The thesis deals with the Lidstone and Hermite interpolating polynomials and Euler's integral identities which extend the well known formula for expansion of function with Bernoulli polynomials, which is given in [31]. The research is presented for the higher order convex functions. After the Introduction, the thesis is divided into five chapters. The basic concepts which are used in the thesis are given in the first chapter. In the second chapter, using Lidstone's and Hermite's interpolating polynomials with conditions on the Green functions, and also Euler's identity, generalizations of Jensen's inequality and converses of Jensen's inequality for regular, real (signed) Borel measure are obtained, and as a consequence Hermite-Hadamard inequality is presented. In the third chapter, using majorization theorems, Lidstone's and Hermite's interpolating polynomials and conditions on the Green function, results concerning Jensen's inequality, its converses and Jensen-Steffensen's inequality is also developed in both the integral and discrete case. Using Chebyshev functionals, bounds for identities related to these inequalities are observed and Grüss type inequalities and Ostrowsky type inequalities for these functionals are obtained. Considering Cauchy's error representation of the interpolation polynomials, corresponding generalizations of the Hermite-Hadamard inequality are also obtained. As a special case, for Hermite's interpolating polynomials, generalization of the Hermite-Hadamard inequality for the zeros of orthogonal polynomials is considered. By using these generalizations, at the end of second, third and fourth chapter, linear functionals are constructed and mean value theorems, n-exponential convexity, exponential convexity and log-convexity for these functionals are discussed. Furthermore, some families of functions which enable us to construct a large families of functions that are exponentially convex and also Stolarsky type means with their monotonicity are given. In the last chapter, integral error representation related to Hermite's interpolating polynomial is also considered and some new estimations for the remainder in quadrature formulae of Hermite type is derived by using Hölder's inequality and some inequalities for the Chebyshev functional.