On Generalized Holder Inequality

A FAMILY of inequalities concerning inner products of vectors and functions began with Cauchy. The extensions and generalizations later led to the inequalities of Schwarz, Minkowski and Holder. The well known Holder inequality involves the inner product of vectors measured by Minkowski norms. In this paper, another step of extension is taken so that a Holder type inequality may apply to general, paired non-Euclidean norms. We restrict the discussion to finite dimensional spaces. A theorem is established for such generalized Holder inequality together with its sharpness condition. The new sharpness condition involves the gradient of the norm function. The previous inequalities of this type and their associated sharpness conditions become special cases of this new inequality which is a fundamental primal-dual relation of two spaces defined by paired norms on their respective elements. Given an arbitrarily defined primal norm in one space, a method is presented for constructing the dual norm in the other. The generalized Holder inequality is the basis for many known minimax (duality) theorems in applied fields. Particular examples of this duality are found in mathematical models of plastic behavior. Inequalities appear frequently in algebra, geometry and analysis. This form of mathematical statements is so prevalent that it is difficult to state their relevance and applications exhaustively. Inequalities are used to demarcate numbers, vectors, matrices and functions, e.g. to be positive definite; to define sets, norm measures, convexity of sets and functions; to compare and bound functions with another function just to name a few. There are books devoted exclusively to inequalities [ 11. A class of inequalities concerning inner products of vectors and functions can be grouped into two frequently encountered ones in literature although one is a special case of the other. The Schwarz inequality applies to the Euclidean and Hilbert spaces [2]. The Holder inequality extends the concept to certain non-Euclidean and non-Hilbert spaces associated with a specific family of norm measures defined by Minkowski [3]. We shall restrict our discussion to finite dimensional spaces. A further extension of the Holder inequality is presented in this paper for general paired non-Euclidean spaces. The familiar Schwarz inequality for the Euclidean space R” can be stated for the inner product IX’YI 5 Il~ll*llYll29 (1)