The Proofs of Product Inequalities in Vector Spaces

Abstract

In this paper, we introduce the proofs of product inequalities:
u v ≤ u + v , for all u, v ∈ [0, 2], and u + v ≤ u v , for all
u, v ∈ [2, ∞). The first product inequality u v ≤ u + v holds for
any two vectors in the interval [0, 1] in Holder’s space and also valid any
two vectors in the interval [1, 2] in the Euclidean space. On the other
hand, the second product inequality u + v ≤ u v ∀u, v ∈ [2, ∞)
only in Euclidean space. By applying the first product inequality to the
L p spaces, we observed that if f : Ω → [0, 1], and g : Ω → R, then
f p g p ≤ f p + g p . Also, if f, g : Ω → R, then f p + g p ≤
f p g p .