Abstract
In this article, we are interested in some well-known dynamic inequalities on time scales. For this reason, we will prove some new Hermite–Hadamard (H-H) and Opial dynamic inequalities on time scales. The main results here will be derived via the dynamic integration by parts and chain rule formulas on time scales. In addition, we will extend and unify the inequalities for the convex functions.
Highlights
1 Introduction In 1893, the H-H inequality was established for a convex function ℘ on a given interval [d1, d2] in [1]:
The study of the H-H inequality have been attracted the attention of many scholars
It is well known that the theory of time scales was originated by Hilger in his Ph.D. thesis [23]
Summary
In 1893, the H-H inequality was established for a convex function ℘ on a given interval [d1, d2] in [1]:. It is well known that the theory of time scales was originated by Hilger in his Ph.D. thesis [23] After that, this setting was evolved by many researchers, for more details refer to [24, 25]. Over the recent couple of years, there has been growing interest in the study of dynamic inequalities on time scales and this has become an important field in applied and pure mathematics; see for details [25,26,27,28,29,30]. This article is devoted to establishing some dynamic H-H and Opial inequalities on time scales. Let ℘1, ℘2 : [d1, d2]T0 → R be integrable functions such that ℘1 of one sign and decreasing and 0 ≤ ℘2(s) ≤ 1 for each s ∈ [d1, d2]T0. Holds for each θ ∈ T0[d1,d2] ⊆ [0, 1]
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