An Inequality for the Intermediate Point in Lagrange’s Theorem, Using Quadrature Formulas

Abstract

Starting from an inequality presented by professor A. Lupas (1973), which shown that the intermediate point of the logarithmic function in Lagrange’s theorem is between the arithmetic mean and the geometric mean of the the limits of the interval on which the function is defined, in this paper we will find a sharp estimation of the intermediate point, using a quadrature formula found by I. Popa in 2005, which was generalized by A.N. Branga in 2010.