Fractal and Fractional

Research

15 pages, 337 KiB

Open AccessArticle

Applications of q-Hermite Polynomials to Subclasses of Analytic and Bi-Univalent Functions

by Caihuan Zhang, Bilal Khan, Timilehin Gideon Shaba, Jong-Suk Ro, Serkan Araci and Muhammad Ghaffar Khan

Fractal Fract. 2022, 6(8), 420; https://doi.org/10.3390/fractalfract6080420 - 30 Jul 2022

Cited by 13 | Viewed by 1385

Abstract

In mathematics, physics, and engineering, orthogonal polynomials and special functions play a vital role in the development of numerical and analytical approaches. This field of study has received a lot of attention in recent decades, and it is gaining traction in current fields, [...] Read more.

In mathematics, physics, and engineering, orthogonal polynomials and special functions play a vital role in the development of numerical and analytical approaches. This field of study has received a lot of attention in recent decades, and it is gaining traction in current fields, including computational fluid dynamics, computational probability, data assimilation, statistics, numerical analysis, and image and signal processing. In this paper, using q-Hermite polynomials, we define a new subclass of bi-univalent functions. We then obtain a number of important results such as bonds for the initial coefficients of

| a_{2} |

,

| a_{3} |

, and

| a_{4} |

, results related to Fekete–Szegö functional, and the upper bounds of the second Hankel determinant for our defined functions class. Full article

(This article belongs to the Special Issue New Trends in Geometric Function Theory)

15 pages, 546 KiB

Open AccessArticle

Fractional Integral of the Confluent Hypergeometric Function Related to Fuzzy Differential Subordination Theory

by Mugur Acu, Gheorghe Oros and Ancuța Maria Rus

Fractal Fract. 2022, 6(8), 413; https://doi.org/10.3390/fractalfract6080413 - 27 Jul 2022

Cited by 9 | Viewed by 1145

Abstract

The fuzzy differential subordination concept was introduced in 2011, generalizing the concept of differential subordination following a recent trend of adapting fuzzy sets theory to other already-established theories. A prolific tool in obtaining new results related to operators is the fractional integral applied [...] Read more.

The fuzzy differential subordination concept was introduced in 2011, generalizing the concept of differential subordination following a recent trend of adapting fuzzy sets theory to other already-established theories. A prolific tool in obtaining new results related to operators is the fractional integral applied to different functions. The fractional integral of the confluent hypergeometric function was previously investigated using means of the classical theory of subordination. In this paper, we give new applications of this function using the theory of fuzzy differential subordination. Fuzzy differential subordinations are established and their best dominants are also provided. Corollaries are written using particular functions, in which the conditions for the univalence of the fractional integral of the confluent hypergeometric function are given. An example is constructed as a specific application of the results obtained in this paper. Full article

(This article belongs to the Special Issue New Trends in Geometric Function Theory)

14 pages, 319 KiB

Open AccessArticle

Certain New Subclass of Multivalent Q-Starlike Functions Associated with Q-Symmetric Calculus

by Mohammad Faisal Khan, Anjali Goswami and Shahid Khan

Fractal Fract. 2022, 6(7), 367; https://doi.org/10.3390/fractalfract6070367 - 30 Jun 2022

Cited by 9 | Viewed by 1218

Abstract

In our present investigation, we extend the idea of q-symmetric derivative operators to multivalent functions and then define a new subclass of multivalent q-starlike functions. For this newly defined function class, we discuss some useful properties of multivalent functions, such as [...] Read more.

In our present investigation, we extend the idea of q-symmetric derivative operators to multivalent functions and then define a new subclass of multivalent q-starlike functions. For this newly defined function class, we discuss some useful properties of multivalent functions, such as the Hankel determinant, symmetric Toeplitz matrices, the Fekete–Szego problem, and upper bounds of the functional

|a_{p + 1} - μ a_{p + 1}^{2}|

and investigate some new lemmas for our main results. In addition, we consider the q-Bernardi integral operator along with q-symmetric calculus and discuss some applications of our main results. Full article

(This article belongs to the Special Issue New Trends in Geometric Function Theory)

13 pages, 333 KiB

Open AccessArticle

A Special Family of m-Fold Symmetric Bi-Univalent Functions Satisfying Subordination Condition

by Ibtisam Aldawish, Sondekola Rudra Swamy and Basem Aref Frasin

Fractal Fract. 2022, 6(5), 271; https://doi.org/10.3390/fractalfract6050271 - 17 May 2022

Cited by 9 | Viewed by 1400

Abstract

In this paper, we introduce a special family

M_{σ_{m}} (τ, ν, η, φ)

of the function family

σ_{m}

of m-fold symmetric bi-univalent functions defined in the open unit disc

D

and obtain estimates of [...] Read more.

In this paper, we introduce a special family

M_{σ_{m}} (τ, ν, η, φ)

of the function family

σ_{m}

of m-fold symmetric bi-univalent functions defined in the open unit disc

D

and obtain estimates of the first two Taylor–Maclaurin coefficients for functions in the special family. Further, the Fekete–Szegö functional for functions in this special family is also estimated. The results presented in this paper not only generalize and improve some recent works, but also give new results as special cases. Full article

(This article belongs to the Special Issue New Trends in Geometric Function Theory)

11 pages, 300 KiB

Open AccessArticle

Third Hankel Determinant for the Logarithmic Coefficients of Starlike Functions Associated with Sine Function

by Bilal Khan, Ibtisam Aldawish, Serkan Araci and Muhammad Ghaffar Khan

Fractal Fract. 2022, 6(5), 261; https://doi.org/10.3390/fractalfract6050261 - 9 May 2022

Cited by 15 | Viewed by 2015

Abstract

The logarithmic functions have been used in a verity of areas of mathematics and other sciences. As far as we know, no one has used the coefficients of logarithmic functions to determine the bounds for the third Hankel determinant. In our present investigation, [...] Read more.

The logarithmic functions have been used in a verity of areas of mathematics and other sciences. As far as we know, no one has used the coefficients of logarithmic functions to determine the bounds for the third Hankel determinant. In our present investigation, we first study some well-known classes of starlike functions and then determine the third Hankel determinant bound for the logarithmic coefficients of certain subclasses of starlike functions that also involve the sine functions. We also obtain a number of coefficient estimates. Some of our results are shown to be sharp. Full article

(This article belongs to the Special Issue New Trends in Geometric Function Theory)

21 pages, 364 KiB

Open AccessArticle

The Sharp Bounds of the Third-Order Hankel Determinant for Certain Analytic Functions Associated with an Eight-Shaped Domain

by Lei Shi, Meshal Shutaywi, Naseer Alreshidi, Muhammad Arif and Syed Muhammad Ghufran

Fractal Fract. 2022, 6(4), 223; https://doi.org/10.3390/fractalfract6040223 - 14 Apr 2022

Cited by 16 | Viewed by 1925

Abstract

The main focus of this research is to solve certain coefficient-related problems for analytic functions that are subordinated to a unique trigonometric function. For the class

S_{sin}^{*}

, with the quantity

\frac{z f^{'} (z)}{f (z)}

[...] Read more.

The main focus of this research is to solve certain coefficient-related problems for analytic functions that are subordinated to a unique trigonometric function. For the class

S_{sin}^{*}

, with the quantity

\frac{z f^{'} (z)}{f (z)}

subordinated to

1 + sin z

, we obtain an estimate on the initial coefficient

a_{4}

and an upper bound of the third Hankel determinant. For functions in the class

{BT}_{sin}

, with

f^{'} (z)

lie in an eight-shaped domain in the right-half plane, we prove that its upper bound of third Hankel determinant is

\frac{1}{16}

. All the results are proven to be sharp. Full article

(This article belongs to the Special Issue New Trends in Geometric Function Theory)

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11 pages, 310 KiB

Open AccessArticle

Generalization of k-Uniformly Starlike and Convex Functions Using q-Difference Operator

by Irfan Ali, Yousaf Ali Khan Malghani, Sardar Muhammad Hussain, Nazar Khan and Jong-Suk Ro

Fractal Fract. 2022, 6(4), 216; https://doi.org/10.3390/fractalfract6040216 - 11 Apr 2022

Cited by 2 | Viewed by 1202

Abstract

In this article we have defined two new subclasses of analytic functions

k - S_{q} [A, B]

and

k - K_{q} [A, B]

by using q-difference operator in an open unit disk. Furthermore, the [...] Read more.

In this article we have defined two new subclasses of analytic functions

k - S_{q} [A, B]

and

k - K_{q} [A, B]

by using q-difference operator in an open unit disk. Furthermore, the necessary and sufficient conditions along with certain other useful properties of these newly defined subclasses have been calculated by using q-difference operator. Full article

(This article belongs to the Special Issue New Trends in Geometric Function Theory)

19 pages, 366 KiB

Open AccessArticle

Second Hankel Determinant for the Subclass of Bi-Univalent Functions Using q-Chebyshev Polynomial and Hohlov Operator

by Isra Al-Shbeil, Timilehin Gideon Shaba and Adriana Cătaş

Fractal Fract. 2022, 6(4), 186; https://doi.org/10.3390/fractalfract6040186 - 25 Mar 2022

Cited by 15 | Viewed by 2031

Abstract

The q-derivative and Hohlov operators have seen much use in recent years. First, numerous well-known principles of the q-derivative operator are highlighted and explained in this research. We then build a novel subclass of analytic and bi-univalent functions using the Hohlov [...] Read more.

The q-derivative and Hohlov operators have seen much use in recent years. First, numerous well-known principles of the q-derivative operator are highlighted and explained in this research. We then build a novel subclass of analytic and bi-univalent functions using the Hohlov operator and certain q-Chebyshev polynomials. A number of coefficient bounds, as well as the Fekete–Szegö inequalities and the second Hankel determinant are provided for these newly specified function classes. Full article

(This article belongs to the Special Issue New Trends in Geometric Function Theory)

7 pages, 254 KiB

Open AccessArticle

On Geometric Properties of a Certain Analytic Function with Negative Coefficients

by Matthew Olanrewaju Oluwayemi, Esther O. Davids and Adriana Cătaş

Fractal Fract. 2022, 6(3), 172; https://doi.org/10.3390/fractalfract6030172 - 21 Mar 2022

Viewed by 1374

Abstract

Various function theorists have successfully defined and investigated different kinds of analytic functions. The applications of such functions have played significant roles in geometry function theory as a field of complex analysis. In this work, therefore, a certain subclass of univalent analytic functions [...] Read more.

Various function theorists have successfully defined and investigated different kinds of analytic functions. The applications of such functions have played significant roles in geometry function theory as a field of complex analysis. In this work, therefore, a certain subclass of univalent analytic functions of the form

f (z) = z - \sum_{m = 2}^{t} \frac{[ω (2 + β) + c γ - σ] C_{m}}{[m σ - c ω (2 + β) + c γ] K^{n}} z^{m} - \sum_{k = t + 1}^{\infty} a_{k} z^{k}

is defined using a generalized differential operator. Furthermore, some geometric properties for the class were established. Full article

(This article belongs to the Special Issue New Trends in Geometric Function Theory)

12 pages, 300 KiB

Open AccessArticle

Inclusion Relations for Dini Functions Involving Certain Conic Domains

by Bilal Khan, Shahid Khan, Jong-Suk Ro, Serkan Araci, Nazar Khan and Nasir Khan

Fractal Fract. 2022, 6(2), 118; https://doi.org/10.3390/fractalfract6020118 - 17 Feb 2022

Cited by 2 | Viewed by 1886

Abstract

In recent years, special functions such as Bessel functions have been widely used in many areas of mathematics and physics. We are essentially motivated by the recent development; in our present investigation, we make use of certain conic domains and define a new [...] Read more.

In recent years, special functions such as Bessel functions have been widely used in many areas of mathematics and physics. We are essentially motivated by the recent development; in our present investigation, we make use of certain conic domains and define a new class of analytic functions associated with the Dini functions. We derive inclusion relationships and certain integral preserving properties. By applying the Bernardi-Libera-Livingston integral operator, we obtain some remarkable applications of our main results. Finally, in the concluding section, we recall the attention of curious readers to studying the q-generalizations of the results presented in this paper. Furthermore, based on the suggested extension, the

(p, q)

-extension will be a relatively minor and unimportant change, as the new parameter

p

is redundant. Full article

(This article belongs to the Special Issue New Trends in Geometric Function Theory)

21 pages, 3125 KiB

Open AccessArticle

Starlike Functions of Complex Order with Respect to Symmetric Points Defined Using Higher Order Derivatives

by Kadhavoor R. Karthikeyan, Sakkarai Lakshmi, Seetharam Varadharajan, Dharmaraj Mohankumar and Elangho Umadevi

Fractal Fract. 2022, 6(2), 116; https://doi.org/10.3390/fractalfract6020116 - 17 Feb 2022

Cited by 17 | Viewed by 1926

Abstract

In this paper, we introduce and study a new subclass of multivalent functions with respect to symmetric points involving higher order derivatives. In order to unify and extend various well-known results, we have defined the class subordinate to a conic region impacted by [...] Read more.

In this paper, we introduce and study a new subclass of multivalent functions with respect to symmetric points involving higher order derivatives. In order to unify and extend various well-known results, we have defined the class subordinate to a conic region impacted by Janowski functions. We focused on conic regions when it pertained to applications of our main results. Inclusion results, subordination property and coefficient inequality of the defined class are the main results of this paper. The applications of our results which are extensions of those given in earlier works are presented here as corollaries. Full article

(This article belongs to the Special Issue New Trends in Geometric Function Theory)

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8 pages, 261 KiB

Open AccessArticle

Third Hankel Determinant for a Subclass of Univalent Functions Associated with Lemniscate of Bernoulli

by Najeeb Ullah, Irfan Ali, Sardar Muhammad Hussain, Jong-Suk Ro, Nazar Khan and Bilal Khan

Fractal Fract. 2022, 6(1), 48; https://doi.org/10.3390/fractalfract6010048 - 16 Jan 2022

Cited by 3 | Viewed by 2020

Abstract

This paper deals with a new subclass of univalent function associated with the right half of the lemniscate of Bernoulli. We find the upper bound of the Hankel determinant

H_{3} (1)

for this subclass by applying the Carlson–Shaffer operator to [...] Read more.

This paper deals with a new subclass of univalent function associated with the right half of the lemniscate of Bernoulli. We find the upper bound of the Hankel determinant

H_{3} (1)

for this subclass by applying the Carlson–Shaffer operator to it. The present work also deals with certain properties of this newly defined subclass, such as the upper bound of the Hankel determinant of order 3, coefficient estimates, etc. Full article

(This article belongs to the Special Issue New Trends in Geometric Function Theory)

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