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17 pages, 354 KiB

Open AccessArticle

On Voigt-Type Functions Extended by Neumann Function in Kernels and Their Bounding Inequalities

by Rakesh K. Parmar, Tibor K. Pogány and Uthara Sabu

Axioms 2024, 13(8), 534; https://doi.org/10.3390/axioms13080534 - 7 Aug 2024

Viewed by 654

The principal aim of this paper is to introduce the extended Voigt-type function

V_{μ, ν} (x, y)

and its counterpart extension

W_{μ, ν} (x, y)

, involving the Neumann function

Y_{ν}

in [...] Read more.

The principal aim of this paper is to introduce the extended Voigt-type function

V_{μ, ν} (x, y)

and its counterpart extension

W_{μ, ν} (x, y)

, involving the Neumann function

Y_{ν}

in the kernel of the representing integral. The newly defined integral reduces to the classical Voigt functions

K (x, y)

and

L (x, y)

, and to their generalizations by Srivastava and Miller, by the unification of Klusch. Following an approach by Srivastava and Pogány, we also present the multiparameter and multivariable versions

V_{μ, ν}^{(r)} (x, y), W_{μ, ν}^{(r)} (x, y)

and the r positive integer of the initial extensions

V_{μ, ν} (x, y), W_{μ, ν} (x, y)

. Several computable series expansions are obtained for the discussed Voigt-type functions in terms of Humbert confluent hypergeometric functions

Ψ_{2}^{(r)}

. Furthermore, by transforming the input extended Voigt-type functions by the Grünwald–Letnikov fractional derivative, we establish representation formulae in terms of the associated Legendre functions of the second kind

Q_{η}^{- ν}

in the two-parameter and two-variable cases. Finally, functional bounding inequalities are given for

V_{μ, ν} (x, y)

and

W_{μ, ν} (x, y)

. Particularly interesting results are presented for the Neumann function

Y_{ν}

and for the Struve

H_{ν}

function in the form of several functional bounds. The article ends with a thorough discussion and closing remarks. Full article

(This article belongs to the Special Issue Orthogonal Polynomials, Special Functions and Applications: 2nd Edition)

15 pages, 309 KiB

Open AccessArticle

Radii of γ-Spirallike of q-Special Functions

by Sercan Kazımoğlu

Mathematics 2024, 12(14), 2261; https://doi.org/10.3390/math12142261 - 19 Jul 2024

Viewed by 550

Abstract

The geometric properties of q-Bessel and q-Bessel-Struve functions are examined in this study. For each of them, three different normalizations are applied in such a way that the resulting functions are analytic in the unit disk of the complex plane. For [...] Read more.

The geometric properties of q-Bessel and q-Bessel-Struve functions are examined in this study. For each of them, three different normalizations are applied in such a way that the resulting functions are analytic in the unit disk of the complex plane. For these normalized functions, the radii of

γ

-spirallike and convex

γ

-spirallike of order

σ

are determined using their Hadamard factorization. These findings extend the known results for Bessel and Struve functions. The characterization of entire functions from the Laguerre-Pólya class plays an important role in our proofs. Additionally, the interlacing property of zeros of q-Bessel and q-Bessel-Struve functions and their derivatives is useful in the proof of our main theorems. Full article

(This article belongs to the Special Issue Geometric Function Theory and Applications―Festschrift for Grigore-Stefan Salagean's 75th Birthday)

25 pages, 3191 KiB

Open AccessArticle

Fractal Operators Abstracted from Arterial Blood Flow

by Tianyi Zhou, Yajun Yin, Gang Peng, Chaoqian Luo and Zhimo Jian

Fractal Fract. 2024, 8(7), 420; https://doi.org/10.3390/fractalfract8070420 - 18 Jul 2024

Cited by 1 | Viewed by 588

Abstract

In this paper, we present a re-established functional fractal circuit model of arterial blood flow that incorporates the shunt effect of the branch vessels. Under the background of hemodynamics, we abstracted a family of fractal operators and investigate the kernel function and properties [...] Read more.

In this paper, we present a re-established functional fractal circuit model of arterial blood flow that incorporates the shunt effect of the branch vessels. Under the background of hemodynamics, we abstracted a family of fractal operators and investigate the kernel function and properties thereof. Based on fractal operators, the intrinsic relation between Bessel function and Struve function was revealed, and some new special functions were found. The results provide mathematical tools for biomechanics and automatic control. Full article

(This article belongs to the Special Issue Fractals in Biophysics and Their Applications)

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24 pages, 824 KiB

Open AccessArticle

Geometric Properties of Normalized Galué Type Struve Function

by Samanway Sarkar, Sourav Das and Saiful R. Mondal

Symmetry 2024, 16(2), 211; https://doi.org/10.3390/sym16020211 - 9 Feb 2024

Viewed by 1084

Abstract

The field of geometric function theory has thoroughly investigated starlike functions concerning symmetric points. The main objective of this work is to derive certain geometric properties, such as the starlikeness of order

δ

, convexity of order

δ

, k-starlikeness, k-uniform [...] Read more.

The field of geometric function theory has thoroughly investigated starlike functions concerning symmetric points. The main objective of this work is to derive certain geometric properties, such as the starlikeness of order

δ

, convexity of order

δ

, k-starlikeness, k-uniform convexity, lemniscate starlikeness and convexity, exponential starlikeness and convexity, and pre-starlikeness for the Galué type Struve function (GTSF). Furthermore, the conditions for GTSF belonging to the Hardy space are also derived. The results obtained in this work generalize several results available in the literature. Full article

(This article belongs to the Special Issue Symmetries of Difference Equations, Special Functions and Orthogonal Polynomials II)

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21 pages, 381 KiB

Open AccessArticle

Geometric Nature of Special Functions on Domain Enclosed by Nephroid and Leminscate Curve

by Reem Alzahrani and Saiful R. Mondal

Symmetry 2024, 16(1), 19; https://doi.org/10.3390/sym16010019 - 22 Dec 2023

Cited by 1 | Viewed by 1709

Abstract

In this work, the geometric nature of solutions to two second-order differential equations,

z y^{''} (z) + a (z) y^{'} (z) + b (z) y (z) = 0

and [...] Read more.

In this work, the geometric nature of solutions to two second-order differential equations,

z y^{''} (z) + a (z) y^{'} (z) + b (z) y (z) = 0

and

z^{2} y^{''} (z) + a (z) y^{'} (z) + b (z) y (z) = d (z)

, is studied. Here,

a (z)

,

b (z)

, and

d (z)

are analytic functions defined on the unit disc. Using differential subordination, we established that the normalized solution

F (z)

(with F(0) = 1) of above differential equations maps the unit disc to the domain bounded by the leminscate curve

\sqrt{1 + z}

. We construct several examples by the judicious choice of

a (z)

,

b (z)

, and

d (z)

. The examples include Bessel functions, Struve functions, the Bessel–Sturve kernel, confluent hypergeometric functions, and many other special functions. We also established a connection with the nephroid domain. Directly using subordination, we construct functions that are subordinated by a nephroid function. Two open problems are also suggested in the conclusion. Full article

(This article belongs to the Special Issue Symmetry in Geometric Theory of Analytic Functions)

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12 pages, 284 KiB

Open AccessArticle

Partial Sums of the Normalized Le Roy-Type Mittag-Leffler Function

by Basem Aref Frasin and Luminiţa-Ioana Cotîrlă

Axioms 2023, 12(5), 441; https://doi.org/10.3390/axioms12050441 - 29 Apr 2023

Viewed by 1032

Abstract

Recently, some researchers determined lower bounds for the normalized version of some special functions to its sequence of partial sums, e.g., Struve and Dini functions, Wright functions and Miller–Ross functions. In this paper, we determine lower bounds for the normalized Le Roy-type Mittag-Leffler [...] Read more.

Recently, some researchers determined lower bounds for the normalized version of some special functions to its sequence of partial sums, e.g., Struve and Dini functions, Wright functions and Miller–Ross functions. In this paper, we determine lower bounds for the normalized Le Roy-type Mittag-Leffler function

F_{α, β}^{(γ)} (z)

= z + \sum_{n = 1}^{\infty} A_{n} z^{n + 1}

, where

A_{n} = {[\frac{Γ (β)}{Γ (α (n - 1) + β)}]}^{γ}

and its sequence of partial sums

{(F_{α, β}^{(γ)} (z))}_{m} (z) = z + \sum_{n = 1}^{m} A_{n} z^{n + 1}

. Several examples of the main results are also considered. Full article

(This article belongs to the Special Issue Recent Advances in Complex Analysis and Applications)

17 pages, 329 KiB

Open AccessArticle

Redheffer-Type Bounds of Special Functions

by Reem Alzahrani and Saiful R. Mondal

Mathematics 2023, 11(2), 379; https://doi.org/10.3390/math11020379 - 11 Jan 2023

Cited by 1 | Viewed by 1409

Abstract

In this paper, we aim to construct inequalities of the Redheffer type for certain functions defined by the infinite product involving the zeroes of these functions. The key tools used in our proofs are classical results on the monotonicity of the ratio of [...] Read more.

In this paper, we aim to construct inequalities of the Redheffer type for certain functions defined by the infinite product involving the zeroes of these functions. The key tools used in our proofs are classical results on the monotonicity of the ratio of differentiable functions. The results are proved using the

n^{th}

positive zero, denoted by

b_{n} (ν)

. Special cases lead to several examples involving special functions, namely, Bessel, Struve, and Hurwitz functions, as well as several other trigonometric functions. Full article

(This article belongs to the Special Issue Orthogonal Polynomials and Special Functions: Recent Trends and Their Applications)

11 pages, 288 KiB

Open AccessArticle

Fast Calculation of the Derivatives of Bessel Functions with Respect to the Parameter and Applications

by Aijuan Li and Huizeng Qin

Symmetry 2023, 15(1), 64; https://doi.org/10.3390/sym15010064 - 26 Dec 2022

Cited by 2 | Viewed by 2327

Abstract

In this paper, the fast algorithms of the derivatives of Bessel functions with respect to the parameter are obtained. Based on these fast algorithms, we discuss the calculations of the derivatives of the functions related to the heterogeneous Bessel differential equation, such as [...] Read more.

In this paper, the fast algorithms of the derivatives of Bessel functions with respect to the parameter are obtained. Based on these fast algorithms, we discuss the calculations of the derivatives of the functions related to the heterogeneous Bessel differential equation, such as Anger, Weber, Struve and modified Struve functions. In addition, the fast calculation of some integrals related to these functions are obtained. At last, numerical examples show the algorithms given in this paper are fast and high precision. Full article

(This article belongs to the Section Mathematics)

12 pages, 333 KiB

Open AccessArticle

Briot–Bouquet Differential Subordinations for Analytic Functions Involving the Struve Function

by Asena Çetinkaya and Luminita-Ioana Cotîrlă

Fractal Fract. 2022, 6(10), 540; https://doi.org/10.3390/fractalfract6100540 - 25 Sep 2022

Cited by 3 | Viewed by 1391

Abstract

We define a new class of exponential starlike functions constructed by a linear operator involving normalized form of the generalized Struve function. Making use of a technique of differential subordination introduced by Miller and Mocanu, we investigate several new results related to the [...] Read more.

We define a new class of exponential starlike functions constructed by a linear operator involving normalized form of the generalized Struve function. Making use of a technique of differential subordination introduced by Miller and Mocanu, we investigate several new results related to the Briot–Bouquet differential subordinations for the linear operator involving the normalized form of the generalized Struve function. We also obtain univalent solutions to the Briot–Bouquet differential equations and observe that these solutions are the best dominant of the Briot–Bouquet differential subordinations for the exponential starlike function class. Moreover, we give an application of fractional integral operator for a complex-valued function associated with the generalized Struve function. The significance of this paper is due to the technique employed in proving the results and novelty of these results for the Struve functions. The approach used in this paper can lead to several new problems in geometric function theory associated with special functions. Full article

(This article belongs to the Special Issue Fractional Operators and Their Applications)

15 pages, 331 KiB

Open AccessArticle

On Geometric Properties of Bessel–Struve Kernel Functions in Unit Disc

by Najla M. Alarifi and Saiful R. Mondal

Mathematics 2022, 10(14), 2516; https://doi.org/10.3390/math10142516 - 19 Jul 2022

Cited by 3 | Viewed by 1811

Abstract

The Bessel–Struve kernel function defined in the unit disc is used in this study. The Bessel–Struve kernel functions are generalized in this article, and differential equations are derived. We found conditions under which the generalized Bessel–Struve function is Lemniscate convex by using a [...] Read more.

The Bessel–Struve kernel function defined in the unit disc is used in this study. The Bessel–Struve kernel functions are generalized in this article, and differential equations are derived. We found conditions under which the generalized Bessel–Struve function is Lemniscate convex by using a subordination technique. The relation between the Janowski class and exponential class is also derived. Full article

(This article belongs to the Special Issue New Trends in Complex Analysis Researches)

13 pages, 1570 KiB

Open AccessArticle

Theoretical Study on Non-Improvement of the Multi-Frequency Direct Sampling Method in Inverse Scattering Problems

by Won-Kwang Park

Mathematics 2022, 10(10), 1674; https://doi.org/10.3390/math10101674 - 13 May 2022

Cited by 2 | Viewed by 1427

Abstract

Generally, it has been confirmed that applying multiple frequencies guarantees a successful imaging result for various non-iterative imaging algorithms in inverse scattering problems. However, the application of multiple frequencies does not yield good results for direct sampling methods (DSMs), which has been confirmed [...] Read more.

Generally, it has been confirmed that applying multiple frequencies guarantees a successful imaging result for various non-iterative imaging algorithms in inverse scattering problems. However, the application of multiple frequencies does not yield good results for direct sampling methods (DSMs), which has been confirmed through simulation but not theoretically. This study proves this premise theoretically by showing that the indicator function with multi-frequency can be expressed by the Bessel and Struve functions and the propagation direction of the incident field. This is based on the fact that the indicator function with single frequency can be expressed by the exponential and Bessel function of order zero of the first kind. Various simulation outcomes are shown to support the theoretical result. Full article

(This article belongs to the Section Computational and Applied Mathematics)

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19 pages, 386 KiB

Open AccessArticle

Radius of k-Parabolic Starlikeness for Some Entire Functions

by Saiful R. Mondal

Symmetry 2022, 14(4), 637; https://doi.org/10.3390/sym14040637 - 22 Mar 2022

Cited by 4 | Viewed by 1845

Abstract

This article considers three types of analytic functions based on their infinite product representation. The radius of the

k

-parabolic starlikeness of the functions of these classes is studied. The optimal parameter values for

k

-parabolic starlike functions are determined in the unit [...] Read more.

This article considers three types of analytic functions based on their infinite product representation. The radius of the

k

-parabolic starlikeness of the functions of these classes is studied. The optimal parameter values for

k

-parabolic starlike functions are determined in the unit disk. Several examples are provided that include special functions such as Bessel, Struve, Lommel, and q-Bessel functions. Full article

(This article belongs to the Special Issue Applications of Symmetric Functions Theory to Certain Fields)

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8 pages, 3200 KiB

Open AccessArticle

A Quadruple Integral Involving Product of the Struve H_v(βt) and Parabolic Cylinder D_u(αx) Functions

by Robert Reynolds and Allan Stauffer

Symmetry 2022, 14(1), 9; https://doi.org/10.3390/sym14010009 - 22 Dec 2021

Cited by 1 | Viewed by 2362

Abstract

The objective of the present paper is to obtain a quadruple infinite integral. This integral involves the product of the Struve and parabolic cylinder functions and expresses it in terms of the Hurwitz–Lerch Zeta function. Almost all Hurwitz-Lerch Zeta functions have an asymmetrical [...] Read more.

The objective of the present paper is to obtain a quadruple infinite integral. This integral involves the product of the Struve and parabolic cylinder functions and expresses it in terms of the Hurwitz–Lerch Zeta function. Almost all Hurwitz-Lerch Zeta functions have an asymmetrical zero distributionSpecial cases in terms fundamental constants and other special functions are produced. All the results in the work are new. Full article

(This article belongs to the Special Issue Symmetric Methods and Analysis for Time-Dependent Partial Differential Equations)

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16 pages, 522 KiB

Open AccessArticle

Self-Inductance of the Circular Coils of the Rectangular Cross-Section with the Radial and Azimuthal Current Densities

by Slobodan Babic and Cevdet Akyel

Physics 2020, 2(3), 352-367; https://doi.org/10.3390/physics2030019 - 13 Jul 2020

Cited by 5 | Viewed by 3779

Abstract

In this paper, we give new formulas for calculating the self-inductance for circular coils of the rectangular cross-sections with the radial and the azimuthal current densities. These formulas are given by the single integration of the elementary functions which are integrable on the [...] Read more.

In this paper, we give new formulas for calculating the self-inductance for circular coils of the rectangular cross-sections with the radial and the azimuthal current densities. These formulas are given by the single integration of the elementary functions which are integrable on the interval of the integration. From these new expressions, we can obtain the special cases for the self-inductance of the thin-disk pancake and the thin-wall solenoids that confirm the validity of this approach. For the asymptotic cases, the new formula for the self-inductance of the thin-wall solenoid is obtained for the first time in the literature. In this paper, we do not use special functions such as the elliptical integrals of the first, second and third kind, nor Struve and Bessel functions because that is very tedious work. The results of this work are compared with already different known methods and all results are in excellent agreement. We consider this approach novel because of its simplicity in the self-inductance calculation of the previously-mentioned configurations. Full article

(This article belongs to the Section Applied Physics)

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16 pages, 276 KiB

Open AccessArticle

A Certain Family of Integral Operators Associated with the Struve Functions

by Shahid Mahmood, H.M. Srivastava, Sarfraz Nawaz Malik, Mohsan Raza, Neelam Shahzadi and Saira Zainab

Symmetry 2019, 11(4), 463; https://doi.org/10.3390/sym11040463 - 2 Apr 2019

Cited by 5 | Viewed by 2590

Abstract

This article presents the study of Struve functions and certain integral operators associated with the Struve functions. It contains the investigation of certain geometric properties like the strong starlikeness and strong convexity of the Struve functions. It also includes the criteria of univalence [...] Read more.

This article presents the study of Struve functions and certain integral operators associated with the Struve functions. It contains the investigation of certain geometric properties like the strong starlikeness and strong convexity of the Struve functions. It also includes the criteria of univalence for a family of certain integral operators associated with the generalized Struve functions. The starlikeness and uniform convexity of the said integral operators are also part of this research. Full article

(This article belongs to the Special Issue Integral Transforms and Operational Calculus)

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