Turán-type inequalities for generalized k-Bessel functions

We propose an approach to the generalized $\rm{k}$-Bessel function defined by&nbsp;$$\rm{U}_{p,q,r}^{\rm{k}}(z)=\sum\limits_{n=0}^{\infty}\frac{(-r)^{n}}{\Gamma_{\rm{k}}\left(n\rm{k}+p+\dfrac{q+1}{2}\rm{k}\right)n!}\left(\dfrac{z}{2}\right)&nbsp; ^{2n+\frac{p}{\rm{k}}},$$&nbs...

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Дата:	2024
Автор:	Zayed, Hanaa M.
Формат:	Стаття
Мова:	Англійська
Опубліковано:	Institute of Mathematics, NAS of Ukraine 2024
Онлайн доступ:	https://umj.imath.kiev.ua/index.php/umj/article/view/7375
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Назва журналу:	Ukrains’kyi Matematychnyi Zhurnal
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Ukrains’kyi Matematychnyi Zhurnal

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author	Zayed, Hanaa M. Zayed, Hanaa M.
author_facet	Zayed, Hanaa M. Zayed, Hanaa M.
author_sort	Zayed, Hanaa M.
baseUrl_str	https://umj.imath.kiev.ua/index.php/umj/oai
collection	OJS
datestamp_date	2024-06-19T00:35:12Z
description	We propose an approach to the generalized $\rm{k}$-Bessel function defined by&nbsp;$$\rm{U}_{p,q,r}^{\rm{k}}(z)=\sum\limits_{n=0}^{\infty}\frac{(-r)^{n}}{\Gamma_{\rm{k}}\left(n\rm{k}+p+\dfrac{q+1}{2}\rm{k}\right)n!}\left(\dfrac{z}{2}\right)&nbsp; ^{2n+\frac{p}{\rm{k}}},$$&nbsp;where $\rm{k}&gt;0$ and $p,q,r\in\mathbb{C}$.&nbsp;We discuss the uniform convergence of $\rm{U}_{p,q,r}^{\rm{k}}(z).$&nbsp;Moreover, we prove that the analyzed function is entire and determine its growth order and type.&nbsp;We also find its Weierstrass factorization, which turns out to be an infinite product uniformly convergent on a compact subset of the complex plane.&nbsp;The integral representation for $\rm{U}_{p,q,r}^{\rm{k}}(z)$ is found by using the representation for&nbsp; $\rm{k}$-beta functions.&nbsp;&nbsp;We also prove that the specified function is a solution of a second-order differential equation that generalizes certain well-known differential equations for the classical Bessel functions.&nbsp;In addition, some interesting properties, such as&nbsp; recurrence and differential relations, are demonstrated.&nbsp;Some of these properties can be used to establish some Turán-type&nbsp;inequalities for this function.&nbsp;&nbsp;Ultimately, we study the monotonicity and log-convexity of the normalized form of the modified \textrm{k}-Bessel function $\rm{T}_{p,q,1}^{\rm{k}}$ defined by $\rm{T}_{p,q,1}^{\rm{k}}(z)=i^{-\frac{p}{k}}\rm{U}_{p,q,1}^{\rm{k}}(iz),$ as well as the quotient of the modified \textrm{k}\textit{-}Bessel function, exponential, and \textrm{k}\textit{-}hypergeometric functions.&nbsp;In this case, the leading concept of the proofs comes from the monotonicity of the ratio of two power series.
doi_str_mv	10.3842/umzh.v76i2.7375
first_indexed	2026-03-24T03:32:27Z
format	Article
fulltext	Skip to main content Advertisement Log in Menu Find a journal Publish with us Track your research Search Saved research Cart Home Ukrainian Mathematical Journal Article Turán-Type Inequalities for Generalized k-Bessel Functions Published: 17 August 2024 Volume 76, pages 254–279, (2024) Cite this article Save article View saved research Ukrainian Mathematical Journal Aims and scope Submit manuscript Hanaa M. Zayed1 95 Accesses 1 Citation Explore all metrics We propose an approach to the generalized k-Bessel function defined by ${\text{U}}_{p,q,r}^{\text{k}}\left(z\right)=\sum_{n=0}^{\infty }\frac{{\left(-r\right)}^{n}}{{\Gamma }_{k}\left(nk+p+\frac{q+1}{2}\text{k}\right)n!}{\left(\frac{z}{2}\right)}^{2n+\frac{p}{\text{k}}},$ where k > 0 and p, q, r ∈ ${\mathbb{C}}$. We discuss the uniform convergence of ${\text{U}}_{p,q,r}^{\text{k}}$ (z). Moreover, we prove that the analyzed function is entire and determine its growth order and type. We also find its Weierstrass factorization, which turns out to be an infinite product uniformly convergent on a compact subset of the complex plane. The integral representation for ${\text{U}}_{p,q,r}^{\text{k}}$ (z) is found by using the representation for k-beta functions. We also prove that the specified function is a solution of a second-order differential equation that generalizes certain well-known differential equations for the classical Bessel functions. In addition, some interesting properties, such as recurrence and differential relations, are demonstrated. Some of these properties can be used to establish Turán-type inequalities for this function. Ultimately, we study the monotonicity and log-convexity of the normalized form of the modified k-Bessel function ${\text{T}}_{p,q,1}^{\text{k}}$ defined by ${\text{T}}_{p,q,1}^{\text{k}}$ (z) = $i{-}^\frac{p}{k}{\text{U}}_{p,q,1}^{\text{k}}$ (iz), as well as the quotient of the modified k-Bessel function, exponential, and k-hypergeometric functions. In this case, the leading concept of the proofs comes from the monotonicity of the ratio of two power series. This is a preview of subscription content, log in via an institution to check access. Access this article Log in via an institution Subscribe and save Springer+ from €37.37 /Month Starting from 10 chapters or articles per month Access and download chapters and articles from more than 300k books and 2,500 journals Cancel anytime View plans Buy Now Buy article PDF 39,95 € Price includes VAT (Ukraine) Instant access to the full article PDF. Institutional subscriptions Similar content being viewed by others Differential equation and inequalities of the generalized k-Bessel functions Article Open access 16 July 2018 Inequalities on an extended Bessel function Article Open access 27 March 2018 The Capacity of Sets of Divergence of Certain Taylor Series on the Unit Circle Article 09 May 2019 Explore related subjects Discover the latest articles, books and news in related subjects, suggested using machine learning. Fourier Analysis Functions of a Complex Variable Real Functions Special Functions Functional Analysis Integral Transforms and Operational Calculus Inequalities and Integral Operators in Mathematical Analysis References M. Abramowitz and I. A. Stegun (editors), Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, Dover, New York (1965). Google Scholar P. Agarwal, M. Chand, and G. 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institution	Ukrains’kyi Matematychnyi Zhurnal
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last_indexed	2026-03-24T03:32:27Z
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spelling	umjimathkievua-article-73752024-06-19T00:35:12Z Turán-type inequalities for generalized k-Bessel functions Turán-type inequalities for generalized k-Bessel functions Zayed, Hanaa M. Zayed, Hanaa M. Generalized k-Bessel functions, integral representation, differential properties, turán type inequalities, monotonicity We propose an approach to the generalized $\rm{k}$-Bessel function defined by&nbsp;$$\rm{U}_{p,q,r}^{\rm{k}}(z)=\sum\limits_{n=0}^{\infty}\frac{(-r)^{n}}{\Gamma_{\rm{k}}\left(n\rm{k}+p+\dfrac{q+1}{2}\rm{k}\right)n!}\left(\dfrac{z}{2}\right)&nbsp; ^{2n+\frac{p}{\rm{k}}},$$&nbsp;where $\rm{k}&gt;0$ and $p,q,r\in\mathbb{C}$.&nbsp;We discuss the uniform convergence of $\rm{U}_{p,q,r}^{\rm{k}}(z).$&nbsp;Moreover, we prove that the analyzed function is entire and determine its growth order and type.&nbsp;We also find its Weierstrass factorization, which turns out to be an infinite product uniformly convergent on a compact subset of the complex plane.&nbsp;The integral representation for $\rm{U}_{p,q,r}^{\rm{k}}(z)$ is found by using the representation for&nbsp; $\rm{k}$-beta functions.&nbsp;&nbsp;We also prove that the specified function is a solution of a second-order differential equation that generalizes certain well-known differential equations for the classical Bessel functions.&nbsp;In addition, some interesting properties, such as&nbsp; recurrence and differential relations, are demonstrated.&nbsp;Some of these properties can be used to establish some Turán-type&nbsp;inequalities for this function.&nbsp;&nbsp;Ultimately, we study the monotonicity and log-convexity of the normalized form of the modified \textrm{k}-Bessel function $\rm{T}_{p,q,1}^{\rm{k}}$ defined by $\rm{T}_{p,q,1}^{\rm{k}}(z)=i^{-\frac{p}{k}}\rm{U}_{p,q,1}^{\rm{k}}(iz),$ as well as the quotient of the modified \textrm{k}\textit{-}Bessel function, exponential, and \textrm{k}\textit{-}hypergeometric functions.&nbsp;In this case, the leading concept of the proofs comes from the monotonicity of the ratio of two power series. УДК 517.5 Нерівності типу Турана&nbsp; для узагальнених k-функцій Бесселя Запропоновано підхід до вивчення узагальненої $\rm{k}$-функції Бесселя, що визначена рівністю&nbsp;$$\rm{U}_{p,q,r}^{\rm{k}}(z)=\sum\limits_{n=0}^{\infty}\frac{(-r)^{n}}{\Gamma_{\rm{k}}\left(n\rm{k}+p+\dfrac{q+1}{2}\rm{k}\right)n!}\left(\dfrac{z}{2}\right)&nbsp; ^{2n+\frac{p}{\rm{k}}},$$&nbsp;де $\rm{k}&gt;0$ і $p,q,r\in\mathbb{C}$.&nbsp;Ми обговорюємо рівномірну збіжність $\rm{U}_{p, q, r}^{\rm {k}}(z). $&nbsp;&nbsp;Крім того, доведено, що дана функція є цілою, і визначено порядок її зростання і тип.&nbsp;І навіть більше,&nbsp; знайдено її факторизацію Веєрштрасса у вигляді&nbsp; нескінченного добутку, рівномірно збіжного в компактній підмножині комплексної площини.&nbsp;Інтегральне зображення для $\rm{U}_{p, q, r}^{\rm{k}}(z) $ знайдено за допомогою зображення для $\rm{k}$-бета-функцій.&nbsp;&nbsp;Також доведено, що вказана функція є розв'язком диференціального рівняння другого порядку, яке узагальнює певні відомі диференціальні рівняння для класичних функцій Бесселя. І навіть більше, продемонстровано деякі цікаві властивості, такі як рекурентність, та диференціальні співвідношення.&nbsp;&nbsp;Деякі з цих властивостей можуть бути корисними при встановленні певних нерівностей туранівського типу&nbsp; для цієї функції.&nbsp;&nbsp;Зрештою, ми також вивчаємо монотонність та log-опуклість нормалізованої форми модифікованої \textrm{k}-функції Бесселя&nbsp;&nbsp;$\rm{T}_{p,q,1}^{\rm{k}}(z)=i^{-\frac{p}{k}}\rm{U}_{p,q,1}^{\rm{k}}(iz)$, а також частку модифікованої \textrm{k}-функції Бесселя, експоненціальної та \textrm{k}-гіпергеометричної функцій.&nbsp;У цьому випадку основна ідея доведення базується на&nbsp; монотонності відношення двох степеневих рядів. Institute of Mathematics, NAS of Ukraine 2024-02-28 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/7375 10.3842/umzh.v76i2.7375 Ukrains’kyi Matematychnyi Zhurnal; Vol. 76 No. 2 (2024); 234-256 Український математичний журнал; Том 76 № 2 (2024); 234-256 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/7375/9727 Copyright (c) 2024 Hanaa Zayed
spellingShingle	Zayed, Hanaa M. Zayed, Hanaa M. Turán-type inequalities for generalized k-Bessel functions
title	Turán-type inequalities for generalized k-Bessel functions
title_alt	Turán-type inequalities for generalized k-Bessel functions
title_full	Turán-type inequalities for generalized k-Bessel functions
title_fullStr	Turán-type inequalities for generalized k-Bessel functions
title_full_unstemmed	Turán-type inequalities for generalized k-Bessel functions
title_short	Turán-type inequalities for generalized k-Bessel functions
title_sort	turán-type inequalities for generalized k-bessel functions
topic_facet	Generalized k-Bessel functions integral representation differential properties turán type inequalities monotonicity
url	https://umj.imath.kiev.ua/index.php/umj/article/view/7375
work_keys_str_mv	AT zayedhanaam turantypeinequalitiesforgeneralizedkbesselfunctions AT zayedhanaam turantypeinequalitiesforgeneralizedkbesselfunctions

Turán-type inequalities for generalized k-Bessel functions

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