By using the Levinson inequality we give the extension for 3-convex functions
of Wulbert's result from Favard's Inequality on Average Values of Convex
Functions, Math. Comput. Model. 37 (2003), 1383{1391. Also, we obtain
inequalities with divided differences, and as a consequence, the convexity of
higher order for function defined by divided difference is proved. Further, we
construct a new family of exponentially convex functions and Cauchy-type
means by looking at linear functionals associated with these new
inequalities.