INEQUALITY OF HARDY-TYPE FOR n-CONVEX FUNCTION VIA INTERPOLATION POLYNOMIAL AND GREEN FUNCTIONS

We obtain new results on the Hardy-type inequality in the general context, in terms of measure spaces with positive sigma-finite measures. The connection is made between the difference operator derived from the Hardy-type inequality on the one hand and the expression containing the interpolating polynomial of Abel-Gontscharoff and the four Green functions on the other hand. We discuss the n-convexity of the function and consider the result depending on the parity of the indexes n and m. Further results are obtained by using the Holder inequality for conjugate exponents p and q. Finally, we derive upper bounds for the remainder, obtained from the main result, using the Cebysev functional. The Ostrowski-type bound for the generalized Hardy inequality is also given.