Neuman–Sándor Mean¶

Aggregate two positive numbers by dividing their difference by twice the inverse hyperbolic sine of their normalized difference, producing a symmetric homogeneous mean strictly between the arithmetic and second Seiffert means.

Version: v2 · 2026-08-30 · History
Domain-specific #: 2367
Origin domain: mathematical analysis
Subdomain: bivariate means and inequalities
Aliases: Neuman-Sandor mean, Neuman–Sándor mean M, NS mean

Core Idea¶

For positive real numbers (a) and (b), the Neuman–Sándor mean is the symmetric bivariate mean

\[ M(a,b)=\frac{a-b}{2\operatorname{arsinh}\!\left(\frac{a-b}{a+b}\right)} \qquad (a\ne b), \]

with the continuous extension (M(a,a)=a). Here \(operatorname{arsinh}x=\log(x+\sqrt{1+x^2})\). Neuman and Sándor introduced the mean through their study of the Schwab–Borchardt mean, and subsequent work developed sharp comparisons with classical bivariate means.^[1]^[2]

Its cleanest representation separates scale from contrast. Let

\[ A=\frac{a+b}{2},\qquad x=\frac{a-b}{a+b}. \]

For positive inputs, (|x|<1), and

\[ M(a,b)=A\,\frac{x}{\operatorname{arsinh}x}. \]

The even correction factor \(x/\operatorname{arsinh}x\) equals (1) at (x=0) by continuity and increases the arithmetic mean as the relative disparity of the inputs grows. For unequal inputs the standard comparison chain includes

\[ A(a,b)<M(a,b)<T(a,b), \]

where \(T(a,b)=(a-b)/[2\arctan((a-b)/(a+b))]\) is the second Seiffert mean.^[2] The abstraction is therefore not merely “take an average.” It is a particular inverse-hyperbolic-sine aggregation whose normalized formula, order relations, endpoint behavior, and optimal bounds form a recognized object in the theory of means.

Structural Signature¶

The defining package has these roles:

Input pair: (a,b>0).
Scale statistic: the arithmetic mean (A=(a+b)/2).
Normalized contrast: (x=(a-b)/(a+b)), constrained to ((-1,1)).
Nonlinear response: the inverse hyperbolic sine (operatorname{arsinh}x).
Correction factor: (x/operatorname{arsinh}x), extended continuously to (1) at zero.
Output: (M=A,x/operatorname{arsinh}x).

Several invariants follow directly. Swapping the inputs changes the signs of both (x) and (operatorname{arsinh}x), so (M(a,b)=M(b,a)). Multiplying both inputs by (c>0) leaves (x) fixed and multiplies (A) by ©, so (M(ca,cb)=cM(a,b)). The equality extension gives idempotence, (M(a,a)=a). The inequality (A<M<T) makes the output internal because (T), like every strict mean here, lies between the positive inputs.

Near equality, the Maclaurin expansion

\[ \frac{x}{\operatorname{arsinh}x} =1+\frac{x^2}{6}-\frac{17x^4}{360}+O(x^6) \]

shows that the first departure from the arithmetic mean is quadratic in relative contrast. Asymptotic-expansion methods exploit this structure to derive and test comparisons among (M), harmonic, geometric, logarithmic, quadratic, centroidal, and contraharmonic means.^[3]

What It Is Not¶

The Neuman–Sándor mean is not the arithmetic mean. They agree only on equal inputs; for unequal positive inputs, (M>A).

It is not the second Seiffert mean. That mean replaces (operatorname{arsinh}) with (arctan); because the functions differ, (M<T) for unequal inputs.

It is not the first Seiffert, logarithmic, identric, quadratic, or contraharmonic mean. Those are neighboring bivariate means with different defining functions and different positions in comparison chains. Bounds relating them do not make them interchangeable.

It is not the Schwab–Borchardt mean itself. The Neuman–Sándor mean was derived and studied through that broader construction, but the displayed two-variable arsinh formula fixes this named specialization.

It is not a sample average, estimator, or consensus procedure. The definition combines exactly two positive numbers and makes no probabilistic claim about unbiasedness, robustness, population parameters, or decision legitimacy.

It is not the Neuman mean as an unrestricted label. Edward Neuman studied several families of bivariate means; a shared author and overlapping inequality literature do not collapse the named objects.

Scope of Application¶

The home domain is mathematical analysis of bivariate means. Researchers compare (M) with classical means, seek best constants in double inequalities, establish convex or geometric-combination bounds, and study reciprocals, ratios, and generalized logarithmic or Seiffert bounds.^[2]^[4]^[5]

The normalized form makes the problem essentially one-dimensional. Symmetry and homogeneity allow many global inequalities on (a,b>0) to be reduced to a function of (x=|a-b|/(a+b)), or to an equivalent hyperbolic parameter. Monotonicity, power-series coefficients, and endpoint limits can then determine whether a proposed constant works for every positive unequal pair and whether it is sharp.

Approximation theory supplies another use: finite expansions and combinations of simpler means can approximate (M) near equality or across the full contrast interval. This is an analytical comparison problem, not evidence that the mean is a standard numerical estimator in applied data analysis.

Clarity¶

The scale–contrast factorization clarifies apparent singularities and naming boundaries. The original difference quotient is (0/0) at (a=b), but the normalized factor has limit one, so the singularity is removable. The same representation makes symmetry visible, whereas it is less obvious when numerator and denominator both change sign in the original formula.

It also distinguishes local from global claims. A truncated series explains behavior when (a) and (b) are close, but it does not by itself prove an inequality for every (|x|<1). Conversely, an optimal global bound must account for both the equality limit \(x\to0\) and the extreme-contrast limit \(|x|\to1\). Stating the domain, normalization, and limiting convention prevents a formula comparison from being mistaken for a verified universal bound.

Manages Complexity¶

The abstraction packages an extensive comparison landscape into one response function. Rather than manipulating two independent positive variables, analysts factor out (A) and study \(x/\operatorname{arsinh}x\). Homogeneity removes absolute scale; symmetry permits \(x\ge0\); continuous extension closes the equality case. A two-variable inequality can often be converted into sign or monotonicity analysis on a bounded interval.

Sharp bounds further replace a transcendental expression with combinations of familiar means. Such bounds can expose ordering, provide tractable estimates, and identify the exact constants at which an inequality changes from globally valid to false. They manage symbolic complexity without changing the identity of (M): the bounding expression remains an approximation or comparator, not a new definition.

Abstract Reasoning¶

The normalized representation licenses several diagnostic moves:

Recognition: verify positive inputs and the exact (operatorname{arsinh}) formula or its algebraically equivalent normalized form.
Symmetry: reduce to (a>b), because exchanging inputs leaves the result unchanged.
Scale reduction: set (a+b) or one input to a convenient value, because positive homogeneity preserves ratios.
Equality handling: use the continuous extension rather than evaluating the displayed quotient at (a=b).
Ordering: compare the response functions. For (0<x<1), (operatorname{arsinh}xA); established comparison results locate it below (T).
Local approximation: use the even series to see that unequal-input effects begin at second order.
Sharpness testing: examine \(x\to0\), \(x\to1\), and interior monotonicity before calling a constant optimal.

These moves apply to the mathematical object. They do not license claims that (M) is preferable for empirical averaging without a separate loss function or application model.

Knowledge Transfer¶

The main transferable residue is a general mathematical tactic: separate overall scale from dimensionless contrast, express a symmetric homogeneous operation as scale times a one-variable response, and study that response by calculus or series. The tactic travels to other bivariate means, constitutive laws, similarity reductions, and dimensionless analysis.

The named mean itself transfers more narrowly. Its arsinh correction appears in the literature on means, special functions, and inequalities. A domain that merely normalizes a difference or uses an inverse hyperbolic function has not thereby instantiated the Neuman–Sándor mean. Exact transfer requires the full aggregation relation and its positive-input domain.

Examples¶

Unequal inputs¶

For (a=2) and (b=1), (A=1.5) and (x=⅓). Thus

\[ M(2,1)=\frac{1}{2\operatorname{arsinh}(1/3)}\approx1.52695. \]

The second Seiffert mean is \(T(2,1)=1/[2\arctan(1/3)]\approx1.55400\). The example exhibits the strict chain (1.5<M(2,1)<1.55400), while keeping the result between the original inputs.

Equal-input limit¶

For (a=b=5), the displayed difference quotient is indeterminate. Taking \(x\to0\) gives \(x/\operatorname{arsinh}x\to1\), hence (M(5,5)=5). The extension is part of treating the formula as a mean on all positive pairs.

Scale invariance¶

Scaling the first example by ten gives \(M(20,10)=10M(2,1)\approx15.2695\). The relative correction \(M/A\approx1.01797\) is unchanged because it depends only on (x).

Near-equality approximation¶

If (a=101) and (b=99), then (A=100) and (x=0.01). The series predicts \(M\approx100(1+0.01^2/6)=100.0016667\), with the next correction of order (10^{-8}) in the factor. This shows why (M) can be extremely close to (A) even though their strict inequality remains true.

Structural Tensions¶

Closed form versus comparison form. The arsinh quotient gives exact identity, while bounds in terms of familiar means may be easier to manipulate. Replacing the definition with a bound loses exactness; retaining only the exact form may obscure order and approximation.

Local series versus global validity. Series reveal equality-neighborhood behavior and candidate sharp constants. A global inequality still needs control over the entire interval (0<x<1).

Equality convention versus literal formula. The quotient excludes (a=b) syntactically, yet a mean should reproduce equal inputs. Continuous extension resolves the tension; direct substitution does not.

Named object versus family representation. The mean can be represented through the Schwab–Borchardt framework and compared within parameterized families. Those representations illuminate it without dissolving its fixed arsinh identity.

Structural aggregation versus application choice. The formula is a legitimate mathematical aggregation. Whether it is useful for a concrete dataset is a separate modeling decision that requires an error criterion and domain evidence.

Structural–Framed Character¶

The Neuman–Sándor Mean is strongly framed by mathematical means and inequalities.

Vocabulary travels: 0.80 framed. Mean, homogeneity, sharp bound, Seiffert mean, and arsinh response belong to specialized analysis vocabulary.
Evaluative weight: 0.00 framed. The identity is descriptive and formal.
Institutional origin: 0.00 framed. No legal or organizational authority fixes the formula.
Human-practice bound: 0.00 framed. The definition is mathematical and substrate-neutral.
Import versus recognize: 1.00 framed. Outside its literature, the exact named object is recognized only by importing the formula.

Aggregate: 0.36 framed. The mechanism is formal, but the narrow named identity and comparison ecology keep the abstraction domain-specific rather than prime.

Structural Core vs. Domain Accent¶

The portable core is symmetric homogeneous aggregation through a dimensionless disparity correction. One removes scale, applies an even nonlinear response to contrast, and restores scale. This core connects to Aggregation and to broader techniques of normalization and dimensional analysis.

The domain accent fixes positive real inputs, the arithmetic scale (A), the contrast ((a-b)/(a+b)), the inverse hyperbolic sine, the equality extension, and the comparison literature for bivariate means. Remove those commitments and only a generic nonlinear average remains. Retain them and the Neuman–Sándor mean is recognizable exactly.

The mean most directly specializes Aggregation. It maps two positive values to one representative value and deliberately suppresses their individual distinction, but fixes one exact nonlinear rule rather than the general many-to-one pattern.

It relates to Normalization through (x=(a-b)/(a+b)), which makes contrast dimensionless, and to Dimensional Analysis through positive homogeneity. It relates to Approximation and Bounding through the sharp-inequality literature. Those are supporting reasoning patterns, not additional direct parents.

Prime qualification fails because the exact arsinh formula and its named web of mathematical means do not recur as a substrate-independent abstraction across unrelated domains. What transfers broadly is aggregation and normalization, already represented by live primes.

Relationships to Other Abstractions¶

Current abstraction Neuman–Sándor Mean Domain-specific

Parents (1) — more general patterns this builds on

Neuman–Sándor Mean is a kind of Aggregation Prime

The mean most directly specializes Aggregation.

Hierarchy path (1) — routes to 1 parentless root

Neuman–Sándor Mean → Aggregation → Micro Macro Linkage

Neighborhood in Abstraction Space¶

Neuman–Sándor Mean sits in a sparse region of the domain-specific corpus (86^th percentile for distinctiveness): few abstractions share its structure, so a faithful description tends to retrieve it precisely.

Family — Unclustered & Miscellaneous (1565 abstractions)

Nearest neighbors

Computed from structural-signature embeddings · 2026-09-08

Not to Be Confused With¶

Arithmetic mean: (A=(a+b)/2); strictly below (M) for unequal positive inputs.
Second Seiffert mean: uses (arctan), not (operatorname{arsinh}), and lies above (M).
First Seiffert mean: a different inverse-trigonometric mean.
Logarithmic mean: uses a logarithmic difference quotient with a different normalization.
Identric mean: exponential-logarithmic mean with a distinct formula.
Quadratic mean: root mean square; lies above the second Seiffert mean in the standard strict chain.
Contraharmonic mean: ((a^2+b2)/(a+b)), another upper comparator.
Schwab–Borchardt mean: broader source construction from which related means can be obtained.
Neuman means: other bivariate means studied by Edward Neuman, not unrestricted aliases.
Sample mean: statistical estimator over observations, not this fixed two-input special-function mean.
Arithmetic progression: an additive sequence, unrelated despite the word arithmetic.
Von Neumann paradox: a geometric group-action result and only a lexical false neighbor.

References¶

[1] Edward Neuman and József Sándor, “On the Schwab–Borchardt Mean,” Mathematica Pannonica 14, no. 2 (2003): 253–266, http://www.kurims.kyoto-u.ac.jp/EMIS/journals/MP/index_elemei/mp14-2/mp14-2-253-266.pdf. registry ↩

[2] Hui Sun, Tiehong Zhao, Yuming Chu, and Baoyu Liu, “A Note on the Neuman–Sándor Mean,” Journal of Mathematical Inequalities 8, no. 2 (2014): 287–297, https://doi.org/10.7153/jmi-08-20. registry ↩a ↩b ↩c

[3] Neven Elezović and Lenka Vukšić, “Neuman–Sándor Mean, Asymptotic Expansions and Related Inequalities,” Journal of Mathematical Inequalities 9, no. 4 (2015): 1337–1348, https://doi.org/10.7153/jmi-09-102. registry ↩

[4] Yuming Chu, Baoyu Long, Weiming Gong, and Yaqiong Song, “Sharp Bounds for Seiffert and Neuman–Sándor Means in Terms of Generalized Logarithmic Means,” Journal of Inequalities and Applications 2013, article 10 (2013), https://doi.org/10.1186/1029-242X-2013-10. registry ↩

[5] Haoyong Huang, Nan Wang, and Biyuan Long, “Optimal Bounds for Neuman–Sándor Mean in Terms of the Geometric Convex Combination of Two Seiffert Means,” Journal of Inequalities and Applications 2016, article 14 (2016), https://doi.org/10.1186/s13660-015-0955-2. registry ↩