Some Opial Dynamic Inequalities Involving Higher Order Derivatives on Time Scales

We will prove some new Opial dynamic inequalities involving higher order derivatives on time scales. The results will be proved by making use of H？lder's inequality, a simple consequence of Keller's chain rule and Taylor monomials on time scales. Some continuous and discrete inequalities will be derived from our results as special cases. 1. Introduction In the past decade a number of Opial dynamic inequalities have been established by some authors which are motivated by some applications; we refer to the papers [1–3]. The general idea is to prove a result for a dynamic inequality where the domain of the unknown function is a so-called time scale , which may be an arbitrary closed subset of the real numbers , to avoid proving results twice, once on a continuous time scale which leads to a differential inequality and once again on a discrete time scale which leads to a difference inequality. The three most popular examples of calculus on time scales are differential calculus, difference calculus, and quantum calculus (see [4]), that is, when , and where . A cover story article in New Scientist [5] discusses several possible applications of time scales. In this paper, we will assume that and define the time scale interval by . Since the continuous and discrete inequalities involving higher order derivatives are important in the analysis of qualitative properties of solutions of differential and difference equations [6–8], we also believe that the dynamic inequalities involving higher order derivatives on time scales will play the same effective act in the analysis of qualitative properties of solutions of dynamic equations [2, 3, 9]. To the best of the author’s knowledge there are few inequalities involving higher order derivatives established in the literature [10–13]. In the following, we recall some of these results that serve and motivate the contents of this paper. In [13] the authors proved that if is delta differentiable times with , for , and is a positive rd-continuous function on , then In [10] it is proved that if is delta differentiable times ( odd) with , for , then where and satisfy . Also in [10] it is proved that if is delta differentiable times with , for , and is increasing, then where and satisfy . As a generalization of (1.3) it is proved in [10] that if is delta differentiable times with , for , and is increasing, then where and satisfy . In [12] the authors proved that if and are positive rd-continuous functions on such that is nonincreasing, and is delta differentiable times with , for , then where and . For contributions of