Mathematics

Research

16 pages, 295 KiB

Open AccessArticle

On an Extension of a Hardy–Hilbert-Type Inequality with Multi-Parameters

by Bicheng Yang, Michael Th. Rassias and Andrei Raigorodskii

Mathematics 2021, 9(19), 2432; https://doi.org/10.3390/math9192432 - 30 Sep 2021

Cited by 4 | Viewed by 1292

Making use of weight coefficients as well as real/complex analytic methods, an extension of a Hardy–Hilbert-type inequality with a best possible constant factor and multiparameters is established. Equivalent forms, reverses, operator expression with the norm, and a few particular cases are also considered. [...] Read more.

Making use of weight coefficients as well as real/complex analytic methods, an extension of a Hardy–Hilbert-type inequality with a best possible constant factor and multiparameters is established. Equivalent forms, reverses, operator expression with the norm, and a few particular cases are also considered. Full article

(This article belongs to the Special Issue Mathematical Analysis and Analytic Number Theory 2020)

13 pages, 321 KiB

Open AccessArticle

Modified Operators Interpolating at Endpoints

by Ana Maria Acu, Ioan Raşa and Rekha Srivastava

Mathematics 2021, 9(17), 2051; https://doi.org/10.3390/math9172051 - 25 Aug 2021

Cited by 1 | Viewed by 1331

Abstract

Some classical operators (e.g., Bernstein) preserve the affine functions and consequently interpolate at the endpoints. Other classical operators (e.g., Bernstein–Durrmeyer) have been modified in order to preserve the affine functions. We propose a simpler modification with the effect that the new operators interpolate [...] Read more.

Some classical operators (e.g., Bernstein) preserve the affine functions and consequently interpolate at the endpoints. Other classical operators (e.g., Bernstein–Durrmeyer) have been modified in order to preserve the affine functions. We propose a simpler modification with the effect that the new operators interpolate at endpoints although they do not preserve the affine functions. We investigate the properties of these modified operators and obtain results concerning iterates and their limits, Voronovskaja-type results and estimates of several differences. Full article

(This article belongs to the Special Issue Mathematical Analysis and Analytic Number Theory 2020)

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16 pages, 297 KiB

Open AccessArticle

A Certain Subclass of Multivalent Analytic Functions Defined by the q-Difference Operator Related to the Janowski Functions

by Bo Wang, Rekha Srivastava and Jin-Lin Liu

Mathematics 2021, 9(14), 1706; https://doi.org/10.3390/math9141706 - 20 Jul 2021

Cited by 9 | Viewed by 1546

Abstract

A class of p-valent analytic functions is introduced using the q-difference operator and the familiar Janowski functions. Several properties of functions in the class, such as the Fekete–Szegö inequality, coefficient estimates, necessary and sufficient conditions, distortion and growth theorems, radii of [...] Read more.

A class of p-valent analytic functions is introduced using the q-difference operator and the familiar Janowski functions. Several properties of functions in the class, such as the Fekete–Szegö inequality, coefficient estimates, necessary and sufficient conditions, distortion and growth theorems, radii of convexity and starlikeness, closure theorems and partial sums, are discussed in this paper. Full article

(This article belongs to the Special Issue Mathematical Analysis and Analytic Number Theory 2020)

10 pages, 273 KiB

Open AccessArticle

Some Families of Apéry-Like Fibonacci and Lucas Series

by Robert Frontczak, Hari Mohan Srivastava and Živorad Tomovski

Mathematics 2021, 9(14), 1621; https://doi.org/10.3390/math9141621 - 09 Jul 2021

Cited by 5 | Viewed by 1788

Abstract

In this paper, the authors investigate two special families of series involving the reciprocal central binomial coefficients and Lucas numbers. Connections with several familiar sums representing the integer-valued Riemann zeta function are also pointed out. Full article

(This article belongs to the Special Issue Mathematical Analysis and Analytic Number Theory 2020)

20 pages, 294 KiB

Open AccessEditor’s ChoiceArticle

Qualitative Analyses of Differential Systems with Time-Varying Delays via Lyapunov–Krasovskiĭ Approach

by Cemil Tunç, Osman Tunç, Yuanheng Wang and Jen-Chih Yao

Mathematics 2021, 9(11), 1196; https://doi.org/10.3390/math9111196 - 25 May 2021

Cited by 22 | Viewed by 1791

Abstract

In this paper, a class of systems of linear and non-linear delay differential equations (DDEs) of first order with time-varying delay is considered. We obtain new sufficient conditions for uniform asymptotic stability of zero solution, integrability of solutions of an unperturbed system and [...] Read more.

In this paper, a class of systems of linear and non-linear delay differential equations (DDEs) of first order with time-varying delay is considered. We obtain new sufficient conditions for uniform asymptotic stability of zero solution, integrability of solutions of an unperturbed system and boundedness of solutions of a perturbed system. We construct two appropriate Lyapunov–Krasovskiĭ functionals (LKFs) as the main tools in proofs. The technique of the proofs depends upon the Lyapunov–Krasovskiĭ method. For illustration, two examples are provided in particular cases. An advantage of the new LKFs used here is that they allow to eliminate using Gronwall’s inequality. When we compare our results with recent results in the literature, the established conditions are more general, less restrictive and optimal for applications. Full article

(This article belongs to the Special Issue Mathematical Analysis and Analytic Number Theory 2020)

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10 pages, 282 KiB

Open AccessArticle

Repdigits as Product of Terms of k-Bonacci Sequences

by Petr Coufal and Pavel Trojovský

Mathematics 2021, 9(6), 682; https://doi.org/10.3390/math9060682 - 22 Mar 2021

Cited by 6 | Viewed by 1852

Abstract

For any integer

k \geq 2

, the sequence of the k-generalized Fibonacci numbers (or k-bonacci numbers) is defined by the k initial values [...] Read more.

For any integer

k \geq 2

, the sequence of the k-generalized Fibonacci numbers (or k-bonacci numbers) is defined by the k initial values

F_{- (k - 2)}^{(k)} = \dots = F_{0}^{(k)} = 0

and

F_{1}^{(k)} = 1

and such that each term afterwards is the sum of the k preceding ones. In this paper, we search for repdigits (i.e., a number whose decimal expansion is of the form

a a \dots a

, with

a \in [1, 9]

) in the sequence

{(F_{n}^{(k)} F_{n}^{(k + m)})}_{n}

, for

m \in [1, 9]

. This result generalizes a recent work of Bednařík and Trojovská (the case in which

(k, m) = (2, 1)

). Our main tools are the transcendental method (for Diophantine equations) together with the theory of continued fractions (reduction method). Full article

(This article belongs to the Special Issue Mathematical Analysis and Analytic Number Theory 2020)

17 pages, 337 KiB

Open AccessArticle

On Four Classical Measure Theorems

by Salvador López-Alfonso, Manuel López-Pellicer and Santiago Moll-López

Mathematics 2021, 9(5), 526; https://doi.org/10.3390/math9050526 - 03 Mar 2021

Cited by 2 | Viewed by 1319

Abstract

A subset

B

of an algebra

A

of subsets of a set

Ω

has property

(N)

if each

B

-pointwise bounded sequence of the Banach space

b a (A)

is bounded in

b a (A)

, where [...] Read more.

A subset

B

of an algebra

A

of subsets of a set

Ω

has property

(N)

if each

B

-pointwise bounded sequence of the Banach space

b a (A)

is bounded in

b a (A)

, where

b a (A)

is the Banach space of real or complex bounded finitely additive measures defined on

A

endowed with the variation norm.

B

has property

(G)

[

(V H S)

] if for each bounded sequence [if for each sequence] in

b a (A)

the

B

-pointwise convergence implies its weak convergence.

B

has property

(s N)

[

(s G)

or

(s V H S)

] if every increasing covering

{B_{n} : n \in N}

of

B

contains a set

B_{p}

with property

(N)

[

(G)

or

(V H S)

], and

B

has property

(w N)

[

(w G)

or

(w V H S)

] if every increasing web

{B_{n_{1} n_{2} \dots n_{m}} : n_{i} \in N, 1 \leq i \leq m, m \in N}

of

B

contains a strand

{B_{p_{1} p_{2} \dots p_{m}} : m \in N}

formed by elements

B_{p_{1} p_{2} \dots p_{m}}

with property

(N)

[

(G)

or

(V H S)

] for every

m \in N

. The classical theorems of Nikodým–Grothendieck, Valdivia, Grothendieck and Vitali–Hahn–Saks say, respectively, that every

σ

-algebra has properties

(N)

,

(s N)

,

(G)

and

(V H S)

. Valdivia’s theorem was obtained through theorems of barrelled spaces. Recently, it has been proved that every

σ

-algebra has property

(w N)

and several applications of this strong Nikodým type property have been provided. In this survey paper we obtain a proof of the property

(w N)

of a

σ

-algebra independent of the theory of locally convex barrelled spaces which depends on elementary basic results of Measure theory and Banach space theory. Moreover we prove that a subset

B

of an algebra

A

has property

(w W H S)

if and only if

B

has property

(w N)

and

A

has property

(G)

. Full article

(This article belongs to the Special Issue Mathematical Analysis and Analytic Number Theory 2020)

24 pages, 379 KiB

Open AccessFeature PaperArticle

On Coding by (2,q)-Distance Fibonacci Numbers

by Ivana Matoušová and Pavel Trojovský

Mathematics 2020, 8(11), 2058; https://doi.org/10.3390/math8112058 - 18 Nov 2020

Cited by 4 | Viewed by 2475

Abstract

In 2006, A. Stakhov introduced a new coding/decoding process based on generating matrices of the Fibonacci p-numbers, which he called the Fibonacci coding/decoding method. Stakhov’s papers have motivated many other scientists to seek certain generalizations by introducing new additional coefficients into recurrence [...] Read more.

In 2006, A. Stakhov introduced a new coding/decoding process based on generating matrices of the Fibonacci p-numbers, which he called the Fibonacci coding/decoding method. Stakhov’s papers have motivated many other scientists to seek certain generalizations by introducing new additional coefficients into recurrence of Fibonacci p-numbers. In 2013, I. Włoch et al. studied

(2, q)

-distance Fibonacci numbers

F_{2} (q, n)

and found some of their combinatorial properties. In this paper, we state a new coding theory based on the sequence

{(T_{q} (n))}_{n = - \infty}^{\infty}

, which is an extension of Włoch’s sequence

{(F_{2} (q, n))}_{n = 0}^{\infty}

. Full article

(This article belongs to the Special Issue Mathematical Analysis and Analytic Number Theory 2020)

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17 pages, 326 KiB

Open AccessArticle

A New Representation of the Generalized Krätzel Function

by Asifa Tassaddiq

Mathematics 2020, 8(11), 2009; https://doi.org/10.3390/math8112009 - 11 Nov 2020

Cited by 6 | Viewed by 1862

Abstract

The confluence of distributions (generalized functions) with integral transforms has become a remarkably powerful tool to address important unsolved problems. The purpose of the present study is to investigate a distributional representation of the generalized Krätzel function. Hence, a new definition of these [...] Read more.

The confluence of distributions (generalized functions) with integral transforms has become a remarkably powerful tool to address important unsolved problems. The purpose of the present study is to investigate a distributional representation of the generalized Krätzel function. Hence, a new definition of these functions is formulated over a particular set of test functions. This is validated using the classical Fourier transform. The results lead to a novel extension of Krätzel functions by introducing distributions in terms of the delta function. A new version of the generalized Krätzel integral transform emerges as a natural consequence of this research. The relationship between the Krätzel function and the

H

-function is also explored to study new identities. Full article

(This article belongs to the Special Issue Mathematical Analysis and Analytic Number Theory 2020)

4 pages, 702 KiB

Open AccessArticle

An Alternating Sum of Fibonacci and Lucas Numbers of Order k

by Spiros D. Dafnis, Andreas N. Philippou and Ioannis E. Livieris

Mathematics 2020, 8(9), 1487; https://doi.org/10.3390/math8091487 - 03 Sep 2020

Cited by 3 | Viewed by 2139

Abstract

During the last decade, many researchers have focused on proving identities that reveal the relation between Fibonacci and Lucas numbers. Very recently, one of these identities has been generalized to the case of Fibonacci and Lucas numbers of order k. In the [...] Read more.

During the last decade, many researchers have focused on proving identities that reveal the relation between Fibonacci and Lucas numbers. Very recently, one of these identities has been generalized to the case of Fibonacci and Lucas numbers of order k. In the present work, we state and prove a new identity regarding an alternating sum of Fibonacci and Lucas numbers of order k. Our result generalizes recent works in this direction. Full article

(This article belongs to the Special Issue Mathematical Analysis and Analytic Number Theory 2020)

10 pages, 230 KiB

Open AccessArticle

On the Quartic Residues and Their New Distribution Properties

by Juanli Su and Jiafan Zhang

Mathematics 2020, 8(8), 1337; https://doi.org/10.3390/math8081337 - 11 Aug 2020

Cited by 2 | Viewed by 1592

Abstract

In this paper, we use the analytic methods, the properties of the fourth-order characters, and the estimate for character sums to study the computational problems of one kind of special quartic residues modulo p, and give an exact calculation formula and asymptotic [...] Read more.

In this paper, we use the analytic methods, the properties of the fourth-order characters, and the estimate for character sums to study the computational problems of one kind of special quartic residues modulo p, and give an exact calculation formula and asymptotic formula for their counting functions. Full article

(This article belongs to the Special Issue Mathematical Analysis and Analytic Number Theory 2020)

22 pages, 320 KiB

Open AccessArticle

On the Generalized Riesz Derivative

by Chenkuan Li and Joshua Beaudin

Mathematics 2020, 8(7), 1089; https://doi.org/10.3390/math8071089 - 03 Jul 2020

Cited by 2 | Viewed by 2181

Abstract

The goal of this paper is to construct an integral representation for the generalized Riesz derivative

_{R Z} D_{x}^{2 s} u (x)

for

k < s < k + 1

with

k = 0, 1, \dots

, [...] Read more.

The goal of this paper is to construct an integral representation for the generalized Riesz derivative

_{R Z} D_{x}^{2 s} u (x)

for

k < s < k + 1

with

k = 0, 1, \dots

, which is proved to be a one-to-one and linearly continuous mapping from the normed space

W_{k + 1} (R)

to the Banach space

C (R)

. In addition, we show that

_{R Z} D_{x}^{2 s} u (x)

is continuous at the end points and well defined for

s = \frac{1}{2} + k

. Furthermore, we extend the generalized Riesz derivative

_{R Z} D_{x}^{2 s} u (x)

to the space

C_{k} (R^{n})

, where k is an n-tuple of nonnegative integers, based on the normalization of distribution and surface integrals over the unit sphere. Finally, several examples are presented to demonstrate computations for obtaining the generalized Riesz derivatives. Full article

(This article belongs to the Special Issue Mathematical Analysis and Analytic Number Theory 2020)

11 pages, 255 KiB

Open AccessArticle

A Study of the Second-Kind Multivariate Pseudo-Chebyshev Functions of Fractional Degree

by Paolo Emilio Ricci and Rekha Srivastava

Mathematics 2020, 8(6), 978; https://doi.org/10.3390/math8060978 - 16 Jun 2020

Cited by 3 | Viewed by 1456

Abstract

Here, in this paper, the second-kind multivariate pseudo-Chebyshev functions of fractional degree are introduced by using the Dunford–Taylor integral. As an application, the problem of finding matrix roots for a wide class of non-singular complex matrices has been considered. The principal value of [...] Read more.

Here, in this paper, the second-kind multivariate pseudo-Chebyshev functions of fractional degree are introduced by using the Dunford–Taylor integral. As an application, the problem of finding matrix roots for a wide class of non-singular complex matrices has been considered. The principal value of the fixed matrix root is determined. In general, by changing the determinations of the numerical roots involved, we could find

n^{r}

roots for the n-th root of an

r \times r

matrix. The exceptional cases for which there are infinitely many roots, or no roots at all, are obviously excluded. Full article

(This article belongs to the Special Issue Mathematical Analysis and Analytic Number Theory 2020)

11 pages, 276 KiB

Open AccessArticle

Better Approaches for n-Times Differentiable Convex Functions

by Praveen Agarwal, Mahir Kadakal, İmdat İşcan and Yu-Ming Chu

Mathematics 2020, 8(6), 950; https://doi.org/10.3390/math8060950 - 10 Jun 2020

Cited by 30 | Viewed by 1856

Abstract

In this work, by using an integral identity together with the Hölder–İşcan inequality we establish several new inequalities for n-times differentiable convex and concave mappings. Furthermore, various applications for some special means as arithmetic, geometric, and logarithmic are given. Full article

(This article belongs to the Special Issue Mathematical Analysis and Analytic Number Theory 2020)

11 pages, 287 KiB

Open AccessArticle

A Certain Mean Square Value Involving Dirichlet L-Functions

by Wenpeng Zhang and Di Han

Mathematics 2020, 8(6), 948; https://doi.org/10.3390/math8060948 - 09 Jun 2020

Cited by 1 | Viewed by 2007

Abstract

The main purpose of this article is using the elementary methods, the properties of Dirichlet L-functions to study the computational problem of a certain mean square value involving Dirichlet L-functions at positive integer points, and give some exact calculating formulae. As [...] Read more.

The main purpose of this article is using the elementary methods, the properties of Dirichlet L-functions to study the computational problem of a certain mean square value involving Dirichlet L-functions at positive integer points, and give some exact calculating formulae. As some applications, we obtain some interesting identities and inequalities involving character sums and trigonometric sums. Full article

(This article belongs to the Special Issue Mathematical Analysis and Analytic Number Theory 2020)

18 pages, 331 KiB

Open AccessArticle

Fekete-Szegö Type Problems and Their Applications for a Subclass of q-Starlike Functions with Respect to Symmetrical Points

by Hari Mohan Srivastava, Nazar Khan, Maslina Darus, Shahid Khan, Qazi Zahoor Ahmad and Saqib Hussain

Mathematics 2020, 8(5), 842; https://doi.org/10.3390/math8050842 - 22 May 2020

Cited by 32 | Viewed by 2775

Abstract

In this article, by using the concept of the quantum (or q-) calculus and a general conic domain

Ω_{k, q}

, we study a new subclass of normalized analytic functions with respect to symmetrical points in an open unit disk. [...] Read more.

In this article, by using the concept of the quantum (or q-) calculus and a general conic domain

Ω_{k, q}

, we study a new subclass of normalized analytic functions with respect to symmetrical points in an open unit disk. We solve the Fekete-Szegö type problems for this newly-defined subclass of analytic functions. We also discuss some applications of the main results by using a q-Bernardi integral operator. Full article

(This article belongs to the Special Issue Mathematical Analysis and Analytic Number Theory 2020)

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