The Proofs of Product Inequalities in Vector Spaces

Authors

  • Benedict Barnes KWAME NKRUMAH UNIVERSITY OF SCIENCE AND TECHNOLOGY
  • E.D.J. Owusu-Ansah
  • S.K. Amponsah
  • C. Sebil

DOI:

https://doi.org/10.29020/nybg.ejpam.v11i2.3209

Keywords:

product inequality, first product nequality, second product inequality, Holder’s space, Euclidean space, Cauchy-Schwarz inequality

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 .

Author Biography

  • Benedict Barnes, KWAME NKRUMAH UNIVERSITY OF SCIENCE AND TECHNOLOGY

    MATHEMATICS DEPARTMENT

    LECTURER

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Published

2018-04-27

Issue

Section

Functional Analysis

How to Cite

The Proofs of Product Inequalities in Vector Spaces. (2018). European Journal of Pure and Applied Mathematics, 11(2), 375-389. https://doi.org/10.29020/nybg.ejpam.v11i2.3209

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