Supplements of Hölder's Inequality

Abstract

Given vectors and (or functions f(x) and g(x)) we define the Hölder Quotient Hpq by1or in case of functions by2Here ‖·‖p and ‖·‖q are the usual Lp and Lq norms. We assume throughout thatIf p and q are both greater than one then they are positive but if we allow p and q to be less than one then one of them must be positive and the other one must be negative. This may cause a problem if for example, some value ai is zero and p is negative. In this case we use the convention that and