Important Study on the ∇ Dynamic Hardy–Hilbert-Type Inequalities on Time Scales with Applications

Abstract

The main objective of the present article is to prove some new ∇ dynamic inequalities of Hardy–Hilbert-type on time scales. We present and prove very important generalized results with the help of the Fenchel–Legendre transform, submultiplicative functions, and Hölder’s and Jensen’s inequality on time scales. We obtain some well-known time scale inequalities due to Hardy–Hilbert inequalities. For some specific time scales, we further show some relevant inequalities as special cases: integral inequalities and discrete inequalities. Symmetry plays an essential role in determining the correct methods for solutions to dynamic inequalities

New Results on Boas–Bellman-Type Inequalities in Semi-Hilbert Spaces with Applications

“…Inequalities play a crucial role in analysis and find applications in various areas of mathematics (see [1][2][3][4][5][6][7] and related sources). Among these, Bessel's inequality and the Boas-Bellman inequality hold significant importance and are widely used in the study of operators on Hilbert spaces.…”