The Proofs of Product Inequalities in Vector Spaces
Keywords:product inequality, first product nequality, second product inequality, Holderâ€™s space, Euclidean space, Cauchy-Schwarz inequality
AbstractIn this paper, we introduce the proofs of product inequalities:
u v â‰¤ u + v , for all u, v âˆˆ [0, 2], and u + v â‰¤ u v , for all
u, v âˆˆ [2, âˆž). The first product inequality u v â‰¤ u + v holds for
any two vectors in the interval [0, 1] in Holderâ€™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 â‰¤ u v âˆ€u, v âˆˆ [2, âˆž)
only in Euclidean space. By applying the first product inequality to the
L p spaces, we observed that if f : Î© â†’ [0, 1], and g : Î© â†’ R, then
f p g p â‰¤ f p + g p . Also, if f, g : Î© â†’ R, then f p + g p â‰¤
f p g p .
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