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On Levinson's inequality involving averages of 3- convex functions

By using the integral arithmetic mean and the Levinson inequality we give the extension for $3$-convex function of Wulbert's result from \cite{;wulbert};. Also, we obtain inequalities with divided differences using the Levinson inequality. As a consequence, the convexity of higher order for function define by divided difference is proved. Further, we construct a new family of exponentially convex functions and Cauchy-type means by exploring at linear functionals with the obtained inequalities.

Levinson's inequality, divided differences, $n$-convex function, $(n, m)$-convex function, means, exponential convexity

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