Lerch Zeta-Function

by Firas Ghanim, Khalifa Al-Shaqsi, Maslina Darus and Hiba Fawzi Al-Janaby

Mathematics 2021, 9(2), 192; https://0-doi-org.brum.beds.ac.uk/10.3390/math9020192 - 19 Jan 2021

Cited by 13 | Viewed by 1543

Abstract

Recently, Special Function Theory (SPFT) and Operator Theory (OPT) have acquired a lot of concern due to their considerable applications in disciplines of pure and applied mathematics. The Hurwitz-Lerch Zeta type functions, as a part of Special Function Theory (SPFT), are significant in developing and providing further new studies. In complex domain, the convolution tool is a salutary technique for systematic analytical characterization of geometric functions. The analytic functions in the punctured unit disk are the so-called meromorphic functions. In this present analysis, a new convolution complex operator defined on meromorphic functions related with the Hurwitz-Lerch Zeta type functions and Kummer functions is considered. Certain sufficient stipulations are stated for several formulas of this defining operator to attain subordination. Indeed, these outcomes are an extension of known outcomes of starlikeness, convexity, and close to convexity. Full article

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

8 pages, 234 KiB

Open AccessArticle

Best Subordinant for Differential Superordinations of Harmonic Complex-Valued Functions

by Georgia Irina Oros

Mathematics 2020, 8(11), 2041; https://0-doi-org.brum.beds.ac.uk/10.3390/math8112041 - 16 Nov 2020

Cited by 5 | Viewed by 1237

Abstract

The theory of differential subordinations has been extended from the analytic functions to the harmonic complex-valued functions in 2015. In a recent paper published in 2019, the authors have considered the dual problem of the differential subordination for the harmonic complex-valued functions and have defined the differential superordination for harmonic complex-valued functions. Finding the best subordinant of a differential superordination is among the main purposes in this research subject. In this article, conditions for a harmonic complex-valued function p to be the best subordinant of a differential superordination for harmonic complex-valued functions are given. Examples are also provided to show how the theoretical findings can be used and also to prove the connection with the results obtained in 2015. Full article

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

8 pages, 231 KiB

Open AccessArticle

Coefficient Estimates for Bi-Univalent Functions in Connection with Symmetric Conjugate Points Related to Horadam Polynomial

by S. Melike Aydoğan and Zeliha Karahüseyin

Mathematics 2020, 8(11), 1888; https://0-doi-org.brum.beds.ac.uk/10.3390/math8111888 - 31 Oct 2020

Cited by 2 | Viewed by 1282

Abstract

In the current study, we construct a new subclass of bi-univalent functions with respect to symmetric conjugate points in the open disc E, described by Horadam polynomials. For this subclass, initial Maclaurin coefficient bounds are acquired. The Fekete–Szegö problem of this subclass is [...] Read more.

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

11 pages, 881 KiB

Open AccessArticle

On the Connection Problem for Painlevé Differential Equation in View of Geometric Function Theory

by Rabha W. Ibrahim, Rafida M. Elobaid and Suzan J. Obaiys

Mathematics 2020, 8(7), 1198; https://0-doi-org.brum.beds.ac.uk/10.3390/math8071198 - 21 Jul 2020

Cited by 4 | Viewed by 1886

Abstract

Asymptotic analysis is a branch of mathematical analysis that describes the limiting behavior of the function. This behavior appears when we study the solution of differential equations analytically. The recent work deals with a special class of third type of Painlevé differential equation (PV). Our aim is to find asymptotic, symmetric univalent solution of this class in a symmetric domain with respect to the real axis. As a result that the most important problem in the asymptotic expansion is the connections bound (coefficients bound), we introduce a study of this problem. Full article

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

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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://0-doi-org.brum.beds.ac.uk/10.3390/math8071149 - 14 Jul 2020

Cited by 1 | Viewed by 1487

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)

16 pages, 298 KiB

Open AccessArticle

Starlikness Associated with Cosine Hyperbolic Function

by Abdullah Alotaibi, Muhammad Arif, Mohammed A. Alghamdi and Shehzad Hussain

Mathematics 2020, 8(7), 1118; https://0-doi-org.brum.beds.ac.uk/10.3390/math8071118 - 08 Jul 2020

Cited by 41 | Viewed by 2284

Abstract

The main contribution of this article is to define a family of starlike functions associated with a cosine hyperbolic function. We investigate convolution conditions, integral preserving properties, and coefficient sufficiency criteria for this family. We also study the differential subordinations problems which relate the Janowski and cosine hyperbolic functions. Furthermore, we use these results to obtain sufficient conditions for starlike functions connected with cosine hyperbolic function. Full article

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

13 pages, 301 KiB

Open AccessArticle

by Ágnes Orsolya Páll-Szabó and Georgia Irina Oros

Mathematics 2020, 8(7), 1110; https://0-doi-org.brum.beds.ac.uk/10.3390/math8071110 - 06 Jul 2020

Cited by 31 | Viewed by 2098

Abstract

Using the recently introduced Sălăgean integro-differential operator, three new classes of bi-univalent functions are introduced in this paper. In the study of bi-univalent functions, estimates on the first two Taylor–Maclaurin coefficients are usually given. We go further in the present paper and bounds [...] Read more.

|a_{2}|

|a_{3}|

and

|a_{4}|

of the functions in the newly defined classes are given. Obtaining Fekete–Szegő inequalities for different classes of functions is a topic of interest at this time as it will be shown later by citing recent papers. So, continuing the study on the coefficients of those classes, the well-known Fekete–Szegő functional is obtained for each of the three classes. Full article

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

13 pages, 287 KiB

Open AccessArticle

New Criteria for Meromorphic Starlikeness and Close-to-Convexity

by Ali Ebadian, Nak Eun Cho, Ebrahim Analouei Adegani and Sibel Yalçın

Mathematics 2020, 8(5), 847; https://0-doi-org.brum.beds.ac.uk/10.3390/math8050847 - 23 May 2020

Cited by 3 | Viewed by 1805

Abstract

The main purpose of current paper is to obtain some new criteria for meromorphic strongly starlike functions of order

α

and strongly close-to-convexity of order

α

. Furthermore, the main results presented here are compared with the previous outcomes obtained in this area. [...] Read more.

The main purpose of current paper is to obtain some new criteria for meromorphic strongly starlike functions of order

α

and strongly close-to-convexity of order

α

. Furthermore, the main results presented here are compared with the previous outcomes obtained in this area. Full article

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

21 pages, 402 KiB

Open AccessArticle

On The Third-Order Complex Differential Inequalities of ξ-Generalized-Hurwitz–Lerch Zeta Functions

by Hiba Al-Janaby, Firas Ghanim and Maslina Darus

Mathematics 2020, 8(5), 845; https://0-doi-org.brum.beds.ac.uk/10.3390/math8050845 - 23 May 2020

Cited by 13 | Viewed by 2002

Abstract

In the z- domain, differential subordination is a complex technique of geometric function theory based on the idea of differential inequality. It has formulas in terms of the first, second and third derivatives. In this study, we introduce some applications of the third-order [...] Read more.

ξ

-Generalized-Hurwitz–Lerch Zeta functions (GHLZF). These outcomes are derived by investigating the appropriate classes of admissible functions. Full article

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

11 pages, 307 KiB

Open AccessArticle

Subclasses of Bi-Univalent Functions Defined by Frasin Differential Operator

by Ibtisam Aldawish, Tariq Al-Hawary and B. A. Frasin

Mathematics 2020, 8(5), 783; https://0-doi-org.brum.beds.ac.uk/10.3390/math8050783 - 13 May 2020

Cited by 32 | Viewed by 2523

Abstract

Let

Ω

denote the class of functions

f (z) = z + a_{2} z^{2} + a_{3} z^{3} + \dots

belonging to the normalized analytic function class

A

in the open unit disk [...] Read more.

Let

Ω

denote the class of functions

f (z) = z + a_{2} z^{2} + a_{3} z^{3} + \dots

belonging to the normalized analytic function class

A

in the open unit disk

U = \{z : |z| < 1\}

, which are bi-univalent in

U

, that is, both the function f and its inverse

f^{- 1}

are univalent in

U

. In this paper, we introduce and investigate two new subclasses of the function class

Ω

of bi-univalent functions defined in the open unit disc

U

, which are associated with a new differential operator of analytic functions involving binomial series. Furthermore, we find estimates on the Taylor–Maclaurin coefficients

| a_{2} |

and

| a_{3} |

for functions in these new subclasses. Several (known or new) consequences of the results are also pointed out. Full article

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

9 pages, 235 KiB

Open AccessArticle

Starlikeness Condition for a New Differential-Integral Operator

by Mugur Acu and Gheorghe Oros

Mathematics 2020, 8(5), 694; https://0-doi-org.brum.beds.ac.uk/10.3390/math8050694 - 02 May 2020

Cited by 5 | Viewed by 1358

Abstract

A new differential-integral operator of the form [...] Read more.

A new differential-integral operator of the form

I^{n} f (z) = (1 - λ) S^{n} f (z) + λ L^{n} f (z), z \in U, f \in A, 0 \leq λ \leq 1, n \in N

is introduced in this paper, where

S^{n}

is the Sălăgean differential operator and

L^{n}

is the Alexander integral operator. Using this operator, a new integral operator is defined as:

F (z) = {[\frac{β + γ}{z^{γ}} \int_{0}^{z} I^{n} f (z) \cdot t^{β + γ - 2} d t]}^{\frac{1}{β}},

where

I^{n} f (z)

is the differential-integral operator given above. Using a differential subordination, we prove that the integral operator

F (z)

is starlike. Full article

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

17 pages, 12928 KiB

Open AccessArticle

Taming the Natural Boundary of Centered Polygonal Lacunary Functions—Restriction to the Symmetry Angle Space

by Leah K. Mork, Keith Sullivan and Darin J. Ulness

Mathematics 2020, 8(4), 568; https://0-doi-org.brum.beds.ac.uk/10.3390/math8040568 - 11 Apr 2020

Cited by 2 | Viewed by 1820

Abstract

This work investigates centered polygonal lacunary functions restricted from the unit disk onto symmetry angle space which is defined by the symmetry angles of a given centered polygonal lacunary function. This restriction allows for one to consider only the p-sequences of the centered polygonal lacunary functions which are bounded, but not convergent, at the natural boundary. The periodicity of the p-sequences naturally gives rise to a convergent subsequence, which can be used as a grounds for decomposition of the restricted centered polygonal lacunary functions. A mapping of the unit disk to the sphere allows for the study of the line integrals of restricted centered polygonal that includes analytic progress towards closed form representations. Obvious closures of the domain obtained from the spherical map lead to four distinct topological spaces of the “broom topology” type. Full article

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

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Figure 1

14 pages, 297 KiB

Open AccessArticle

Maclaurin Coefficient Estimates of Bi-Univalent Functions Connected with the q-Derivative

by Sheza M. El-Deeb, Teodor Bulboacă and Bassant M. El-Matary

Mathematics 2020, 8(3), 418; https://0-doi-org.brum.beds.ac.uk/10.3390/math8030418 - 14 Mar 2020

Cited by 44 | Viewed by 2290

Abstract

In this paper we introduce a new subclass of the bi-univalent functions defined in the open unit disc and connected with a q-analogue derivative. We find estimates for the first two Taylor-Maclaurin coefficients

|a_{2}|

and

|a_{3}|

for functions in this [...] Read more.

|a_{2}|

and

|a_{3}|

for functions in this subclass, and we obtain an estimation for the Fekete-Szegő problem for this function class. Full article

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

13 pages, 285 KiB

Open AccessArticle

Symmetric Conformable Fractional Derivative of Complex Variables

by Rabha W. Ibrahim, Rafida M. Elobaid and Suzan J. Obaiys

Mathematics 2020, 8(3), 363; https://0-doi-org.brum.beds.ac.uk/10.3390/math8030363 - 06 Mar 2020

Cited by 17 | Viewed by 2325

Abstract

It is well known that the conformable and the symmetric differential operators have formulas in terms of the first derivative. In this document, we combine the two definitions to get the symmetric conformable derivative operator (SCDO). The purpose of this effort is to provide a study of SCDO connected with the geometric function theory. These differential operators indicate a generalization of well known differential operator including the Sàlàgean differential operator. Our contribution is to impose two classes of symmetric differential operators in the open unit disk and to describe the further development of these operators by introducing convex linear symmetric operators. In addition, by acting these SCDOs on the class of univalent functions, we display a set of sub-classes of analytic functions having geometric representation, such as starlikeness and convexity properties. Investigations in this direction lead to some applications in the univalent function theory of well known formulas, by defining and studying some sub-classes of analytic functions type Janowski function and convolution structures. Moreover, by using the SCDO, we introduce a generalized class of Briot–Bouquet differential equations to introduce, what is called the symmetric conformable Briot–Bouquet differential equations. We shall show that the upper bound of this class is symmetric in the open unit disk. Full article

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

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