Special Issue "Mathematical Analysis and Analytic Number Theory 2019"

A special issue of Mathematics (ISSN 2227-7390).

Deadline for manuscript submissions: closed (31 January 2020).

Special Issue Editor

Prof. Dr. Rekha Srivastava
Website
Guest Editor
Department of Mathematics and Statistics, University of Victoria, Victoria, British Columbia V8W 3R4, Canada
Interests: mathematical analysis; applied mathematics; fractional calculus and its applications
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Special Issue Information

Dear Colleagues,

Investigations involving the theory and applications of the various tools and techniques of mathematical analysis and analytic number theory are remarkably wide-spread in many diverse areas of the mathematical, biological, physical, chemical, engineering, and statistical sciences. In this Special Issue, we invite and welcome original as well as review-cum-expository research articles dealing with recent and new developments on the topics of mathematical analysis and analytic number theory as well as their multidisciplinary applications.

We look forward to receiving and editorially processing your contributions to this Special Issue.

With kind regards and many thanks in advance for your contributions.

Prof. Dr. Rekha Srivastava
Guest Editor

Manuscript Submission Information

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Keywords

  • Theory and applications of the tools and techniques of mathematical
    analysis
  • Theory and applications of the tools and techniques of analytic number
    theory
  • Applications involving special (or higher transcendental) functions
  • Applications involving fractional-order differential and
    differintegral equations
  • Applications involving q-Series and q-Polynomials
  • Applications involving special functions of mathematical physics and
    applied mathematics
  • Applications involving geometric function theory of complex analysis
  • Applications involving real analysis and operator theory

Published Papers (21 papers)

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Research

Open AccessArticle
Janowski Type q-Convex and q-Close-to-Convex Functions Associated with q-Conic Domain
Mathematics 2020, 8(3), 440; https://doi.org/10.3390/math8030440 - 18 Mar 2020
Cited by 1
Abstract
Certain new classes of q-convex and q-close to convex functions that involve the q-Janowski type functions have been defined by using the concepts of quantum (or q-) calculus as well as q-conic domain Ω k , q [ [...] Read more.
Certain new classes of q-convex and q-close to convex functions that involve the q-Janowski type functions have been defined by using the concepts of quantum (or q-) calculus as well as q-conic domain Ω k , q [ λ , α ] . This study explores some important geometric properties such as coefficient estimates, sufficiency criteria and convolution properties of these classes. A distinction of new findings with those obtained in earlier investigations is also provided, where appropriate. Full article
(This article belongs to the Special Issue Mathematical Analysis and Analytic Number Theory 2019)
Open AccessArticle
Argument and Coefficient Estimates for Certain Analytic Functions
Mathematics 2020, 8(1), 88; https://doi.org/10.3390/math8010088 - 05 Jan 2020
Abstract
The aim of the present paper is to introduce a new class G α , δ of analytic functions in the open unit disk and to study some properties associated with strong starlikeness and close-to-convexity for the class G α , δ . [...] Read more.
The aim of the present paper is to introduce a new class G α , δ of analytic functions in the open unit disk and to study some properties associated with strong starlikeness and close-to-convexity for the class G α , δ . We also consider sharp bounds of logarithmic coefficients and Fekete-Szegö functionals belonging to the class G α , δ . Moreover, we provide some topics related to the results reported here that are relevant to outcomes presented in earlier research. Full article
(This article belongs to the Special Issue Mathematical Analysis and Analytic Number Theory 2019)
Open AccessArticle
Approximation by Generalized Lupaş Operators Based on q-Integers
Mathematics 2020, 8(1), 68; https://doi.org/10.3390/math8010068 - 02 Jan 2020
Cited by 1
Abstract
The purpose of this paper is to introduce q-analogues of generalized Lupaş operators, whose construction depends on a continuously differentiable, increasing, and unbounded function ρ . Depending on the selection of q, these operators provide more flexibility in approximation and the [...] Read more.
The purpose of this paper is to introduce q-analogues of generalized Lupaş operators, whose construction depends on a continuously differentiable, increasing, and unbounded function ρ . Depending on the selection of q, these operators provide more flexibility in approximation and the convergence is at least as fast as the generalized Lupaş operators, while retaining their approximation properties. For these operators, we give weighted approximations, Voronovskaja-type theorems, and quantitative estimates for the local approximation. Full article
(This article belongs to the Special Issue Mathematical Analysis and Analytic Number Theory 2019)
Open AccessArticle
Some Applications of a New Integral Operator in q-Analog for Multivalent Functions
Mathematics 2019, 7(12), 1178; https://doi.org/10.3390/math7121178 - 03 Dec 2019
Cited by 2
Abstract
This paper introduces a new integral operator in q-analog for multivalent functions. Using as an application of this operator, we study a novel class of multivalent functions and define them. Furthermore, we present many new properties of these functions. These include distortion [...] Read more.
This paper introduces a new integral operator in q-analog for multivalent functions. Using as an application of this operator, we study a novel class of multivalent functions and define them. Furthermore, we present many new properties of these functions. These include distortion bounds, sufficiency criteria, extreme points, radius of both starlikness and convexity, weighted mean and partial sum for this newly defined subclass of multivalent functions are discussed. Various integral operators are obtained by putting particular values to the parameters used in the newly defined operator. Full article
(This article belongs to the Special Issue Mathematical Analysis and Analytic Number Theory 2019)
Open AccessArticle
A New Extension of the τ-Gauss Hypergeometric Function and Its Associated Properties
Mathematics 2019, 7(10), 996; https://doi.org/10.3390/math7100996 - 20 Oct 2019
Abstract
In this article, we define an extended version of the Pochhammer symbol and then introduce the corresponding extension of the τ-Gauss hypergeometric function. The basic properties of the extended τ-Gauss hypergeometric function, including integral and derivative formulas involving the Mellin transform [...] Read more.
In this article, we define an extended version of the Pochhammer symbol and then introduce the corresponding extension of the τ-Gauss hypergeometric function. The basic properties of the extended τ-Gauss hypergeometric function, including integral and derivative formulas involving the Mellin transform and the operators of fractional calculus, are derived. We also consider some new and known results as consequences of our proposed extension of the τ-Gauss hypergeometric function. Full article
(This article belongs to the Special Issue Mathematical Analysis and Analytic Number Theory 2019)
Open AccessArticle
The Second Hankel Determinant Problem for a Class of Bi-Close-to-Convex Functions
Mathematics 2019, 7(10), 986; https://doi.org/10.3390/math7100986 - 17 Oct 2019
Cited by 1
Abstract
The purpose of the present work is to determine a bound for the functional H 2 ( 2 ) = a 2 a 4 a 3 2 for functions belonging to the class C Σ of bi-close-to-convex functions. The main result presented [...] Read more.
The purpose of the present work is to determine a bound for the functional H 2 ( 2 ) = a 2 a 4 a 3 2 for functions belonging to the class C Σ of bi-close-to-convex functions. The main result presented here provides much improved estimation compared with the previous result by means of different proof methods than those used by others. Full article
(This article belongs to the Special Issue Mathematical Analysis and Analytic Number Theory 2019)
Open AccessArticle
Interesting Explicit Expressions of Determinants and Inverse Matrices for Foeplitz and Loeplitz Matrices
Mathematics 2019, 7(10), 939; https://doi.org/10.3390/math7100939 - 11 Oct 2019
Abstract
Foeplitz and Loeplitz matrices are Toeplitz matrices with entries being Fibonacci and Lucas numbers, respectively. In this paper, explicit expressions of determinants and inverse matrices of Foeplitz and Loeplitz matrices are studied. Specifically, the determinant of the n × n Foeplitz matrix is [...] Read more.
Foeplitz and Loeplitz matrices are Toeplitz matrices with entries being Fibonacci and Lucas numbers, respectively. In this paper, explicit expressions of determinants and inverse matrices of Foeplitz and Loeplitz matrices are studied. Specifically, the determinant of the n × n Foeplitz matrix is the ( n + 1 ) th Fibonacci number, while the inverse matrix of the n × n Foeplitz matrix is sparse and can be expressed by the nth and the ( n + 1 ) th Fibonacci number. Similarly, the determinant of the n × n Loeplitz matrix can be expressed by use of the ( n + 1 ) th Lucas number, and the inverse matrix of the n × n ( n > 3 ) Loeplitz matrix can be expressed by only seven elements with each element being the explicit expressions of Lucas numbers. Finally, several numerical examples are illustrated to show the effectiveness of our new theoretical results. Full article
(This article belongs to the Special Issue Mathematical Analysis and Analytic Number Theory 2019)
Open AccessArticle
A Study of Determinants and Inverses for Periodic Tridiagonal Toeplitz Matrices with Perturbed Corners Involving Mersenne Numbers
Mathematics 2019, 7(10), 893; https://doi.org/10.3390/math7100893 - 24 Sep 2019
Abstract
In this paper, we study periodic tridiagonal Toeplitz matrices with perturbed corners. By using some matrix transformations, the Schur complement and matrix decompositions techniques, as well as the Sherman-Morrison-Woodbury formula, we derive explicit determinants and inverses of these matrices. One feature of these [...] Read more.
In this paper, we study periodic tridiagonal Toeplitz matrices with perturbed corners. By using some matrix transformations, the Schur complement and matrix decompositions techniques, as well as the Sherman-Morrison-Woodbury formula, we derive explicit determinants and inverses of these matrices. One feature of these formulas is the connection with the famous Mersenne numbers. We also propose two algorithms to illustrate our formulas. Full article
(This article belongs to the Special Issue Mathematical Analysis and Analytic Number Theory 2019)
Open AccessArticle
Upper Bound of the Third Hankel Determinant for a Subclass of Close-to-Convex Functions Associated with the Lemniscate of Bernoulli
Mathematics 2019, 7(9), 848; https://doi.org/10.3390/math7090848 - 14 Sep 2019
Cited by 1
Abstract
In this paper, our aim is to define a new subclass of close-to-convex functions in the open unit disk U that are related with the right half of the lemniscate of Bernoulli. For this function class, we obtain the upper bound of the [...] Read more.
In this paper, our aim is to define a new subclass of close-to-convex functions in the open unit disk U that are related with the right half of the lemniscate of Bernoulli. For this function class, we obtain the upper bound of the third Hankel determinant. Various other related results are also considered. Full article
(This article belongs to the Special Issue Mathematical Analysis and Analytic Number Theory 2019)
Open AccessArticle
Transformation of Some Lambert Series and Cotangent Sums
Mathematics 2019, 7(9), 840; https://doi.org/10.3390/math7090840 - 11 Sep 2019
Abstract
By considering a contour integral of a cotangent sum, we give a simple derivation of a transformation formula of the series A ( τ , s ) = n = 1 σ s 1 ( n ) e 2 π [...] Read more.
By considering a contour integral of a cotangent sum, we give a simple derivation of a transformation formula of the series A ( τ , s ) = n = 1 σ s 1 ( n ) e 2 π i n τ for complex s under the action of the modular group on τ in the upper half plane. Some special cases directly give expressions of generalized Dedekind sums as cotangent sums. Full article
(This article belongs to the Special Issue Mathematical Analysis and Analytic Number Theory 2019)
Open AccessArticle
Euler Sums and Integral Connections
Mathematics 2019, 7(9), 833; https://doi.org/10.3390/math7090833 - 09 Sep 2019
Abstract
In this paper, we present some Euler-like sums involving partial sums of the harmonic and odd harmonic series. First, we give a brief historical account of Euler’s work on the subject followed by notations used in the body of the paper. After discussing [...] Read more.
In this paper, we present some Euler-like sums involving partial sums of the harmonic and odd harmonic series. First, we give a brief historical account of Euler’s work on the subject followed by notations used in the body of the paper. After discussing some alternating Euler sums, we investigate the connection of integrals of inverse trigonometric and hyperbolic type functions to generate many new Euler sum identities. We also give some new identities for Catalan’s constant, Apery’s constant and a fast converging identity for the famous ζ ( 2 ) constant. Full article
(This article belongs to the Special Issue Mathematical Analysis and Analytic Number Theory 2019)
Open AccessArticle
Appell-Type Functions and Chebyshev Polynomials
Mathematics 2019, 7(8), 679; https://doi.org/10.3390/math7080679 - 30 Jul 2019
Cited by 2
Abstract
In a recent article we noted that the first and second kind Cebyshev polynomials can be used to separate the real from the imaginary part of the Appell polynomials. The purpose of this article is to show that the same classic polynomials can [...] Read more.
In a recent article we noted that the first and second kind Cebyshev polynomials can be used to separate the real from the imaginary part of the Appell polynomials. The purpose of this article is to show that the same classic polynomials can also be used to separate the even part from the odd part of the Appell polynomials and of the Appell–Bessel functions. Full article
(This article belongs to the Special Issue Mathematical Analysis and Analytic Number Theory 2019)
Open AccessArticle
A Study of Multivalent q-starlike Functions Connected with Circular Domain
Mathematics 2019, 7(8), 670; https://doi.org/10.3390/math7080670 - 27 Jul 2019
Cited by 6
Abstract
Starlike functions have gained popularity both in literature and in usage over the past decade. In this paper, our aim is to examine some useful problems dealing with q-starlike functions. These include the convolution problem, sufficiency criteria, coefficient estimates, and Fekete–Szegö type [...] Read more.
Starlike functions have gained popularity both in literature and in usage over the past decade. In this paper, our aim is to examine some useful problems dealing with q-starlike functions. These include the convolution problem, sufficiency criteria, coefficient estimates, and Fekete–Szegö type inequalities for a new subfamily of analytic and multivalent functions associated with circular domain. In addition, we also define and study a Bernardi integral operator in its q-extension for multivalent functions. Furthermore, we will show that the class defined in this paper, along with the obtained results, generalizes many known works available in the literature. Full article
(This article belongs to the Special Issue Mathematical Analysis and Analytic Number Theory 2019)
Open AccessArticle
A Novel Integral Equation for the Riemann Zeta Function and Large t-Asymptotics
Mathematics 2019, 7(7), 650; https://doi.org/10.3390/math7070650 - 20 Jul 2019
Abstract
Based on the new approach to Lindelöf hypothesis recently introduced by one of the authors, we first derive a novel integral equation for the square of the absolute value of the Riemann zeta function. Then, we introduce the machinery needed to obtain an [...] Read more.
Based on the new approach to Lindelöf hypothesis recently introduced by one of the authors, we first derive a novel integral equation for the square of the absolute value of the Riemann zeta function. Then, we introduce the machinery needed to obtain an estimate for the solution of this equation. This approach suggests a substantial improvement of the current large t - asymptotics estimate for ζ 1 2 + i t . Full article
(This article belongs to the Special Issue Mathematical Analysis and Analytic Number Theory 2019)
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Open AccessArticle
The Application of Generalized Quasi-Hadamard Products of Certain Subclasses of Analytic Functions with Negative and Missing Coefficients
by En Ao and Shuhai Li
Mathematics 2019, 7(7), 620; https://doi.org/10.3390/math7070620 - 12 Jul 2019
Abstract
In this paper, we introduce a new generalized differential operator using a new generalized quasi-Hadamard product, and certain new classes of analytic functions using subordination. We obtain certain results concerning the closure properties of the generalized quasi-Hadamard products and the generalized differential operators [...] Read more.
In this paper, we introduce a new generalized differential operator using a new generalized quasi-Hadamard product, and certain new classes of analytic functions using subordination. We obtain certain results concerning the closure properties of the generalized quasi-Hadamard products and the generalized differential operators for this new subclasses of analytic functions with negative and missing coefficients. Full article
(This article belongs to the Special Issue Mathematical Analysis and Analytic Number Theory 2019)
Open AccessArticle
Some Properties and Generating Functions of Generalized Harmonic Numbers
Mathematics 2019, 7(7), 577; https://doi.org/10.3390/math7070577 - 28 Jun 2019
Abstract
In this paper, we introduce higher-order harmonic numbers and derive their relevant properties and generating functions by using an umbral-type method. We discuss the link with recent works on the subject, and show that the combinations of umbral and other techniques (such as [...] Read more.
In this paper, we introduce higher-order harmonic numbers and derive their relevant properties and generating functions by using an umbral-type method. We discuss the link with recent works on the subject, and show that the combinations of umbral and other techniques (such as the Laplace and other types of integral transforms) yield a very efficient tool to explore the properties of these numbers. Full article
(This article belongs to the Special Issue Mathematical Analysis and Analytic Number Theory 2019)
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Open AccessArticle
Some Classes of Harmonic Mapping with a Symmetric Conjecture Point Defined by Subordination
Mathematics 2019, 7(6), 548; https://doi.org/10.3390/math7060548 - 16 Jun 2019
Abstract
In the paper, we introduce some subclasses of harmonic mapping, the analytic part of which is related to general starlike (or convex) functions with a symmetric conjecture point defined by subordination. Using the conditions satisfied by the analytic part, we obtain the integral [...] Read more.
In the paper, we introduce some subclasses of harmonic mapping, the analytic part of which is related to general starlike (or convex) functions with a symmetric conjecture point defined by subordination. Using the conditions satisfied by the analytic part, we obtain the integral expressions, the coefficient estimates, distortion estimates and the growth estimates of the co-analytic part g, and Jacobian estimates, the growth estimates and covering theorem of the harmonic function f. Through the above research, the geometric properties of the classes are obtained. In particular, we draw figures of extremum functions to better reflect the geometric properties of the classes. For the first time, we introduce and obtain the properties of harmonic univalent functions with respect to symmetric conjugate points. The conclusion of this paper extends the original research. Full article
(This article belongs to the Special Issue Mathematical Analysis and Analytic Number Theory 2019)
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Open AccessArticle
A Study of Third Hankel Determinant Problem for Certain Subfamilies of Analytic Functions Involving Cardioid Domain
Mathematics 2019, 7(5), 418; https://doi.org/10.3390/math7050418 - 10 May 2019
Cited by 2
Abstract
In the present article, we consider certain subfamilies of analytic functions connected with the cardioid domain in the region of the unit disk. The purpose of this article is to investigate the estimates of the third Hankel determinant for these families. Further, the [...] Read more.
In the present article, we consider certain subfamilies of analytic functions connected with the cardioid domain in the region of the unit disk. The purpose of this article is to investigate the estimates of the third Hankel determinant for these families. Further, the same bounds have been investigated for two-fold and three-fold symmetric functions. Full article
(This article belongs to the Special Issue Mathematical Analysis and Analytic Number Theory 2019)
Open AccessArticle
Remarks on the Generalized Fractional Laplacian Operator
Mathematics 2019, 7(4), 320; https://doi.org/10.3390/math7040320 - 29 Mar 2019
Cited by 2
Abstract
The fractional Laplacian, also known as the Riesz fractional derivative operator, describes an unusual diffusion process due to random displacements executed by jumpers that are able to walk to neighbouring or nearby sites, as well as perform excursions to remote sites by way [...] Read more.
The fractional Laplacian, also known as the Riesz fractional derivative operator, describes an unusual diffusion process due to random displacements executed by jumpers that are able to walk to neighbouring or nearby sites, as well as perform excursions to remote sites by way of Lévy flights. The fractional Laplacian has many applications in the boundary behaviours of solutions to differential equations. The goal of this paper is to investigate the half-order Laplacian operator ( Δ ) 1 2 in the distributional sense, based on the generalized convolution and Temple’s delta sequence. Several interesting examples related to the fractional Laplacian operator of order 1 / 2 are presented with applications to differential equations, some of which cannot be obtained in the classical sense by the standard definition of the fractional Laplacian via Fourier transform. Full article
(This article belongs to the Special Issue Mathematical Analysis and Analytic Number Theory 2019)
Open AccessArticle
On Geometric Properties of Normalized Hyper-Bessel Functions
Mathematics 2019, 7(4), 316; https://doi.org/10.3390/math7040316 - 28 Mar 2019
Cited by 1
Abstract
In this paper, the normalized hyper-Bessel functions are studied. Certain sufficient conditions are determined such that the hyper-Bessel functions are close-to-convex, starlike and convex in the open unit disc. We also study the Hardy spaces of hyper-Bessel functions. Full article
(This article belongs to the Special Issue Mathematical Analysis and Analytic Number Theory 2019)
Open AccessArticle
Some Reciprocal Classes of Close-to-Convex and Quasi-Convex Analytic Functions
Mathematics 2019, 7(4), 309; https://doi.org/10.3390/math7040309 - 27 Mar 2019
Cited by 1
Abstract
The present paper comprises the study of certain functions which are analytic and defined in terms of reciprocal function. The reciprocal classes of close-to-convex functions and quasi-convex functions are defined and studied. Various interesting properties, such as sufficiency criteria, coefficient estimates, distortion results, [...] Read more.
The present paper comprises the study of certain functions which are analytic and defined in terms of reciprocal function. The reciprocal classes of close-to-convex functions and quasi-convex functions are defined and studied. Various interesting properties, such as sufficiency criteria, coefficient estimates, distortion results, and a few others, are investigated for these newly defined sub-classes. Full article
(This article belongs to the Special Issue Mathematical Analysis and Analytic Number Theory 2019)
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