Abstract. In this paper we consider (2n) -convex functions and completely convex functions. Using Lidstone's interpolating polynomials and conditions on Green's functions we present results for Jensen's inequality and converses of Jensen's inequality for signed measure. By using the obtained inequalities, we produce new exponentially convex functions. Finally, we give several examples of the families of functions for which the obtained results can be applied.Mathematics subject classification (2010): 26D15, 26D20, 26D99.

Abstract. In this paper we consider convex functions of higher order. Using the Cauchy's error representation of Hermite interpolating polynomial the results concerning to the HermiteHadamard inequalities are presented. As a special case, generalizations for the zeros of orthogonal polynomials are considered.Mathematics subject classification (2010): Primary 26D15; Secondary 26D07, 26A51.

In this paper, using majorization theorems and Lidstone's interpolating polynomials we obtain results concerning Jensen's and Jensen-Steffensen's inequalities and their converses in both the integral and the discrete case. We give bounds for identities related to these inequalities by using Chebyshev functionals. We also give Grüss type inequalities and Ostrowsky type inequalities for these functionals. Also we use these generalizations to construct a linear functionals 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.

We consider integral error representation related to the Hermite interpolating polynomial and derive some new estimations of the remainder in quadrature formulae of Hermite type, using Holder’s inequality and some inequalities for the Čebyšev functional. As a special case, generalizations for the zeros of orthogonal polynomials are considered.

Abstract. In this paper we consider the Cauchy's error representation of Lidstone interpolating polynomial and as a consequence the results concerning to the Hermite-Hadamard inequalities. Using these inequalities, we produce new exponentially convex functions. Also, we give several examples of the families of functions for which the obtained results can be applied.Mathematics subject classification (2010): Primary 26D15; Secondary 26D07, 26A51.