Research

13 pages, 368 KiB

Open AccessArticle

Unified Integrals of Generalized Mittag–Leffler Functions and Their Graphical Numerical Investigation

by Nabiullah Khan, Mohammad Iqbal Khan, Talha Usman, Kamsing Nonlaopon and Shrideh Al-Omari

Symmetry 2022, 14(5), 869; https://0-doi-org.brum.beds.ac.uk/10.3390/sym14050869 - 23 Apr 2022

Viewed by 1254

In this article, we obtain certain finite integrals concerning generalized Mittag–Leffler functions, which are evaluated in terms of the generalized Fox–Wright function. The integrals of concern are unified in nature and thereby yield some new integral formulas as special cases. Moreover, we numerically [...] Read more.

In this article, we obtain certain finite integrals concerning generalized Mittag–Leffler functions, which are evaluated in terms of the generalized Fox–Wright function. The integrals of concern are unified in nature and thereby yield some new integral formulas as special cases. Moreover, we numerically compute some integrals using the Gaussian quadrature formula and draw a comparison with the main integrals by using graphical numerical investigation. Full article

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

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21 pages, 2599 KiB

Open AccessArticle

Non-Separable Linear Canonical Wavelet Transform

by Hari M. Srivastava, Firdous A. Shah, Tarun K. Garg, Waseem Z. Lone and Huzaifa L. Qadri

Symmetry 2021, 13(11), 2182; https://0-doi-org.brum.beds.ac.uk/10.3390/sym13112182 - 15 Nov 2021

Cited by 18 | Viewed by 1481

Abstract

This study aims to achieve an efficient time-frequency representation of higher-dimensional signals by introducing the notion of a non-separable linear canonical wavelet transform in

L^{2} (R^{n})

. The preliminary analysis encompasses the derivation of fundamental properties of the novel [...] Read more.

This study aims to achieve an efficient time-frequency representation of higher-dimensional signals by introducing the notion of a non-separable linear canonical wavelet transform in

L^{2} (R^{n})

. The preliminary analysis encompasses the derivation of fundamental properties of the novel integral transform including the orthogonality relation, inversion formula, and the range theorem. To extend the scope of the study, we formulate several uncertainty inequalities, including the Heisenberg’s, logarithmic, and Nazorav’s inequalities for the proposed transform in the linear canonical domain. The obtained results are reinforced with illustrative examples. Full article

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

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8 pages, 251 KiB

Open AccessArticle

A Double Logarithmic Transform Involving the Exponential and Polynomial Functions Expressed in Terms of the Hurwitz–Lerch Zeta Function

by Robert Reynolds and Allan Stauffer

Symmetry 2021, 13(11), 1983; https://0-doi-org.brum.beds.ac.uk/10.3390/sym13111983 - 20 Oct 2021

Cited by 2 | Viewed by 1070

Abstract

The object of this paper is to derive a double integral in terms of the Hurwitz–Lerch zeta function. Almost all Hurwitz–Lerch zeta functions have an asymmetrical zero-distribution. Special cases are evaluated in terms of fundamental constants. All the results in this work are [...] Read more.

The object of this paper is to derive a double integral in terms of the Hurwitz–Lerch zeta function. Almost all Hurwitz–Lerch zeta functions have an asymmetrical zero-distribution. Special cases are evaluated in terms of fundamental constants. All the results in this work are new. Full article

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

11 pages, 296 KiB

Open AccessArticle

Double Integral of the Product of the Exponential of an Exponential Function and a Polynomial Expressed in Terms of the Lerch Function

by Robert Reynolds and Allan Stauffer

Symmetry 2021, 13(10), 1962; https://0-doi-org.brum.beds.ac.uk/10.3390/sym13101962 - 18 Oct 2021

Viewed by 1334

Abstract

In this work, the authors use their contour integral method to derive an application of the Fourier integral theorem given by [...] Read more.

In this work, the authors use their contour integral method to derive an application of the Fourier integral theorem given by

\int_{- \infty}^{\infty} \int_{- \infty}^{\infty} e^{m x - m y - e^{x} - e^{y} + y} {(\log (a) + x - y)}^{k} d x d y

in terms of the Lerch function. This integral formula is then used to derive closed solutions in terms of fundamental constants and special functions. Almost all Lerch functions have an asymmetrical zero distribution. There are some useful results relating double integrals of certain kinds of functions to ordinary integrals for which we know no general reference. Thus, a table of integral pairs is given for interested readers. All of the results in this work are new. Full article

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

24 pages, 371 KiB

Open AccessArticle

Fuzzy Mixed Variational-like and Integral Inequalities for Strongly Preinvex Fuzzy Mappings

by Muhammad Bilal Khan, Hari Mohan Srivastava, Pshtiwan Othman Mohammed and Juan L. G. Guirao

Symmetry 2021, 13(10), 1816; https://0-doi-org.brum.beds.ac.uk/10.3390/sym13101816 - 29 Sep 2021

Cited by 9 | Viewed by 1322

Abstract

It is a familiar fact that convex and non-convex fuzzy mappings play a critical role in the study of fuzzy optimization. Due to the behavior of its definition, the idea of convexity plays a significant role in the subject of inequalities. The concepts [...] Read more.

It is a familiar fact that convex and non-convex fuzzy mappings play a critical role in the study of fuzzy optimization. Due to the behavior of its definition, the idea of convexity plays a significant role in the subject of inequalities. The concepts of convexity and symmetry have a tight connection. We may use whatever we learn from one to the other, thanks to the significant correlation that has developed between both in recent years. Our aim is to consider a new class of fuzzy mappings (FMs) known as strongly preinvex fuzzy mappings (strongly preinvex-FMs) on the invex set. These FMs are more general than convex fuzzy mappings (convex-FMs) and preinvex fuzzy mappings (preinvex-FMs), and when generalized differentiable (briefly, G-differentiable), strongly preinvex-FMs are strongly invex fuzzy mappings (strongly invex-FMs). Some new relationships among various concepts of strongly preinvex-FMs are established and verified with the support of some useful examples. We have also shown that optimality conditions of G-differentiable strongly preinvex-FMs and the fuzzy functional, which is the sum of G-differentiable preinvex-FMs and non G-differentiable strongly preinvex-FMs, can be distinguished by strongly fuzzy variational-like inequalities and strongly fuzzy mixed variational-like inequalities, respectively. In the end, we have established and verified a strong relationship between the Hermite–Hadamard inequality and strongly preinvex-FM. Several exceptional cases are also discussed. These inequalities are a very interesting outcome of our main results and appear to be new ones. The results in this research can be seen as refinements and improvements to previously published findings. Full article

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

9 pages, 249 KiB

Open AccessArticle

A Quadruple Definite Integral Expressed in Terms of the Lerch Function

by Robert Reynolds and Allan Stauffer

Symmetry 2021, 13(9), 1638; https://0-doi-org.brum.beds.ac.uk/10.3390/sym13091638 - 06 Sep 2021

Viewed by 1264

Abstract

A quadruple integral involving the logarithmic, exponential and polynomial functions is derived in terms of the Lerch function. Special cases of this integral are evaluated in terms of special functions and fundamental constants. Almost all Lerch functions have an asymmetrical zero-distribution. The majority [...] Read more.

A quadruple integral involving the logarithmic, exponential and polynomial functions is derived in terms of the Lerch function. Special cases of this integral are evaluated in terms of special functions and fundamental constants. Almost all Lerch functions have an asymmetrical zero-distribution. The majority of the results in this work are new. Full article

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

12 pages, 272 KiB

Open AccessArticle

On New Generalized Dunkel Type Integral Inequalities with Applications

by Dong-Sheng Wang, Huan-Nan Shi, Chun-Ru Fu and Wei-Shih Du

Symmetry 2021, 13(9), 1576; https://0-doi-org.brum.beds.ac.uk/10.3390/sym13091576 - 27 Aug 2021

Viewed by 1463

Abstract

In this paper, by applying majorization theory, we study the Schur convexity of functions related to Dunkel integral inequality. We establish some new generalized Dunkel type integral inequalities and their applications to inequality theory. Full article

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

14 pages, 303 KiB

Open AccessArticle

A Subclass of Multivalent Janowski Type q-Starlike Functions and Its Consequences

by Qiuxia Hu, Hari M. Srivastava, Bakhtiar Ahmad, Nazar Khan, Muhammad Ghaffar Khan, Wali Khan Mashwani and Bilal Khan

Symmetry 2021, 13(7), 1275; https://0-doi-org.brum.beds.ac.uk/10.3390/sym13071275 - 16 Jul 2021

Cited by 25 | Viewed by 1743

Abstract

In this article, by utilizing the theory of quantum (or q-) calculus, we define a new subclass of analytic and multivalent (or p-valent) functions class

A_{p}

, where class

A_{p}

is invariant (or symmetric) under rotations. The well-known class [...] Read more.

In this article, by utilizing the theory of quantum (or q-) calculus, we define a new subclass of analytic and multivalent (or p-valent) functions class

A_{p}

, where class

A_{p}

is invariant (or symmetric) under rotations. The well-known class of Janowski functions are used with the help of the principle of subordination between analytic functions in order to define this subclass of analytic and p-valent functions. This function class generalizes various other subclasses of analytic functions, not only in classical Geometric Function Theory setting, but also some q-analogue of analytic multivalent function classes. We study and investigate some interesting properties such as sufficiency criteria, coefficient bounds, distortion problem, growth theorem, radii of starlikeness and convexity for this newly-defined class. Other properties such as those involving convex combination are also discussed for these functions. In the concluding part of the article, we have finally given the well-demonstrated fact that the results presented in this article can be obtained for the

(p, q)

-variations, by making some straightforward simplification and will be an inconsequential exercise simply because the additional parameter

p

is obviously unnecessary. Full article

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

11 pages, 262 KiB

Open AccessArticle

Properties of Certain Subclass of Meromorphic Multivalent Functions Associated with q-Difference Operator

by Cai-Mei Yan, Rekha Srivastava and Jin-Lin Liu

Symmetry 2021, 13(6), 1035; https://0-doi-org.brum.beds.ac.uk/10.3390/sym13061035 - 08 Jun 2021

Cited by 4 | Viewed by 1248

Abstract

A new subclass

Σ_{p, q} (α, A, B)

of meromorphic multivalent functions is defined by means of a q-difference operator. Some properties of the functions in this new subclass, such as sufficient and necessary conditions, coefficient [...] Read more.

A new subclass

Σ_{p, q} (α, A, B)

of meromorphic multivalent functions is defined by means of a q-difference operator. Some properties of the functions in this new subclass, such as sufficient and necessary conditions, coefficient estimates, growth and distortion theorems, radius of starlikeness and convexity, partial sums and closure theorems, are investigated. Full article

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

24 pages, 351 KiB

Open AccessArticle

A Spectral Calculus for Lorentz Invariant Measures on Minkowski Space

by John Mashford

Symmetry 2020, 12(10), 1696; https://0-doi-org.brum.beds.ac.uk/10.3390/sym12101696 - 15 Oct 2020

Cited by 4 | Viewed by 2289

Abstract

This paper presents a spectral calculus for computing the spectra of causal Lorentz invariant Borel complex measures on Minkowski space, thereby enabling one to compute their densities with respect to Lebesque measure. The spectra of certain elementary convolutions involving Feynman propagators of scalar [...] Read more.

This paper presents a spectral calculus for computing the spectra of causal Lorentz invariant Borel complex measures on Minkowski space, thereby enabling one to compute their densities with respect to Lebesque measure. The spectra of certain elementary convolutions involving Feynman propagators of scalar particles are computed. It is proved that the convolution of arbitrary causal Lorentz invariant Borel complex measures exists and the product of such measures exists in a wide class of cases. Techniques for their computation in terms of their spectral representation are presented. Full article

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

17 pages, 327 KiB

Open AccessArticle

An Upper Bound of the Third Hankel Determinant for a Subclass of q-Starlike Functions Associated with k-Fibonacci Numbers

by Muhammad Shafiq, Hari M. Srivastava, Nazar Khan, Qazi Zahoor Ahmad, Maslina Darus and Samiha Kiran

Symmetry 2020, 12(6), 1043; https://0-doi-org.brum.beds.ac.uk/10.3390/sym12061043 - 22 Jun 2020

Cited by 35 | Viewed by 2053

Abstract

In this paper, we use q-derivative operator to define a new class of q-starlike functions associated with k-Fibonacci numbers. This newly defined class is a subclass of class

A

of normalized analytic functions, where class

A

is invariant (or symmetric) [...] Read more.

In this paper, we use q-derivative operator to define a new class of q-starlike functions associated with k-Fibonacci numbers. This newly defined class is a subclass of class

A

of normalized analytic functions, where class

A

is invariant (or symmetric) under rotations. For this function class we obtain an upper bound of the third Hankel determinant. Full article

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

19 pages, 792 KiB

Open AccessArticle

Difference of Some Positive Linear Approximation Operators for Higher-Order Derivatives

by Vijay Gupta, Ana Maria Acu and Hari Mohan Srivastava

Symmetry 2020, 12(6), 915; https://0-doi-org.brum.beds.ac.uk/10.3390/sym12060915 - 02 Jun 2020

Cited by 21 | Viewed by 1949

Abstract

In the present paper, we deal with some general estimates for the difference of operators which are associated with different fundamental functions. In order to exemplify the theoretical results presented in (for example) Theorem 2, we provide the estimates of the differences between [...] Read more.

In the present paper, we deal with some general estimates for the difference of operators which are associated with different fundamental functions. In order to exemplify the theoretical results presented in (for example) Theorem 2, we provide the estimates of the differences between some of the most representative operators used in Approximation Theory in especially the difference between the Baskakov and the Szász–Mirakyan operators, the difference between the Baskakov and the Szász–Mirakyan–Baskakov operators, the difference of two genuine-Durrmeyer type operators, and the difference of the Durrmeyer operators and the Lupaş–Durrmeyer operators. By means of illustrative numerical examples, we show that, for particular cases, our result improves the estimates obtained by using the classical result of Shisha and Mond. We also provide the symmetry aspects of some of these approximations operators which we have studied in this paper. Full article

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

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