Dynamic Hilbert-Type Inequalities with Fenchel-Legendre Transform

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.

New dynamic Hilbert-type inequalities in two independent variables involving Fenchel–Legendre transform

“…These inequalities extend and give more general new forms of several previously established inequalities. For example, we generalize the results given in [7,20] and others.…”

“…Very recently, El-Deeb et al [7] studied the time scales version of the above inequalities. They proved that if A(s) :=…”

On some new double dynamic inequalities associated with Leibniz integral rule on time scales

“…If k denotes the derivative of k with respect to the first variable, then f (t) = Other dynamic inequalities on time scales may be found in [37][38][39][40]. In this manuscript, we will discuss the retarded time scale case of the inequalities obtained in [1] using new techniques by replacing the upper limitς andˆ of the integral by the delay functionα(ς) ≤ ς andβ(ˆ ) ≤ˆ .…”

A Variety of Nabla Hardy’s Type Inequality on Time Scales

“…In [10][11][12][13][14] many authors have studied many new dynamic inequalities. Řehák [14] is the first author proved the version of Hardy inequality on time scales that unifies (1) and (2).…”