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Keywords = Specht’s ratio

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19 pages, 301 KB  
Article
Some New Generalizations of Reverse Hilbert-Type Inequalities via Supermultiplicative Functions
by Haytham M. Rezk, Ahmed I. Saied, Ghada AlNemer and Mohammed Zakarya
Symmetry 2022, 14(10), 2043; https://doi.org/10.3390/sym14102043 - 30 Sep 2022
Viewed by 1277
Abstract
Our work in this paper is based on the reverse Hölder-type dynamic inequalities illustrated by El-Deeb in 2018 and the reverse Hilbert-type dynamic inequalities illustrated by Rezk in 2021 and 2022. With the help of Specht’s ratio, the concept of supermultiplicative functions, chain [...] Read more.
Our work in this paper is based on the reverse Hölder-type dynamic inequalities illustrated by El-Deeb in 2018 and the reverse Hilbert-type dynamic inequalities illustrated by Rezk in 2021 and 2022. With the help of Specht’s ratio, the concept of supermultiplicative functions, chain rule, and Jensen’s inequality on time scales, we can establish some comprehensive and generalize a number of classical reverse Hilbert-type inequalities to a general time scale space. In time scale calculus, results are unified and extended. At the same time, the theory of time scale calculus is applied to unify discrete and continuous analysis and to combine them in one comprehensive form. This hybrid theory is also widely applied on symmetrical properties which play an essential role in determining the correct methods to solve inequalities. As a special case of our results when the supermultiplicative function represents the identity map, we obtain some results that have been recently published. Full article
(This article belongs to the Special Issue Functional Equations and Inequalities 2021)
24 pages, 321 KB  
Article
Some New Generalizations of Reverse Hilbert-Type Inequalities on Time Scales
by Haytham M. Rezk, Ghada AlNemer, Ahmed I. Saied, Omar Bazighifan and Mohammed Zakarya
Symmetry 2022, 14(4), 750; https://doi.org/10.3390/sym14040750 - 6 Apr 2022
Cited by 8 | Viewed by 1679
Abstract
This manuscript develops the study of reverse Hilbert-type inequalities by applying reverse Hölder inequalities on T. We generalize the reverse inequality of Hilbert-type with power two by replacing the power with a new power β,β>1. The main [...] Read more.
This manuscript develops the study of reverse Hilbert-type inequalities by applying reverse Hölder inequalities on T. We generalize the reverse inequality of Hilbert-type with power two by replacing the power with a new power β,β>1. The main results are proved by using Specht’s ratio, chain rule and Jensen’s inequality. Our results (when T=N) are essentially new. Symmetrical properties play an essential role in determining the correct methods to solve inequalities. Full article
20 pages, 817 KB  
Article
Some New Reverse Hilbert’s Inequalities on Time Scales
by Ghada AlNemer, Ahmed I. Saied, Mohammed Zakarya, Hoda A. Abd El-Hamid, Omar Bazighifan and Haytham M. Rezk
Symmetry 2021, 13(12), 2431; https://doi.org/10.3390/sym13122431 - 15 Dec 2021
Cited by 14 | Viewed by 2571
Abstract
This paper is interested in establishing some new reverse Hilbert-type inequalities, by using chain rule on time scales, reverse Jensen’s, and reverse Hölder’s with Specht’s ratio and mean inequalities. To get the results, we used the Specht’s ratio function and its applications for [...] Read more.
This paper is interested in establishing some new reverse Hilbert-type inequalities, by using chain rule on time scales, reverse Jensen’s, and reverse Hölder’s with Specht’s ratio and mean inequalities. To get the results, we used the Specht’s ratio function and its applications for reverse inequalities of Hilbert-type. Symmetrical properties play an essential role in determining the correct methods to solve inequalities. The new inequalities in special cases yield some recent relevance, which also provide new estimates on inequalities of these type. Full article
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