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Generalization of Jensen's Inequality and its Converses by Lidstone's Polynomial and Majorization Theorem

In this paper, using majorization theorems, Lidstone's interpolating polynomials and conditions on Green's function we obtain results concerning Jensen's inequality and converses of this inequality in both the integral and the discrete case. We use these generalizations to construct a linear functional and we present mean value theorems and n-exponential convexity which leads to exponential convexity and then log-convexity for these functionals. We give some families of functions which enable us to construct a large families of functions that are exponentially convex and also give Stolarsky type means with their monotonicity.

Majorization ; Green function ; Jensen inequality ; (2n)-convex function ; Lidstone polynomial ; Cauchy type mean value theorems ; n-exponential convexity ; exponential convexity ; log-convexity ; means.

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