Steffensen's inequality¶
Bounds a capped, fixed-mass weighting of a monotone function between its integrals over equal-length endpoint intervals.
Core Idea¶
Steffensen's inequality bounds how much a capped weight can change the integral of a monotone function. If real-valued \(f\) is integrable and nonincreasing on \([a,b]\), \(g\) is integrable with \(0\leq g\leq 1\), and \(k=\int_a^b g(t)\,dt\), then[^ref-1611015899ac]
The interval at the right end gives the lower bound; the one at the left end gives the upper bound. For nondecreasing \(f\), reverse both inequality signs. The checked theorem does not require \(f\geq 0\), despite that extra restriction in the original screening seed.[^ref-1611015899ac]
Use Within the Domain¶
Use it in real analysis when an ordered interval, a monotone integrand, and a weight of known total mass satisfying the unit cap are given. The weight may select a set of points with values zero or one, or distribute fractional weight across the interval. The theorem replaces the detailed weight profile with two sharp endpoint comparisons.[^ref-1611015899ac]
Steffensen's 1918 paper has actuarial problems in its title, but its full text was not inspected for this entry. The displayed theorem is supported by a fully inspected later publisher article; the original publisher record supports attribution and metadata only.[ref-38ba1078f7c6][ref-1611015899ac]
Clarity¶
Here \(k\) is both the integral of \(g\) and the length of each comparison interval: \(g\) contributes at most one unit of weight per unit length. For decreasing \(f\), the largest values occur at the left, so concentrating weight there gives the upper bound. Indicator weights on the two endpoint intervals attain equality.[^ref-1611015899ac]
Manages Complexity¶
There are infinitely many admissible weight profiles with the same cap and mass. The inequality gives a valid range for all of them by computing just two integrals. More information about a particular \(g\) may give a narrower estimate.
Abstract Reasoning¶
Check that \(f\) has the stated monotonicity and finite integrals, \(g\) stays between zero and one almost everywhere, and \(k=\int_a^b g\). Then integrate \(f\) over the rightmost and leftmost intervals of length \(k\). Do not use the displayed direction when \(f\) increases; reverse it.[^ref-1611015899ac]
Knowledge Transfer¶
The same rule handles a disconnected indicator weight and a continuously varying fractional weight. Both satisfy the same cap and mass condition; the kind of object used for \(g\) does not change the theorem. These are mathematical examples, not documented actuarial or probability case studies.
Example¶
On \([0,1]\), take \(f(x)=1-x\) and \(g(x)=x\). Then \(k=1/2\) and \(\int_0^1 f(x)g(x)\,dx=1/6\). The right and left half-interval integrals are \(1/8\) and \(3/8\), so \(1/8\leq 1/6\leq 3/8\). This is a constructed fractional-weight example.[^ref-1611015899ac]
Mapped back: \([0,1]\) supplies the order; \(1-x\) is decreasing; \(x\) is the capped weight; \(1/2\) is its total mass; \(1/6\) is the weighted target; and \(1/8,3/8\) are the endpoint comparators.
Relationships to Other Abstractions¶
Current abstraction Steffensen's inequality Domain-specific
Parents (2) — more general patterns this builds on
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Steffensen's inequality is part of Boundedness Prime
The unit bound on g is an identity-bearing constituent that makes fixed mass k pack into an endpoint interval of length k.
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Steffensen's inequality is part of Order Prime
The ordered real interval and monotone comparison are identity-bearing constituents of Steffensen’s endpoint bounds.
Hierarchy paths (4) — routes to 4 parentless roots
- Steffensen's inequality → Boundedness
- Steffensen's inequality → Order → Comparison
- Steffensen's inequality → Order → Relation
- Steffensen's inequality → Order → Set and Membership
Neighborhood in Abstraction Space¶
Steffensen's inequality sits in a sparse region of the domain-specific corpus (76th percentile for distinctiveness): few abstractions share its structure, so a faithful description tends to retrieve it precisely.
Family — Numerical Root-Finding & Quadrature Methods (8 abstractions)
Nearest neighbors
- Average Order of an Arithmetic Function — 0.85
- Càdlàg Function — 0.84
- Riemann–Liouville integral — 0.83
- Jensen's Inequality — 0.82
- Maharam Algebra — 0.81
Computed from structural-signature embeddings · 2026-10-17
Not to Be Confused With¶
- A bound for any weight: allowing \(g>1\) can break the upper bound.
- A bound for any integrand: the displayed direction needs monotonicity.
- The Steffensen numerical method: that is a distinct root-finding procedure.
- A nonnegative-only result: the checked finite-integral theorem allows real-valued \(f\).[^ref-1611015899ac]
References¶
[^ref-1611015899ac]: Josip Pečarić, Anamarija Perušić Pribanić, and Ksenija Smoljak Kalamir, Generalizations of Steffensen's Inequality via Taylor's Formula, Journal of Inequalities and Applications 2015, article 207, Introduction Theorem 1.1 and equations (1.1)–(1.3). Full publisher HTML inspected for the classical theorem and later extensions. [^ref-38ba1078f7c6]: J. F. Steffensen, On Certain Inequalities Between Mean Values, and Their Application to Actuarial Problems, Scandinavian Actuarial Journal 1918(1), 82–97, DOI 10.1080/03461238.1918.10405302. Publisher-domain bibliographic record inspected; original full text and detailed actuarial applications were not verified. The title does not contain the later eponym.