Characteristics of nonlinear positive functionals and their applications

Abstract

In 1966-1967, Beckenbach [3,4] extended, among other things, the Holder and Minkowski inequalities by a method of the differentiability of multivariable real functions. In 1979, Wang [21] established the A-G (arithmetic-geometric), Holder, Minkowski, and Beckenbach inequalities by means of the convexity of multivariable real functions. However, their results dealt with only the discrete case of the inequalities. In the development of the theory of inequalities, the discrete and continuous cases of the inequalities have always been examined hand in hand (see Beckenbach and Bellman [S ], Hardy, Littlewood, and Polya [ 111, Iwamoto and Wang [ 131, and Mitrinovic [ 16]), This examination has concerned not only the forms of the inequalities but also the methods used to establish them (e.g., see Iwamoto and Wang [ 131). For this reason, we investigate here the continuous counterparts of the works of Beckenbach and Wang mentioned above. Indeed, the set of classical inequalities for integrals has been established by various other methods (e.g., see [S, 11, 13, 161) as well. It is evident that the forms of inequalities for integrals can be regarded as nonlinear functionals (see [ 7, 14, 191). On the other hand, in the study of nonlinear operators (and/or functionals), the differentiability of operators has been intensively and successfuly used to provide approximate solutions for nonlinear functional equations (e.g., see [7, 14, 19]), while their convexity has been used in a somewhat restricted manner (see Collatz [7, 334-341 I). Although the differentiability and convexity of nonlinear functionals have attracted considerable attention of many investigators (see [7, 9, 14, 15, 17-19]), it seems to have escaped notice that they can be employed to establish the