Open AccessThe Proofs of Product Inequalities in Vector Spaces
Author(s) -
Belinda Barnes,
E.D.J. Owusu-Ansah,
S. K. Amponsah,
C. Sebil
Publication year - 2018
Publication title -
european journal of pure and applied mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.245
H-Index - 5
ISSN - 1307-5543
DOI - 10.29020/nybg.ejpam.v11i2.3209
Subject(s) - mathematics , product (mathematics) , combinatorics , euclidean space , triangle inequality , interval (graph theory) , mathematical proof , product topology , space (punctuation) , euclidean geometry , discrete mathematics , geometry , linguistics , philosophy
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 allu, v ∈ [2, ∞). The first product inequality u v ≤ u + v holds forany two vectors in the interval [0, 1] in Holder’s space and also valid anytwo vectors in the interval [1, 2] in the Euclidean space. On the otherhand, the second product inequality u + v ≤ u v ∀u, v ∈ [2, ∞)only in Euclidean space. By applying the first product inequality to theL p spaces, we observed that if f : Ω → [0, 1], and g : Ω → R, thenf p g p ≤ f p + g p . Also, if f, g : Ω → R, then f p + g p ≤f p g p .