Abstract
Our work is based on the multiple inequalities illustrated in 2020 by Hamiaz and Abuelela. With the help of a Fenchel-Legendre transform, which is used in various problems involving symmetry, we generalize a number of those inequalities to a general time scale. Besides that, in order to get new results as special cases, we will extend our results to continuous and discrete calculus.
Highlights
In 2020, Hamiaz and Abuelela [1] have studied the following discrete inequalities: Theorem 1
We present the Fenchel-Legendre transform and refer, for example, to [11–13], for more details
By using Fubini’s theorem and the Fenchel-Legendre transform, which is used in various problems involving symmetry, we extend the discrete results proved in [1] on time scales
Summary
In 2020, Hamiaz and Abuelela [1] have studied the following discrete inequalities: Theorem 1. Assuming h : Rn −→ R ∪ {+∞} is a function: h 6= +∞ i.e., Dom(h) = { x ∈ Rn , |h( x ) −h( x ), x ∈ Dom(h)}. Let h be a function and h∗ its Fenchel-Legendre transform. We write Fubini’s theorem on time scales. In this manuscript, by using Fubini’s theorem and the Fenchel-Legendre transform, which is used in various problems involving symmetry, we extend the discrete results proved in [1] on time scales. Our results can be applied to give more general forms of some previously proved inequalities through substituting h and h∗ by suitable functions as we will see in the following two sections.
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