Mathematics

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12 pages, 334 KiB

Open AccessArticle

New Properties of Analytic Functions

by Hatun Özlem Güney and Shigeyoshi Owa

Mathematics 2024, 12(10), 1469; https://doi.org/10.3390/math12101469 - 9 May 2024

Viewed by 282

Abstract

In the present paper, we consider the class

\bar{A}

of functions

f (z)

of the form

f (z) = z + \sum_{k = 1}^{\infty} a_{1 + \frac{k}{3}} z^{1 + \frac{k}{3}}

that are [...] Read more.

In the present paper, we consider the class

\bar{A}

of functions

f (z)

of the form

f (z) = z + \sum_{k = 1}^{\infty} a_{1 + \frac{k}{3}} z^{1 + \frac{k}{3}}

that are analytic in the open unit disc

U .

If

a_{1 + \frac{k}{3}} = 0

for

k \neq 3 n

(n = 1, 2, 3, \dots),

then

f (z)

is given by

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

For such functions

f (z) \in \bar{A},

some interesting properties for subordinations and strongly starlike functions are given. Also, some interesting examples for the results are shown. Full article

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

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13 pages, 298 KiB

Open AccessArticle

On the Zeros of the Differential Polynomials

φ f^{l} {(f^{(k)})}^{n} - a

by Jiantang Lu and Junfeng Xu

Mathematics 2024, 12(8), 1196; https://doi.org/10.3390/math12081196 - 16 Apr 2024

Viewed by 336

Abstract

Letting f be a transcendental meromorphic function, we consider the value distribution of the differential polynomials

φ f^{l} {(f^{(k)})}^{n} - a

, where

φ (≢ 0)

is a small function of f, [...] Read more.

Letting f be a transcendental meromorphic function, we consider the value distribution of the differential polynomials

φ f^{l} {(f^{(k)})}^{n} - a

, where

φ (≢ 0)

is a small function of f,

l (\geq 2)

,

n (\geq 1)

,

k (\geq 1)

are integers and a is a non-zero constant, and obtain an important inequality concerning the reduced counting function of

φ f^{l} {(f^{(k)})}^{n} - a

. Our results improve and generalize the results obtained by Xu and Ye, Karmakar and Sahoo, Chakraborty et.al, and Chen and Huang. Full article

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

22 pages, 345 KiB

Open AccessArticle

On the Study of Starlike Functions Associated with the Generalized Sine Hyperbolic Function

by Baseer Gul, Muhammad Arif, Reem K. Alhefthi, Daniel Breaz, Luminiţa-Ioana Cotîrlă and Eleonora Rapeanu

Mathematics 2023, 11(23), 4848; https://doi.org/10.3390/math11234848 - 1 Dec 2023

Viewed by 762

Abstract

Geometric function theory, a subfield of complex analysis that examines the geometrical characteristics of analytic functions, has seen a sharp increase in research in recent years. In particular, by employing subordination notions, the contributions of different subclasses of analytic functions associated with innovative [...] Read more.

Geometric function theory, a subfield of complex analysis that examines the geometrical characteristics of analytic functions, has seen a sharp increase in research in recent years. In particular, by employing subordination notions, the contributions of different subclasses of analytic functions associated with innovative image domains are of significant interest and are extensively investigated. Since

ℜ (1 + sinh (z)) ≯ 0,

it implies that the class

S_{sinh}^{*}

introduced in reference third by Kumar et al. is not a subclass of starlike functions. Now, we have introduced a parameter

λ

with the restriction

0 \leq λ \leq ln (1 + \sqrt{2}),

and by doing that,

ℜ (1 + sinh (λ z)) > 0 .

The present research intends to provide a novel subclass of starlike functions in the open unit disk

U,

denoted as

S_{sinh λ}^{*}

, and investigate its geometric nature. For this newly defined subclass, we obtain sharp upper bounds of the coefficients

a_{n}

for

n = 2, 3, 4, 5 .

Then, we prove a lemma, in which the largest disk contained in the image domain of

q_{0} (z) = 1 + sinh (λ z)

and the smallest disk containing

q_{0} (U)

are investigated. This lemma has a central role in proving our radius problems. We discuss radius problems of various known classes, including

S^{*} (β)

and

K (β)

of starlike functions of order

β

and convex functions of order

β

. Investigating

S_{sinh λ}^{*}

radii for several geometrically known classes and some classes of functions defined as ratios of functions are also part of the present research. The methodology used for finding

S_{sinh λ}^{*}

radii of different subclasses is the calculation of that value of the radius

r < 1

for which the image domain of any function belonging to a specified class is contained in the largest disk of this lemma. A new representation of functions in this class, but for a more restricted range of

λ

, is also obtained. Full article

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

27 pages, 361 KiB

Open AccessArticle

Boundedness and Compactness of Weighted Composition Operators from α-Bloch Spaces to Bers-Type Spaces on Generalized Hua Domains of the First Kind

by Jiaqi Wang and Jianbing Su

Mathematics 2023, 11(20), 4403; https://doi.org/10.3390/math11204403 - 23 Oct 2023

Viewed by 539

Abstract

We address weighted composition operators

ψ C_{ϕ}

from

α

-Bloch spaces to Bers-type spaces of bounded holomorphic functions on Y, where Y is a generalized Hua domain of the first kind, and obtain some necessary and sufficient conditions for the boundedness and [...] Read more.

We address weighted composition operators

ψ C_{ϕ}

from

α

-Bloch spaces to Bers-type spaces of bounded holomorphic functions on Y, where Y is a generalized Hua domain of the first kind, and obtain some necessary and sufficient conditions for the boundedness and compactness of those operators. Full article

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

10 pages, 297 KiB

Open AccessArticle

Geometric Properties of Certain Classes of Analytic Functions with Respect to (x,y)-Symmetric Points

by Fuad Alsarari, Muhammad Imran Faisal and Alaa Awad Alzulaibani

Mathematics 2023, 11(19), 4180; https://doi.org/10.3390/math11194180 - 6 Oct 2023

Viewed by 603

Abstract

In this article, the present study employs the utilization of the concepts pertaining to

(x, y)

-symmetrical functions, Janowski type functions, and q-calculus in order to establish a novel subclass within the open unit disk. Specifically, we delve into [...] Read more.

In this article, the present study employs the utilization of the concepts pertaining to

(x, y)

-symmetrical functions, Janowski type functions, and q-calculus in order to establish a novel subclass within the open unit disk. Specifically, we delve into the examination of convolution properties, which serve as a tool for investigating and inferring adequate and equivalent conditions. Moreover, we also explore specific characteristics of the class

{\tilde{S}}_{q}^{x, y} (α, β, λ)

, thereby further scrutinizing the convolution properties of these newly defined classes. Full article

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

14 pages, 446 KiB

Open AccessFeature PaperArticle

Zeros of Convex Combinations of Elementary Families of Harmonic Functions

by Jennifer Brooks, Megan Dixon, Michael Dorff, Alexander Lee and Rebekah Ottinger

Mathematics 2023, 11(19), 4057; https://doi.org/10.3390/math11194057 - 25 Sep 2023

Viewed by 589

Abstract

Brilleslyper et al. investigated how the number of zeros of a one-parameter family of harmonic trinomials varies with a real parameter. Brooks and Lee obtained a similar theorem for an analogous family of harmonic trinomials with poles. In this paper, we investigate the [...] Read more.

Brilleslyper et al. investigated how the number of zeros of a one-parameter family of harmonic trinomials varies with a real parameter. Brooks and Lee obtained a similar theorem for an analogous family of harmonic trinomials with poles. In this paper, we investigate the number of zeros of convex combinations of members of these families and show that it is possible for a convex combination of two members of a family to have more zeros than either of its constituent parts. Our main tool to prove these results is the harmonic analog of Rouché’s theorem. Full article

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

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10 pages, 293 KiB

Open AccessArticle

On Miller–Ross-Type Poisson Distribution Series

by Basem Aref Frasin and Luminiţa-Ioana Cotîrlă

Mathematics 2023, 11(18), 3989; https://doi.org/10.3390/math11183989 - 20 Sep 2023

Viewed by 697

Abstract

The objective of the current paper is to find the necessary and sufficient conditions for Miller–Ross-type Poisson distribution series to be in the classes

S_{T}^{*} (γ, β)

and

K_{T} (γ, β)

of analytic functions [...] Read more.

The objective of the current paper is to find the necessary and sufficient conditions for Miller–Ross-type Poisson distribution series to be in the classes

S_{T}^{*} (γ, β)

and

K_{T} (γ, β)

of analytic functions with negative coefficients. Furthermore, we investigate several inclusion properties of the class

Y^{σ} (V, W)

associated of the operator

I_{α, c}^{ε}

defined by this distribution. We also take into consideration an integral operator connected to series of Miller–Ross-type Poisson distributions. Special cases of the main results are also considered. Full article

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

16 pages, 472 KiB

Open AccessArticle

Initial Coefficient Bounds for Bi-Univalent Functions Related to Gregory Coefficients

by Gangadharan Murugusundaramoorthy, Kaliappan Vijaya and Teodor Bulboacă

Mathematics 2023, 11(13), 2857; https://doi.org/10.3390/math11132857 - 26 Jun 2023

Cited by 3 | Viewed by 914

Abstract

In this article we introduce three new subclasses of the class of bi-univalent functions

Σ

, namely

{HG}_{Σ}

,

{GM}_{Σ} (μ)

and

G_{Σ} (λ)

, by using the subordinations with the functions whose coefficients are Gregory [...] Read more.

In this article we introduce three new subclasses of the class of bi-univalent functions

Σ

, namely

{HG}_{Σ}

,

{GM}_{Σ} (μ)

and

G_{Σ} (λ)

, by using the subordinations with the functions whose coefficients are Gregory numbers. First, we evidence that these classes are not empty, i.e., they contain other functions besides the identity one. For functions in each of these three bi-univalent function classes, we investigate the estimates

|a_{2}|

and

|a_{3}|

of the Taylor–Maclaurin coefficients and Fekete–Szegő functional problems. The main results are followed by some particular cases, and the novelty of the characterizations and the proofs may lead to further studies of such types of similarly defined subclasses of analytic bi-univalent functions. Full article

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

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15 pages, 316 KiB

Open AccessArticle

Applications of q-Calculus Multiplier Operators and Subordination for the Study of Particular Analytic Function Subclasses

by Ekram E. Ali, Georgia Irina Oros, Shujaat Ali Shah and Abeer M. Albalahi

Mathematics 2023, 11(12), 2705; https://doi.org/10.3390/math11122705 - 14 Jun 2023

Cited by 3 | Viewed by 1019

Abstract

In this article, a new linear extended multiplier operator is defined utilizing the q-Choi–Saigo–Srivastava operator and the q-derivative. Two generalized subclasses of q—uniformly convex and starlike functions of order

δ

—are defined and studied using this new operator. Necessary conditions [...] Read more.

In this article, a new linear extended multiplier operator is defined utilizing the q-Choi–Saigo–Srivastava operator and the q-derivative. Two generalized subclasses of q—uniformly convex and starlike functions of order

δ

—are defined and studied using this new operator. Necessary conditions are derived for functions to belong in each of the two subclasses, and subordination theorems involving the Hadamard product of such particular functions are stated and proven. As applications of those findings using specific values for the parameters of the new subclasses, associated corollaries are provided. Additionally, examples are created to demonstrate the conclusions’ applicability in relation to the functions from the newly introduced subclasses. Full article

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

14 pages, 310 KiB

Open AccessArticle

Norm Estimates of the Pre-Schwarzian Derivatives for Functions with Conic-like Domains

by Sidra Zafar, Abbas Kareem Wanas, Mohamed Abdalla and Syed Zakar Hussain Bukhari

Mathematics 2023, 11(11), 2490; https://doi.org/10.3390/math11112490 - 29 May 2023

Viewed by 1384

Abstract

The pre-Schwarzianand Schwarzian derivatives of analytic functions f are defined in

U

, where

U

is the open unit disk. The pre-Schwarzian as well as Schwarzian derivatives are popular tools for studying the geometric properties of analytic mappings. These can also be used [...] Read more.

The pre-Schwarzianand Schwarzian derivatives of analytic functions f are defined in

U

, where

U

is the open unit disk. The pre-Schwarzian as well as Schwarzian derivatives are popular tools for studying the geometric properties of analytic mappings. These can also be used to obtain either necessary or sufficient conditions for the univalence of a function f. Because of the computational difficulty, the pre-Schwarzian norm has received more attention than the Schwarzian norm. It has applications in the theory of hypergeometric functions, conformal mappings, Teichmüller spaces, and univalent functions. In this paper, we find sharp norm estimates of the pre-Schwarzian derivatives of certain subfamilies of analytic functions involving some conic-like image domains. These results may also be extended to the families of strongly starlike, convex, as well as to functions with symmetric and conjugate symmetric points. Full article

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

10 pages, 345 KiB

Open AccessArticle

Logarithmic Coefficients Inequality for the Family of Functions Convex in One Direction

by Ebrahim Analouei Adegani, Ahmad Motamednezhad, Mostafa Jafari and Teodor Bulboacă

Mathematics 2023, 11(9), 2140; https://doi.org/10.3390/math11092140 - 3 May 2023

Cited by 3 | Viewed by 947

Abstract

The logarithmic coefficients play an important role for different estimates in the theory of univalent functions. Due to the significance of the recent studies about the logarithmic coefficients, the problem of obtaining the sharp bounds for the modulus of these coefficients has received [...] Read more.

The logarithmic coefficients play an important role for different estimates in the theory of univalent functions. Due to the significance of the recent studies about the logarithmic coefficients, the problem of obtaining the sharp bounds for the modulus of these coefficients has received attention. In this research, we obtain sharp bounds of the inequality involving the logarithmic coefficients for the functions of the well-known class

G

and investigate a majorization problem for the functions belonging to this family. To prove our main results, we use the Briot–Bouquet differential subordination obtained by J.A. Antonino and S.S. Miller and the result of T.J. Suffridge connected to the Alexander integral. Combining these results, we give sharp inequalities for two types of sums involving the modules of the logarithmical coefficients of the functions of the class

G

indicating also the extremal function. In addition, we prove an inequality for the modulus of the derivative of two majorized functions of the class

G

, followed by an application. Full article

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

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9 pages, 289 KiB

Open AccessArticle

Estimates for the Coefficients of Subclasses Defined by the Bell Distribution of Bi-Univalent Functions Subordinate to Gegenbauer Polynomials

by Ala Amourah, Omar Alnajar, Maslina Darus, Ala Shdouh and Osama Ogilat

Mathematics 2023, 11(8), 1799; https://doi.org/10.3390/math11081799 - 10 Apr 2023

Cited by 4 | Viewed by 931

Abstract

In the real world there are many applications that find the Bell distribution to be a useful and relevant model. One of these is the normal distribution. In this paper, we develop a new subclass of analytic bi-univalent functions by making use of [...] Read more.

In the real world there are many applications that find the Bell distribution to be a useful and relevant model. One of these is the normal distribution. In this paper, we develop a new subclass of analytic bi-univalent functions by making use of the Bell distribution as a building block. These functions involve the Gegenbauer polynomials, and we use them to establish our new subclass. In this study, we solve the Fekete–Szegö functional problem and analyse various different estimates of the Maclaurin coefficients

|D_{2}|

and

|D_{3}|

for functions that belong to the built class. Full article

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

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