Inequalities on time scales: a survey. (English) Zbl 1021.34005

This paper surveys a variety of classical inequalities for real-valued functions defined on time scales, i.e., closed subsets of the reals. More detailed, the authors present inequalities between integrals (Hölder, Cauchy-Schwarz, Minkowski), for convex functions (Jensen’s inequality), the Bernoulli inequality, tools for linear and nonlinear integral inequalities (Gronwall and Bihari, respectively), different versions of Wirtinger’s inequality, as well as Lyapunov’s inequality for Sturm-Liouville and linear Hamiltonian dynamic equations.
All the above considerations are based on the concept of delta derivatives and the corresponding Cauchy integral (cf. S. Hilger [Result. Math. 18, No. 1/2, 18–56 (1990; Zbl 0722.39001)] or the monograph of the second and third author [Dynamic equations on time scales. An introduction with applications. Basel: Birkhäuser (2001; Zbl 0978.39001)]). Hence, the integrands are assumed to be rd-continuous throughout the paper. Finally, some results are illustrated by examples on different time scales.


34A40 Differential inequalities involving functions of a single real variable
39A13 Difference equations, scaling (\(q\)-differences)
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