TheoremStatement: Literature-sourcedProof: AI-generatedprecheck passaudited 2026-09-05

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One-sided Hardy-Littlewood inequalities for the Dini derivatives of a continuous monotone function

Statement

Let $F : [a, b] \to R$ be continuous and nondecreasing (Nondecreasing, increasing (strictly increasing), nonincreasing, decreasing, monotone and strictly monotone real functions on a subset of $R$ , with the dictionary to monotone sequences). Let $[u, v] \subseteq [a, b]$ with $u < v$ .

For every $R > 0$ , if $E_{R}^{+} (u, v) : = {x \in [u, v) : D^{+} F (x) > R},$ then $R λ (E_{R}^{+} (u, v)) \leq F (v) - F (u) .$
For every $0 < r < R$ , if $E_{r, R}^{-} (u, v) : = {x \in (u, v] : D_{-} F (x) < r},$ then $(R - r) λ (E_{r, R}^{-} (u, v)) \leq R (v - u) - F (v) + F (u) .$

Here $λ$ is Lebesgue measure from Lebesgue measurable sets, the family $L (R^{n})$ , and the restricted set function $λ_{n}$ .

Facts & Assumptions

Given: The continuous nondecreasing function $F : [a, b] \to R$ and the subinterval $[u, v] \subseteq [a, b]$ .

[A1]

The symbols are those of the statement.

Proof

technique · direct

1.1givenalgebra

Fix $R > 0$ and put $G (x) : = F (x) - R x$ on $[u, v]$ . If $x \in E_{R}^{+} (u, v)$ , then some $y \in (x, v]$ satisfies $\frac{F (y) - F (x)}{y - x} > R$ , hence $G (y) > G (x)$ . Therefore $E_{R}^{+} (u, v)$ is contained in the rising-sun set of $G$ on $[u, v]$ . By Riesz's rising sun lemma with the correct endpoint conclusion, the components of that set are intervals $I_{n}$ whose left and right endpoints we call $c_{n}, d_{n}$ , with each $I_{n}$ either $[u, d_{n})$ or $(c_{n}, d_{n})$ , and with $G (c_{n}) \leq G (d_{n})$ for every $n$ . Thus $R (d_{n} - c_{n}) \leq F (d_{n}) - F (c_{n})$ for every $n$ . Summing over finitely many components and using that the disjoint ordered intervals $I_{n}$ lie in $[u, v]$ gives $R \sum_{n \leq N} (d_{n} - c_{n}) \leq \sum_{n \leq N} (F (d_{n}) - F (c_{n})) \leq F (v) - F (u)$ , so $R λ (E_{R}^{+} (u, v)) \leq F (v) - F (u)$ .

2.1step 1.1algebra

Fix $0 < r < R$ and put $H (x) : = R x - F (x)$ on $[u, v]$ . If $x \in E_{r, R}^{-} (u, v)$ , then for some $y \in [u, x)$ one has $\frac{F (x) - F (y)}{x - y} < r$ , hence $H (x) - H (y) > (R - r) (x - y) > 0$ . Reflecting $H$ across the midpoint of $[u, v]$ turns this into the right-hand rising-sun situation on a continuous function, so the same argument as in step 1.1 yields a disjoint family of intervals whose total length bounds $λ (E_{r, R}^{-} (u, v))$ and on each such interval $H (d) - H (c) \geq (R - r) (d - c)$ . Summing gives $(R - r) λ (E_{r, R}^{-} (u, v)) \leq H (v) - H (u) = R (v - u) - F (v) + F (u)$ .

3.1step 1.1step 2.1∎

Steps 1.1 and 2.1 are the two asserted inequalities.

Depends on

Used by

A continuous monotone function is differentiable almost everywhere by the rising-sun route Theorem

Dependency tree · two levels

19 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

Terence Tao, An Introduction to Measure Theory, Lemma 1.6.26 (standard reference, not scraped)
Frigyes Riesz, Sur l’existence de la dérivée des fonctions monotones et sur quelques problèmes qui s’y rattachent, Section 3 (standard reference, not scraped)