The diamond-alpha Riemann integral and mean value theorems on time scales

时间尺度上带有 Nabla 与 Diamond-Alpha 导数的 Y 函数及 L'Hospital 型单调性法则

Y-function and L'Hospital-type Monotonicity Rules with Nabla and Diamond-Alpha Derivatives on Time Scales

该研究聚焦时间尺度微积分，建立结合 nabla 与 diamond-alpha 导数的 $Y$-函数及 L'Hospital 型单调性法则。

Holder's and Hardy's Two Dimensional Diamond-alpha Inequalities on Time Scales

We prove a two dimensional Holder and reverse-Holder inequality on time scales via the diamond-alpha integral. Other integral inequalities are established as well, which have as corollaries some recent proved Hardy-type inequalities on time scales.

Diamond-alpha Polynomial Series on Time Scales

The objective of this paper is twofold: (i) to survey existing results of generalized polynomials on time scales, covering definitions and properties for both delta and nabla derivatives; (ii) to extend previous results by using the more general notion of diamond-alpha derivative on time scales. We introduce a new notion of combined-polynomial series on a time scale, as a convex linear combination of delta and nabla generalized series. Main results are formulated for homogenous time scales. As an example, we compute diamond-alpha derivatives on time scales for delta and nabla exponential funct…

Diamond-alpha Integral Inequalities on Time Scales

The theory of the calculus of variations was recently extended to the more general time scales setting, both for delta and nabla integrals. The primary purpose of this paper is to further extend the theory on time scales, by establishing some basic diamond-alpha dynamic integral inequalities. We prove generalized versions of Hölder, Cauchy-Schwarz, Minkowski, and Jensen's inequalities. For the particular case when alpha is equal to one or alpha is equal to zero one gets, respectively, correspondent delta and nabla inequalities. If we further restrict ourselves by fixing the time scale to the r…

A Study of Diamond-alpha dynamic equations on regular time scales

We introduce the diamond-alpha exponential function on time scales. As particular cases, one gets both delta and nabla exponential functions. A method of solution of a homogenous linear dynamic diamond-alpha equation on a regular time scale is investigated, and examples of diamond-alpha exponential functions are presented.

Inequalities of Ostrowski's type for m- and (alpha,m)- logarithmically convex functions via Riemann-Liouville Fractional integrals

In this paper, we establish some new Ostrowski's type inequalities for m- and (alpha,m)- logarithmically convex functions by using the Riemann-Liouville fractional integrals.