TheoremStatement: AI-adaptedProof: AI-generatedprecheck passverified 2026-09-24 (gpt-6-sol)

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The $p$ -power triangle inequality for nonnegative functions when $0 < p < 1$

Statement

Let $0 < p < 1$ and let $f, g \in L^{p} (μ)$ be nonnegative. Then

$∥ f + g ∥_{p}^{p} \leq ∥ f ∥_{p}^{p} + ∥ g ∥_{p}^{p} .$

Equivalently,

$\int (f + g)^{p} d μ \leq \int f^{p} d μ + \int g^{p} d μ .$

Facts & Assumptions

Given: An exponent $0 < p < 1$ and nonnegative functions $f, g \in L^{p} (μ)$ .

[L1]

For $0 < p < 1$ and $0 < u \leq 1$ , $p \log u \geq \log u$ , hence $u^{p} \geq u$ by the definition of real power and strict increase of the exponential. The same inequality holds at $u = 0$ (Real powers for positive bases, with the zero-base positive-exponent convention, Order, continuity, range, and the product, quotient, and reciprocal laws for the natural logarithm, The exponential function is strictly increasing).

[L2]

The nonnegative integral is monotone and additive (Monotonicity and nonnegative homogeneity of the nonnegative integral, Additivity of the nonnegative Lebesgue integral).

Proof

technique · For nonnegative numbers $a,b$ and $0<p<1$ one has $(a+b)^p\le a^p+b^p$. Apply this pointwise to $f$ and $g$ and integrate

1.1L1givenalgebra

For nonnegative reals $a, b$ with $a + b > 0$ , set $s = a + b$ , $u = a / s$ and $v = b / s$ . Then $u, v \in [0, 1]$ , $u + v = 1$ , and [L1] gives $a^{p} + b^{p} = s^{p} (u^{p} + v^{p}) \geq s^{p} = (a + b)^{p}$ ; for $a + b = 0$ equality is immediate. Apply this scalar inequality pointwise to $f, g$ to obtain [L1, given, algebra] $(f + g)^{p} \leq f^{p} + g^{p} .$

2.1L2step 1.1∎

Integrating and using monotonicity and additivity from [L2] yields [step 1.1, L2] $\int (f + g)^{p} d μ \leq \int f^{p} d μ + \int g^{p} d μ .$ This is exactly the displayed $∥ \cdot ∥_{p}^{p}$ inequality.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

22 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Richard L. Wheeden and Antoni Zygmund, Measure and Integral, Theorem 8.16 (standard reference, not scraped)
John K. Hunter, Measure Theory, reverse inequality discussion before Definition 7.6 (standard reference, not scraped)

The pp-power triangle inequality for nonnegative functions when 0<p<10<p<1