Abstract
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.
Highlights
We give further improvements of the Jensen inequality and its converse on time scales, allowing negative weights
The diamond-α integral of f from a to b is defined by b b b
The Jensen inequality, the improvement of the Jensen inequality, and their converses are given for time scale integrals
Summary
The combined dynamic derivative, called diamond-α (⬦α) dynamic derivative (α ∈ [0, 1]), was introduced as a linear convex combination of the well-known delta and nabla dynamic derivatives on time scales. In the following two sections of our paper we give some further generalizations of the Jensen-type inequalities on time scales allowing negative weights, and we give the mean-value theorems of the Lagrange and Cauchy type for the functionals obtained by taking the difference of the lefthand side and right-hand side of these new inequalities. These results generalize the results given in [8] for continuous and discrete cases.
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