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A special issue of Symmetry (ISSN 2073-8994). This special issue belongs to the section "Mathematics".

Deadline for manuscript submissions: closed (31 January 2020) | Viewed by 41794

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Prof. Dr. Hari Mohan Srivastava

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Department of Mathematics and Statistics, University of Victoria, Victoria, BC V8W 3R4, Canada
Interests: real and complex analysis; fractional calculus and its applications; integral equations and transforms; higher transcendental functions and their applications; q-series and q-polynomials; analytic number theory; analytic and geometric Inequalities; probability and statistics; inventory modeling and optimization
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Special Issue Information

Dear Colleagues,

The theory and applications of integral transformations and associated operational calculus are remarkably wide-spread in many diverse areas of the mathematical, physical, chemical, engineering and statistical sciences. In this Special Issue, we invite and welcome review, expository and original research articles dealing with the recent advances on the topics of integral transformations and operational calculus as well as their multidisciplinary applications involving their symmetry properties and characteristics.

Prof. Dr. H. M. Srivastava
Guest Editor

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Keywords

Integral Transformations and Integral Equations as well as Other Related Operators Including Their Symmetry Properties and Characteristics
Applications Involving Mathematical (or Higher Transcendental) Functions Including Their Symmetry Properties and Characteristics
Applications Involving Fractional-Order Differential and Differintegral Equations and Their Associated Symmetry
Applications Involving Symmetrical Aspect of Geometric Function Theory of Complex Analysis
Applications Involving q-Series and q-Polynomials and Their Associated Symmetry
Applications Involving Special Functions of Mathematical Physics and Applied Mathematics and Their Symmetrical Aspect
Applications Involving Analytic Number Theory and Symmetry

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Published Papers (15 papers)

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Editorial

Jump to: Research

4 pages, 181 KiB

Open AccessEditorial

Special Issue of Symmetry: “Integral Transformations, Operational Calculus and Their Applications”

by Hari Mohan Srivastava

Symmetry 2020, 12(7), 1169; https://doi.org/10.3390/sym12071169 - 14 Jul 2020

Cited by 1 | Viewed by 1656

Abstract

This Special Issue consists of a total of 14 accepted submissions (including several invited feature articles) to the Special Issue of the MDPI’s journal, Symmetry on the general subject-area of “Integral Transformations, Operational Calculus and Their Applications” from different parts of the world [...] Read more.

(This article belongs to the Special Issue Integral Transformations, Operational Calculus and Their Applications)

Research

Jump to: Editorial

16 pages, 310 KiB

Open AccessArticle

Construction of Weights for Positive Integral Operators

by Ron Kerman

Symmetry 2020, 12(6), 1004; https://doi.org/10.3390/sym12061004 - 12 Jun 2020

Cited by 1 | Viewed by 1533

Abstract

Let

(X, M, μ)

be a

σ

-finite measure space and denote by

P (X)

the

μ

-measurable functions

f : X \to [0, \infty]

f < \infty

μ

ae. Suppose [...] Read more.

Let

(X, M, μ)

be a

σ

-finite measure space and denote by

P (X)

the

μ

-measurable functions

f : X \to [0, \infty]

f < \infty

μ

ae. Suppose

K : X \times X \to [0, \infty)

μ \times μ

-measurable and define the mutually transposed operators T and

T^{'}

P (X)

(T f) (x) = \int_{X} K (x, y) f (y) d μ (y)

and

(T^{'} g) (y) = \int_{X} K (x, y) g (x) d μ (x), f, g \in P (X), x, y \in X .

Our interest is in inequalities involving a fixed (weight) function

w \in P (X)

and an index

p \in (1, \infty)

such that: (*):

\int_{X} {[w (x) (T f) (x)]}^{p} d μ (x) ≲ C \int_{X} {[w (y) f (y)]}^{p} d μ (y) .

The constant

C > 1

is to be independent of

f \in P (X) .

We wish to construct all w for which (*) holds. Considerations concerning Schur’s Lemma ensure that every such w is within constant multiples of expressions of the form

ϕ_{1}^{1 / p - 1} ϕ_{2}^{1 / p},

where

ϕ_{1}, ϕ_{2} \in P (X)

satisfy

T ϕ_{1} \leq C_{1} ϕ_{1} and T^{'} ϕ_{2} \leq C_{2} ϕ_{2} .

Our fundamental result shows that the

ϕ_{1}

and

ϕ_{2}

above are within constant multiples of (**):

ψ_{1} + \sum_{j = 1}^{\infty} E^{- j} T^{(j)} ψ_{1} and ψ_{2} + \sum_{j = 1}^{\infty} E^{- j} {T^{'}}^{(j)} ψ_{2}

respectively; here

ψ_{1}, ψ_{2} \in P (X)

E > 1

and

T^{(j)}, T^{'}^{(j)}

are the jth iterates of T and

T^{'}

. This result is explored in the context of Poisson, Bessel and Gauss–Weierstrass means and of Hardy averaging operators. All but the Hardy averaging operators are defined through symmetric kernels

K (x, y) = K (y, x)

, so that

T^{'} = T

. This means that only the first series in (**) needs to be studied. Full article

(This article belongs to the Special Issue Integral Transformations, Operational Calculus and Their Applications)

11 pages, 251 KiB

Open AccessArticle

Some Improvements of the Hermite–Hadamard Integral Inequality

by Slavko Simić and Bandar Bin-Mohsin

Symmetry 2020, 12(1), 117; https://doi.org/10.3390/sym12010117 - 7 Jan 2020

Cited by 3 | Viewed by 1967

Abstract

We propose several improvements of the Hermite–Hadamard inequality in the form of linear combination of its end-points and establish best possible constants. Improvements of a second order for the class

Φ (I)

with applications in Analysis and Theory of Means are [...] Read more.

Φ (I)

with applications in Analysis and Theory of Means are also given. Full article

(This article belongs to the Special Issue Integral Transformations, Operational Calculus and Their Applications)

14 pages, 778 KiB

Open AccessArticle

The Principle of Differential Subordination and Its Application to Analytic and p-Valent Functions Defined by a Generalized Fractional Differintegral Operator

by Nak Eun Cho, Mohamed Kamal Aouf and Rekha Srivastava

Symmetry 2019, 11(9), 1083; https://doi.org/10.3390/sym11091083 - 29 Aug 2019

Cited by 12 | Viewed by 2452

Abstract

A useful family of fractional derivative and integral operators plays a crucial role on the study of mathematics and applied science. In this paper, we introduce an operator defined on the family of analytic functions in the open unit disk by using the generalized fractional derivative and integral operator with convolution. For this operator, we study the subordination-preserving properties and their dual problems. Differential sandwich-type results for this operator are also investigated. Full article

(This article belongs to the Special Issue Integral Transformations, Operational Calculus and Their Applications)

15 pages, 272 KiB

Open AccessArticle

Class of Analytic Functions Defined by q-Integral Operator in a Symmetric Region

by Lei Shi, Mohsan Raza, Kashif Javed, Saqib Hussain and Muhammad Arif

Symmetry 2019, 11(8), 1042; https://doi.org/10.3390/sym11081042 - 13 Aug 2019

Cited by 9 | Viewed by 3294

Abstract

The aim of the present paper is to introduce a new class of analytic functions by using a q-integral operator in the conic region. It is worth mentioning that these regions are symmetric along the real axis. We find the coefficient estimates, the Fekete–Szegö inequality, the sufficiency criteria, the distortion result, and the Hankel determinant problem for functions in this class. Furthermore, we study the inverse coefficient estimates for functions in this class. Full article

(This article belongs to the Special Issue Integral Transformations, Operational Calculus and Their Applications)

12 pages, 257 KiB

Open AccessArticle

New Symmetric Differential and Integral Operators Defined in the Complex Domain

by Rabha W. Ibrahim and Maslina Darus

Symmetry 2019, 11(7), 906; https://doi.org/10.3390/sym11070906 - 12 Jul 2019

Cited by 19 | Viewed by 2622

Abstract

The symmetric differential operator is a generalization operating of the well-known ordinary derivative. These operators have advantages in boundary value problems, statistical studies and spectral theory. In this effort, we introduce a new symmetric differential operator (SDO) and its integral in the open unit disk. This operator is a generalization of the Sàlàgean differential operator. Our study is based on geometric function theory and its applications in the open unit disk. We formulate new classes of analytic functions using SDO depending on the symmetry properties. Moreover, we define a linear combination operator containing SDO and the Ruscheweyh derivative. We illustrate some inclusion properties and other inequalities involving SDO and its integral. Full article

(This article belongs to the Special Issue Integral Transformations, Operational Calculus and Their Applications)

7 pages, 258 KiB

Open AccessArticle

Sufficiency Criterion for A Subfamily of Meromorphic Multivalent Functions of Reciprocal Order with Respect to Symmetric Points

by Shahid Mahmood, Gautam Srivastava, Hari Mohan Srivastava, Eman S.A. Abujarad, Muhammad Arif and Fazal Ghani

Symmetry 2019, 11(6), 764; https://doi.org/10.3390/sym11060764 - 5 Jun 2019

Cited by 10 | Viewed by 2362

Abstract

In the present research paper, our aim is to introduce a new subfamily of meromorphic p-valent (multivalent) functions. Moreover, we investigate sufficiency criterion for such defined family. Full article

(This article belongs to the Special Issue Integral Transformations, Operational Calculus and Their Applications)

14 pages, 331 KiB

Open AccessArticle

Geometric Properties of Certain Classes of Analytic Functions Associated with a q-Integral Operator

by Shahid Mahmood, Nusrat Raza, Eman S. A. AbuJarad, Gautam Srivastava, H. M. Srivastava and Sarfraz Nawaz Malik

Symmetry 2019, 11(5), 719; https://doi.org/10.3390/sym11050719 - 27 May 2019

Cited by 69 | Viewed by 3778

Abstract

This article presents certain families of analytic functions regarding q-starlikeness and q-convexity of complex order

γ (γ \in C \ \{0\})

. This introduced a q-integral operator and certain subclasses of the newly introduced classes are defined by [...] Read more.

This article presents certain families of analytic functions regarding q-starlikeness and q-convexity of complex order

γ (γ \in C \ \{0\})

. This introduced a q-integral operator and certain subclasses of the newly introduced classes are defined by using this q-integral operator. Coefficient bounds for these subclasses are obtained. Furthermore, the

(δ, q)

-neighborhood of analytic functions are introduced and the inclusion relations between the

(δ, q)

-neighborhood and these subclasses of analytic functions are established. Moreover, the generalized hyper-Bessel function is defined, and application of main results are discussed. Full article

(This article belongs to the Special Issue Integral Transformations, Operational Calculus and Their Applications)

16 pages, 274 KiB

Open AccessArticle

Existence of Solution for Non-Linear Functional Integral Equations of Two Variables in Banach Algebra

by Hari M. Srivastava, Anupam Das, Bipan Hazarika and S. A. Mohiuddine

Symmetry 2019, 11(5), 674; https://doi.org/10.3390/sym11050674 - 16 May 2019

Cited by 26 | Viewed by 2528

Abstract

The aim of this article is to establish the existence of the solution of non-linear functional integral equations [...] Read more.

The aim of this article is to establish the existence of the solution of non-linear functional integral equations

x (l, h) = (U (l, h, x (l, h)) + F (l, h, \int_{0}^{l} \int_{0}^{h} P (l, h, r, u, x (r, u)) d r d u, x (l, h))) \times G (l, h, \int_{0}^{a} \int_{0}^{a} Q (l, h, r, u, x (r, u)) d r d u, x (l, h))

of two variables, which is of the form of two operators in the setting of Banach algebra

C ([0, a] \times [0, a]), a > 0 .

Our methodology relies upon the measure of noncompactness related to the fixed point hypothesis. We have used the measure of noncompactness on

C ([0, a] \times [0, a])

and a fixed point theorem, which is a generalization of Darbo’s fixed point theorem for the product of operators. We additionally illustrate our outcome with the help of an interesting example. Full article

(This article belongs to the Special Issue Integral Transformations, Operational Calculus and Their Applications)

12 pages, 293 KiB

Open AccessArticle

Generalized Mittag-Leffler Input Stability of the Fractional Differential Equations

by Ndolane Sene and Gautam Srivastava

Symmetry 2019, 11(5), 608; https://doi.org/10.3390/sym11050608 - 1 May 2019

Cited by 38 | Viewed by 3271

Abstract

The behavior of the analytical solutions of the fractional differential equation described by the fractional order derivative operators is the main subject in many stability problems. In this paper, we present a new stability notion of the fractional differential equations with exogenous input. Motivated by the success of the applications of the Mittag-Leffler functions in many areas of science and engineering, we present our work here. Applications of Mittag-Leffler functions in certain areas of physical and applied sciences are also very common. During the last two decades, this class of functions has come into prominence after about nine decades of its discovery by a Swedish Mathematician Mittag-Leffler, due to the vast potential of its applications in solving the problems of physical, biological, engineering, and earth sciences, to name just a few. Moreover, we propose the generalized Mittag-Leffler input stability conditions. The left generalized fractional differential equation has been used to help create this new notion. We investigate in depth here the Lyapunov characterizations of the generalized Mittag-Leffler input stability of the fractional differential equation with input. Full article

(This article belongs to the Special Issue Integral Transformations, Operational Calculus and Their Applications)

14 pages, 838 KiB

Open AccessArticle

An Investigation of the Third Hankel Determinant Problem for Certain Subfamilies of Univalent Functions Involving the Exponential Function

by Lei Shi, Hari Mohan Srivastava, Muhammad Arif, Shehzad Hussain and Hassan Khan

Symmetry 2019, 11(5), 598; https://doi.org/10.3390/sym11050598 - 26 Apr 2019

Cited by 71 | Viewed by 3887

Abstract

In the current article, we consider certain subfamilies

S_{e}^{*}

and

C_{e}

of univalent functions associated with exponential functions which are symmetric along real axis in the region of open unit disk. For these classes our aim is to find the [...] Read more.

In the current article, we consider certain subfamilies

S_{e}^{*}

and

C_{e}

of univalent functions associated with exponential functions which are symmetric along real axis in the region of open unit disk. For these classes our aim is to find the bounds of Hankel determinant of order three. Further, the estimate of third Hankel determinant for the family

S_{e}^{*}

in this work improve the bounds which was investigated recently. Moreover, the same bounds have been investigated for 2-fold symmetric and 3-fold symmetric functions. Full article

(This article belongs to the Special Issue Integral Transformations, Operational Calculus and Their Applications)

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12 pages, 243 KiB

Open AccessArticle

On Periodic Solutions of Delay Differential Equations with Impulses

by Mostafa Bachar

Symmetry 2019, 11(4), 523; https://doi.org/10.3390/sym11040523 - 11 Apr 2019

Cited by 5 | Viewed by 2311

Abstract

The purpose of this paper is to study the nonlinear distributed delay differential equations with impulses effects in the vectorial regulated Banach spaces

R ([- r, 0], R^{n}) .

The existence of the periodic solution of [...] Read more.

The purpose of this paper is to study the nonlinear distributed delay differential equations with impulses effects in the vectorial regulated Banach spaces

R ([- r, 0], R^{n}) .

The existence of the periodic solution of impulsive delay differential equations is obtained by using the Schäffer fixed point theorem in regulated space

R ([- r, 0], R^{n}) .

Full article

(This article belongs to the Special Issue Integral Transformations, Operational Calculus and Their Applications)

17 pages, 476 KiB

Open AccessArticle

Collaborative Content Downloading in VANETs with Fuzzy Comprehensive Evaluation

by Tianyu Huang, Xijuan Guo, Yue Zhang and Zheng Chang

Symmetry 2019, 11(4), 502; https://doi.org/10.3390/sym11040502 - 6 Apr 2019

Cited by 4 | Viewed by 2653

Abstract

Vehicle collaborative content downloading has become a hotspot in current vehicular ad-hoc network (VANET) research. However, in reality, the highly dynamic nature of VANET makes users lose resources easily, and the transmission of invalid segment data also wastes valuable bandwidth and storage of the users’ vehicles. In addition, the individual need of each customer vehicle should also be taken into consideration when selecting an agent vehicle for downloading. In this paper, a novel scheme is proposed for vehicle selection in the download of cooperative content from the Internet, by considering the basic evaluation information of the vehicle. To maximize the overall throughput of the system, a collaborative content downloading algorithm is proposed, which is based on fuzzy evaluation and a customer’s own expectations, in order to solve the problems of agent vehicle selection. With the premise of ensuring successful downloading and the selection preferences of customer vehicles, linear programming is used to optimize the distribution of agent vehicles and maximize customer’s satisfaction. Simulation results show that the proposed scheme works well in terms of average quality of service, average bandwidth efficiency, failure frequency, and average consumption. Full article

(This article belongs to the Special Issue Integral Transformations, Operational Calculus and Their Applications)

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20 pages, 339 KiB

Open AccessArticle

Statistically and Relatively Modular Deferred-Weighted Summability and Korovkin-Type Approximation Theorems

by Hari Mohan Srivastava, Bidu Bhusan Jena, Susanta Kumar Paikray and Umakanta Misra

Symmetry 2019, 11(4), 448; https://doi.org/10.3390/sym11040448 - 31 Mar 2019

Cited by 36 | Viewed by 2515

Abstract

The concept of statistically deferred-weighted summability was recently studied by Srivastava et al. (Math. Methods Appl. Sci. 41 (2018), 671–683). The present work is concerned with the deferred-weighted summability mean in various aspects defined over a modular space associated with a generalized double sequence of functions. In fact, herein we introduce the idea of relatively modular deferred-weighted statistical convergence and statistically as well as relatively modular deferred-weighted summability for a double sequence of functions. With these concepts and notions in view, we establish a theorem presenting a connection between them. Moreover, based upon our methods, we prove an approximation theorem of the Korovkin type for a double sequence of functions on a modular space and demonstrate that our theorem effectively extends and improves most (if not all) of the previously existing results. Finally, an illustrative example is provided here by the generalized bivariate Bernstein–Kantorovich operators of double sequences of functions in order to demonstrate that our established theorem is stronger than its traditional and statistical versions. Full article

(This article belongs to the Special Issue Integral Transformations, Operational Calculus and Their Applications)

22 pages, 338 KiB

Open AccessFeature PaperArticle

Construction of Stancu-Type Bernstein Operators Based on Bézier Bases with Shape Parameter λ

by Hari M. Srivastava, Faruk Özger and S. A. Mohiuddine

Symmetry 2019, 11(3), 316; https://doi.org/10.3390/sym11030316 - 2 Mar 2019

Cited by 78 | Viewed by 3826

Abstract

We construct Stancu-type Bernstein operators based on Bézier bases with shape parameter

λ \in [- 1, 1]

and calculate their moments. The uniform convergence of the operator and global approximation result by means of Ditzian-Totik modulus of smoothness are established. [...] Read more.

We construct Stancu-type Bernstein operators based on Bézier bases with shape parameter

λ \in [- 1, 1]

and calculate their moments. The uniform convergence of the operator and global approximation result by means of Ditzian-Totik modulus of smoothness are established. Also, we establish the direct approximation theorem with the help of second order modulus of smoothness, calculate the rate of convergence via Lipschitz-type function, and discuss the Voronovskaja-type approximation theorems. Finally, in the last section, we construct the bivariate case of Stancu-type

λ

-Bernstein operators and study their approximation behaviors. Full article

(This article belongs to the Special Issue Integral Transformations, Operational Calculus and Their Applications)

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