In the presented paper, Levinson's inequality for the 3-convex function is generalized by using two Green functions.Čebyšev-, Grüss-and Ostrowski-type new bounds are found for the functionals involving data points of two types. Moreover, the main results are applied to information theory via the f -divergence, the Rényi divergence, the Rényi entropy, the Shannon entropy and the Zipf-Mandelbrot law.

In this article, we present new fractional Hadamard and Fejér-Hadamard inequalities for generalized fractional integral operators containing Mittag-Leffler function via a monotone function. To establish these inequalities we will use exponentially m-convex functions. The presented results in particular contain a number of fractional Hadamard and Fejér-Hadamard inequalities for functions deducible from exponentially m-convex functions.

Abstract. We obtained useful identities via generalized Montgomery identity, by which the inequality of Popoviciu for convex functions is generalized for higher order convex functions. We investigate the bounds for the identities related to the generalization of the Popoviciu inequality using inequalities for theČebyšev functional. Some results relating to the Grüss and Ostrowski type inequalities are constructed. Further, we also construct new families of exponentially convex functions and Cauchy-type means by looking at linear functionals associated with the obtained inequalities.Mathematics subject classification (2010): Primary 26D07, 26D15, 26D20, 26D99.

In this paper, Levinson type inequalities are studied for the class of higher order convex functions by using Abel-Gontscharoff interpolation. Cebyšev, Grüss, and Ostrowski-type new bounds are also found for the functionals involving data points of two types.

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