Generalized Hermite–Hadamard–Mercer-type inequalities and related estimates for multiplicative convex functions with applications
UDC 517.51, 517.16 The theory of multiplicative calculus has recently gained loads of attention. In this framework, the operations of addition and subtraction are systematically replaced with multiplication and division, respectively. The interest in this system is to obtain a multiplicative analog...
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| Date: | 2026 |
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| Main Authors: | , |
| Format: | Article |
| Language: | English |
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Institute of Mathematics, NAS of Ukraine
2026
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| Online Access: | https://umj.imath.kiev.ua/index.php/umj/article/view/9424 |
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| Journal Title: | Ukrains’kyi Matematychnyi Zhurnal |
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1866663723021631488 |
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| author | Nwaeze, Eze R. Fagbemigun, Bosede O. Nwaeze, Eze R. Fagbemigun, Bosede O. |
| author_facet | Nwaeze, Eze R. Fagbemigun, Bosede O. Nwaeze, Eze R. Fagbemigun, Bosede O. |
| author_institution_txt_mv | [
{
"author": "Eze R. Nwaeze",
"institution": "Department of Mathematics and Computer Science, Alabama State University, Montgomery, USA"
},
{
"author": "Bosede O. Fagbemigun",
"institution": "Department of Mathematical Sciences, Faculty of Natural Sciences, Ajayi Crowther University, Oyo, Nigeria"
}
] |
| author_sort | Nwaeze, Eze R. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2026-05-30T12:44:40Z |
| description | UDC 517.51, 517.16
The theory of multiplicative calculus has recently gained loads of attention. In this framework, the operations of addition and subtraction are systematically replaced with multiplication and division, respectively. The interest in this system is to obtain a multiplicative analog of the results obtained in the classical sense. Our aim is to contribute to the body of knowledge in this direction. Specifically, we establish some novel generalizations of the Hermite–Hadamard–Mercer-type inequalities involving new multiplicative tempered fractional integrals. A new lemma is also established. By using this lemma, Hölder's inequality, and the power-mean inequality, we obtain more inequalities of the Trapezoid–Mercer type. Our results generalize many other results available from the literature. We present some numerical examples, utilizing the Wolfram Mathematica software for computations and graphing, to prove the validity of our results by comparing the outcomes for different values of the operating parameter $\lambda\in[0,1]$ with 0.1 as the step size. Finally, we obtain additional estimates by applying our results to special means, such as arithmetic, harmonic, logarithmic, and $p$-logarithmic means. |
| doi_str_mv | 10.3842/umzh.v78i5-6.9424 |
| first_indexed | 2026-05-30T01:00:57Z |
| format | Article |
| fulltext | |
| id | umjimathkievua-article-9424 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-05-31T01:00:49Z |
| publishDate | 2026 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | |
| spelling | umjimathkievua-article-94242026-05-30T12:44:40Z Generalized Hermite–Hadamard–Mercer-type inequalities and related estimates for multiplicative convex functions with applications Generalized Hermite–Hadamard–Mercer-type inequalities and related estimates for multiplicative convex functions with applications Nwaeze, Eze R. Fagbemigun, Bosede O. Nwaeze, Eze R. Fagbemigun, Bosede O. Hermite--Hadamard type inequalities; Mercer type inequalities; Multiplicative convex functions; Trapezoid inequalities; Special means. Hermite--Hadamard type inequalities; Mercer type inequalities; Multiplicative convex functions; Trapezoid inequalities; Special means. UDC 517.51, 517.16 The theory of multiplicative calculus has recently gained loads of attention. In this framework, the operations of addition and subtraction are systematically replaced with multiplication and division, respectively. The interest in this system is to obtain a multiplicative analog of the results obtained in the classical sense. Our aim is to contribute to the body of knowledge in this direction. Specifically, we establish some novel generalizations of the Hermite–Hadamard–Mercer-type inequalities involving new multiplicative tempered fractional integrals. A new lemma is also established. By using this lemma, Hölder's inequality, and the power-mean inequality, we obtain more inequalities of the Trapezoid–Mercer type. Our results generalize many other results available from the literature. We present some numerical examples, utilizing the Wolfram Mathematica software for computations and graphing, to prove the validity of our results by comparing the outcomes for different values of the operating parameter $\lambda\in[0,1]$ with 0.1 as the step size. Finally, we obtain additional estimates by applying our results to special means, such as arithmetic, harmonic, logarithmic, and $p$-logarithmic means. УДК 517.51, 517.16 Узагальнені нерівності типу Ерміта–Адамара–Мерсера та відповідні оцінки для мультиплікативно опуклих функцій із застосуваннями Теорія мультиплікативного числення останнім часом привертає значну увагу дослідників. У цій теорії операції додавання та віднімання систематично замінюють на множення і ділення відповідно. Інтерес до цієї теорії полягає в отриманні мультиплікативних аналогів результатів, що одержані в класичному розумінні. У цій статті ми робимо певний внесок у розвиток цього напряму. Зокрема, встановлено нові узагальнення нерівностей типу Ерміта–Адамара–Мерсера із залученням нових мультиплікативних темперованих дробових інтегралів. Також доведено нову лему. З використанням цієї леми, нерівності Гельдера та нерівності степеневого середнього отримано нові нерівності типу трапецій Мерсера. Здобуті результати є узагальненням багатьох інших опублікованих результатів. Наведено числові приклади, отримані з використанням програмного забезпечення Wolfram Mathematica для обчислень і побудови графіків із метою підтвердження коректності результатів шляхом порівняння відповідних значень для різних параметрів $\lambda\in[0,1]$ з кроком $0{,}1.$ Крім того, одержано нові оцінки шляхом застосування наших результатів до спеціальних середніх, таких як арифметичне, гармонічне, логарифмічне та $p$-логарифмічне середні. Institute of Mathematics, NAS of Ukraine 2026-05-29 Article Article https://umj.imath.kiev.ua/index.php/umj/article/view/9424 10.3842/umzh.v78i5-6.9424 Ukrains’kyi Matematychnyi Zhurnal; Vol. 78 No. 5-6 (2026); 375–376 Український математичний журнал; Том 78 № 5-6 (2026); 375–376 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/9424/10661 Copyright (c) 2026 Eze R. Nwaeze, Bosede O. Fagbemigun |
| spellingShingle | Nwaeze, Eze R. Fagbemigun, Bosede O. Nwaeze, Eze R. Fagbemigun, Bosede O. Generalized Hermite–Hadamard–Mercer-type inequalities and related estimates for multiplicative convex functions with applications |
| title | Generalized Hermite–Hadamard–Mercer-type inequalities and related estimates for multiplicative convex functions with applications |
| title_alt | Generalized Hermite–Hadamard–Mercer-type inequalities and related estimates for multiplicative convex functions with applications |
| title_full | Generalized Hermite–Hadamard–Mercer-type inequalities and related estimates for multiplicative convex functions with applications |
| title_fullStr | Generalized Hermite–Hadamard–Mercer-type inequalities and related estimates for multiplicative convex functions with applications |
| title_full_unstemmed | Generalized Hermite–Hadamard–Mercer-type inequalities and related estimates for multiplicative convex functions with applications |
| title_short | Generalized Hermite–Hadamard–Mercer-type inequalities and related estimates for multiplicative convex functions with applications |
| title_sort | generalized hermite–hadamard–mercer-type inequalities and related estimates for multiplicative convex functions with applications |
| topic_facet | Hermite--Hadamard type inequalities Mercer type inequalities Multiplicative convex functions Trapezoid inequalities Special means. Hermite--Hadamard type inequalities Mercer type inequalities Multiplicative convex functions Trapezoid inequalities Special means. |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/9424 |
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