Holder Inequality Research Articles

On the Arithmetic and Geometric Means and on Holder's Inequality

On the Arithmetic and Geometric Means and on Holder's Inequality

On a Converse of the Holder Inequality

On a Converse of the Holder Inequality

On a converse of the Hölder inequality

Let X be a measure space with measure pt. We consider the associated dual metric spaces LP and L , where 1 M.(Fn) for all n. The stronger condition will be implied if we show that 1u(F) = + oo. If not, then A(E F) = +00, since M (E) = +00. By the assumed weaker condition on X, E-F contains a subset F' of finite but positive measure. Then, if 1u(F) < + oo, we have M< (FU F') < + 0, contrary to the choice of M. To prove necessity of the condition, we note that if X contains a subset E, of measure + oo, whose subsets have measure 0 or + 0o, then the characteristic function of E is evidently not LP. But if g is any Lq function, the set {xIg(x) #0 } is a countable union of sets of finite measure, and so intersects E in a set of measure 0. So the product of g with the characteristic function of E is 0 almost everywhere and thus is integrable, giving a denial of the converse of the Holder inequality. Before proving sufficiency, we shall show that if f is any measurable function whose product with every Lq function is integrable, and if {xjf(x) } #0 is a countable union of sets of finite measure, then fCLP, independent of any condition on X. In this case there is a sequence, {f.,}, of functions each of which is bounded and vanishes except on a set of finite measure, and such that the sequence { Iffl P} converges monotonely to Ifj P. Each of the functions fn, determines a

Dresher's Inequality

Abstract : This note gives an elementary proof of the Dresher inequality, based on the Minkowski inequality and an inequality due to Radon. The Radon inequality is gotten easily by transforming the Holder inequality; thus Dresher's inequality is seen to be a melange of the Holder and Minkowski inequalities. It is appropriate to note here that Dresher's inequality is a generalization of an inequality due to Beckenbach.