Rabia Bibi

Integral Inequalities on Time Scales via the Theory of Isotonic Linear Functionals

In this paper, we present a series of inequalities for the time-scale integral. Among the inequalities presented, we offer time-scale versions of Jensen’s-inequality, Jensen-type inequalities, converses of Jensen’s inequality, inequalities for means, H¨ older’s inequality,

Time scales integral inequalities for superquadratic functions

cari´ Abstract. In this paper, two different methods of proving Jensens in- equality on time scales for superquadratic functions are demonstrated. Some refinements of classical inequalities on time scales are obtained us- ing properties of superquadratic functions and some known results for isotonic linear functionals.

Jensen's Functionals on Time Scales

We consider Jensen’s functionals on time scales and discuss its properties and applications. Further, we define weighted generalized and power means on time scales. By applying the properties of Jensen’s functionals on these means, we obtain several refinements and converses of Holder’s inequality on time scales.

Generalization of Jensen-type linear functional on time scales via lidstone polynomial

In this paper, we prove Jensen’s inequality for diamond integrals on time scales and generalize this inequality for 2n-convex functions via Lidstone’s polynomials. Bounds for Čhebyšhev functional, Grüss-type inequality and Ostrowaski-type inequality related to Jensen’s type functional are also given in the paper. Moreover, our results are valid if we replace diamond integrals with delta integrals, nabla integrals or -diamond integrals on time scales as well (being special cases).

Refinements and Generalizations of Functional Dresher's and Bellman's Inequalities

We give refinements and generalizations of the Dresher and Bellman inequalities for positive linear functionals. We also give reverse of the new obtained generalized version of these inequalities. Finally, we apply our results on time scales integrals to obtain refinements and generalizations of time scales Dresher’s and Bellman’s inequalities.

Improvements of Jensen-Type Inequalities for Diamond-α Integrals.

We give further improvements of the Jensen inequality and its converse on time scales, allowing also negative weights. These results generalize the Jensen inequality and its converse for both discrete and continuous cases. Further, we investigate the exponential and logarithmic convexity of the differences between the left-hand side and the right-hand side of these inequalities and present several families of functions for which these results can be applied.

Jensen Functionals on Time Scales for Several Variables

We define Jensen functionals and concerned generalized means for several variables on time scales. We derive properties of Jensen functionals and apply them to generalized means. In this setting, we obtain generalizations, refinements, and conversions of many remarkable inequalities.