Inequalities that sharpen the triangle inequality for sums of $N$ functions in $L^p$

Cárlen, Eric A.; Frank, Rupert L.; Lieb, Élliott H.

doi:10.4310/arkiv.2020.v58.n1.a4

Cited by 3 publications

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References 8 publications

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“…and, by Lemma 5.2, the right side is bounded above by 2 1/ p (1+γ ) 2−1/ p , which proves the inequality, Note Since this paper was submitted, two further papers by the authors [5,8] have appeared which, in particular, explore extensions of the inequalities discussed here to more than three functions. The results are less complete than for two functions.…”

Section: A Generalization To Schatten Normssupporting

confidence: 52%

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Inequalities for $$L^p$$-Norms that Sharpen the Triangle Inequality and Complement Hanner’s Inequality

et al. 2020

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In 2006 Carbery raised a question about an improvement on the naïve norm inequality f + g p p ≤ 2 p−1 (f p p + g p p) for two functions f and g in L p of any measure space. When f = g this is an equality, but when the supports of f and g are disjoint the factor 2 p−1 is not needed. Carbery's question concerns a proposed interpolation between the two situations for p > 2 with the interpolation parameter measuring the overlap being f g p/2. Carbery proved that his proposed inequality holds in a special case. Here, we prove the inequality for all functions and, in fact, we prove an inequality of this type that is stronger than the one Carbery proposed. Moreover, our stronger inequalities are valid for all real p = 0.

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Section: A Generalization To Schatten Normssupporting

confidence: 52%

“…Indeed, when c ∈ (0, 1/2) this follows from the fact that all its coefficients are positive. When c ∈ (1/2, 1) we observe that the parabola p is minimized on R at t = 2c 2 −1 (c+1)(1+2c) , and its minimal value is (5…”

Section: Proof Of the Inequalitymentioning

confidence: 94%