����������� ������ 51 (1999), 15–19

UDK 517.28

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ASYMPTOTIC LINEARITY OF MEAN VALUES

Momčilo Bjelica

Abstract. The power and some other families of mean values, considered as the functions of parameter, have asymptotically linear distribution. This holds if either all the variables converge to the same value, or under the equal tending to infinity of additive increment of the variables. This limit linearity strengthens asymptotic property of Hoehn and Niven [10] which was investigated in several papers.

Assume one is given a set of positive numbers x = (x_{1}, x_{2},…, x_{n}) and a decomposition of unity, with variables q_{i} ≥ 0, 1 ≤ i ≤ n, ∑_{i=1}^{n} q_{i} = 1. The power mean (average) of order p is defined as

M_{p}(x) = (∑_{i=1}^{n} q_{i}x_{i}^{p})^{1/p}.

Mean values necessarily satisfy idempotency M(a) = a, a = (a, a,…, a), which can be replaced by a stronger condition

min_{1≤i≤n} x_{i} ≤ M(x) ≤ max_{1≤i≤n} x_{i}.

Special power means are M_{−∞} = min x_{i}, M_{−1} = H the harmonic mean, M_{0} = lim_{p→0} M_{p} = x_{1}^{q_{1}} … x_{n}^{q_{n}} = G the geometric mean, M_{1} = A the arithmetic mean, M_{2} = Q the quadratic mean, and M_{+∞} = max x_{i}.

For fixed x the power mean value M_{p}(x) is a strictly increasing and logarithmically convex function of the real variable p unless x = a. All the power mean values lie in the interval [min x_{i}, max x_{i}]. We consider the behaviour of the power and some other means from two perspectives:

asymptotic I: asymptotic II:

M_{p}(x), M_{p}(x + a), x → a a → ∞.

For homogeneous means M(λx) = λM(x), which are necessarily homogeneous of first degree [8], these two cases are equivalent. In [3] the asymptotic behaviour I was conjectured to be

\frac{M_{p}(x) − M_{q}(x)}{M_{r}(x) − M_{s}(x)} → \frac{p − q}{r − s} , x → a,  (1)

AMS Subject Classification: 26 B 05

a property named as asymptotic linearity. Asymptotic behaviour II is considered in several papers mainly quoted in the references.

Extension of Hoehn and Niven property [10] to all the power means asserts that

M_{p}(x + a) − a → A(x), a → ∞.  (2)

We transform (2) to the equivalent form with two components

M_{p}(x + a) − A(x + a) → 0, a → ∞.

The property (2) was firstly proved by L. Hoehn and I. Niven for harmonic, geometric and quadratic means, and later was extended to the other power means and large classes of homogeneous and inhomogeneous means [6,7,8,12]. Observe that the arithmetic mean A(x + a) keeps the same position with respect to min x_{i} + a and max x_{i} + a as a varies. The asymptotic linearity claims that the other means M_{p}(x + a) as a → +∞ take positions with distances more and more proportionate to the values of the parameter p. As a consequence of linearity, it straightforwardly follows that the means M_{p}(x + a), where p ∈ R, converge to A(x + a). Hence, the power mean functions in R_{+}^{n} have asymptotic isometric linear distribution on the diagonal which contains points a, where a > 0, and along its parallels x + a, where a > 0.

Theorem. For the power mean the next asymptotic formula is valid:

M_{p}(x) = M_{1}(x) + (p − 1)(M_{2}(x) − M_{1}(x)) + o(M_{2}(x) − M_{1}(x)), x → a, (3)

or, equivalently,

M_{p}(x + a) =M_{1}(x + a) + (p − 1)(M_{2}(x + a) − M_{1}(x + a))
+ o(M_{2}(x + a) − M_{1}(x + a)), a → ∞.  (4)

Proof. Let x = a + ht, and let M_{h}^{(k)} be the derivative of order k with respect to t in direction h = (h_{1}, h_{2},…, h_{n})

M_{h}^{(k)}(a) = (∑_{i=1}^{n} h_{i} \frac{∂}{∂x_{i}} )^{k}M(a).

Differentiating the both sides of the equality M(x)^{p} = ∑_{i=1}^{n} q_{i}x_{i}^{p} gives

pM(x)^{p−1}M′_{h}(x) = ∑_{i=1}^{n} pq_{i}h_{i}x_{i}^{p−1}, M′_{h}(a) = ∑_{i=1}^{n} q_{i}h_{i}.

Further,

(p − 1)M(x)^{p−2}M′_{h}(x)^{2} + M(x)^{p−1}M′′_{h}(x) = (p − 1) ∑_{i=1}^{n} q_{i}h_{i}^{2}x_{i}^{p−2},

for x = a obtains

M′′_{h}(a) = \frac{p − 1}{a}[∑_{i=1}^{n} q_{i}h_{i}^{2} − (∑_{i=1}^{n} q_{i}h_{i})^{2}] = \frac{p − 1}{a} [Q^{2}(h) − A^{2}(h)] .

By L’Hopital’s theorem for h ≠ 1

lim_{t→0} \frac{M_{p}(x) − M_{q}(x)}{M_{r}(x) − M_{s}(x)} = lim_{t→0} \frac{M′_{p}(x) − M′_{q}(x)}{M′_{r}(x) − M′_{s}(x)} = lim_{t→0} \frac{M′′_{p}(x) − M′′_{q}(x)}{M′′_{r}(x) − M′′_{s}(x)}

= \begin{matrix}(\frac{p−1}{a} − \frac{q−1}{a}) & (Q^{2}(h) − A^{2}(h)) \\ (\frac{r−1}{a} − \frac{s−1}{a}) & (Q^{2}(h) − A^{2}(h)) \end{matrix} = \frac{p − q}{r − s} .

Equations (3) and (4) are immediate consequences of this result.

For the homogeneous means the asymptotic linearity for equal values of variables is equivalent to the asymptotic linearity at infinity. Asymptotic linearity states that the mean values M_{p}, −∞ ≤ p ≤ +∞, on the infinitesimal segment [min x_{i}, max x_{i}], x → a, as well as on the remote segment [min x_{i} + a, max x_{i} + a], a → +∞, tend to the linear distribution in accordance with the order p (linear asymptotic mean loci). The equivalent form of (3) and (4) is given by the next extension of divided differences of means

\frac{M_{p}(x) − M_{1}(x)}{M_{2}(x) − M_{1}(x)} → p − 1, x → a,  (3′)

and

\frac{M_{p}(x + a) − M_{1}(x + a)}{M_{2}(x + a) − M_{1}(x + a)} → p − 1, a → ∞.  (4′)

The continuous extension (3’) is infinitely differentiable; a slightly more general assertion is that the extension of divided differences of means (1) is a continuous infinitely differentiable extension to the points with equal coordinates.

Some other means are also asymptotic(ally) linear functions of parameters. Let μ be a probability measure on the simplex of nonnegative normalized weights u_{1}, u_{2},…, u_{n}, and let ux = ∑_{i=1}^{n} u_{i}x_{i}. A one-parameter family of means is defined by integration with respect to μ over the simplex

(∫ (ux)^{p} dμ(u))^{1/p}, p ≠ 0; exp ∫ log(ux) dμ(u), p = 0.

The logarithmic means L_{p}, p ∈ R, are such integral power means [12]. Since L_{−n} = G and L_{1} = A we have

L_{p}(x) = L_{1}(x) + \frac{p − 1}{n + 1}(L_{1}(x) − L_{−n}(x)) + o(L_{1}(x) − L_{−n}(x))

= A(x) + \frac{p − 1}{n + 1}(A(x) − G(x)) + o(A(x) − G(x))

= G(x) + \frac{p + n}{n + 1}(A(x) − G(x)) + o(A(x) − G(x)), x → a.

Thus we obtain an asymptotic linear correspondence between two parameterized families of means

L_{p}(x) = M_{q}(x) + o(Q(x) − A(x)),  q = \frac{p + n}{n + 1},  x → a.

The ratio (M_{p}(x)−M_{q}(x))/(L_{r}(x)−L_{s}(x)) has an extension to (n+1)(p−q)/(r−s) as x → a.

Dresher means [8]

(∑_{i=1}^{n} q_{i}x_{i}^{s}/ ∑_{j=1}^{n} q_{j}x_{j}^{t})^{1/(s−t)}, s ≠ t

exp(∑_{i=1}^{n} q_{i}x_{i}^{t} log x_{i} / ∑_{j=1}^{n} q_{j}x_{j}^{t} ), s = t

are increasing in each parameter. The asymptotic representation for both cases has one form which can be derived from Taylor’s expansion of elementary functions by a straightforward calculation

D_{s,t}(x) = A(x) + \frac{s + t − 1}{2a}(Q^{2}(x) − A^{2}(x)) + o(Q^{2}(x) − A^{2}(x))
= G(x) + (s + t)(A(x) − G(x)) + o(A(x) − G(x)), x → a.

Tobey means are another two-parameter family

T_{s,t}(x) = (∫ (ux^{s})^{t} dμ)^{1/st}, ux^{s} = ∑_{i=1}^{n} u_{i}x_{i}^{s}, st ≠ 0.

In limiting cases st = 0 the expression is similar to the limit power and Dresher means. The asymptotic representation obtains similarly

T_{s,t}(x) = A(x) + (st − 1)(Q(x) − A(x)) + (s − 1)(K(x) − Q(x))
+ o(Q(x) − A(x)) + o(K(x) − Q(x)), x → a,

where K = T_{2,1} is a mean appearing in the other asymptotics also. If μ is a q-measure, then the Tobey mean is the power mean of order st and K = Q.

The generalization of Muirhead means [8] is

(\frac{1}{n!} ∑ x_{1}^{α_{1}}x_{2}^{α_{2}} … x_{n}^{α_{n}})^{1/r},

where the sum is over the n! permutations of x_{i}, 1 ≤ i ≤ n; and r = ∑ α_{i}, with r ≠ 0. For Muirhead means r = 1. In particular, if α_{i}(1 − α_{i}) = 0, 1 ≤ i ≤ n, one obtains the symmetric mean S_{r} of degree r, r = 1, 2,…, n. In particular, S_{1} = A and S_{n} = G, A and G are means with weights \frac{1}{n}. The generalization of Muirhead means is an integral mean with the geometric nucleus (∫ G^{r} dμ_{α})^{1/r} and measure μ_{α} which at any of n! points, obtained by permutations of the coordinates \frac{1}{r}(α_{1},…, α_{n}), α_{i} ≥ 0, has point measure \frac{1}{n!}. Therefore, Muirhead means are also asymptotically linear.

Some comprehensive classes of means have nonlinear asymptotic behaviour; e.g., the quasi-arithmetic integral means

M_{μ}(x) = φ^{−1}(∫ φ(ux) dμ)

have characteristic function ϕ = φ′′/φ′, which should be included into the limit ratio (1). Note that the power means of functions are also asymptotically linear.

REFERENCES

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[3] M. Bjelica, Fixed point and inequalities, Ph.D. thesis, University of Belgrade, 1990.

[4] M. Bjelica, Extension of ratio of differences of means and asymptotic expansion of means from mean value theorems, 9th Congress of Yugoslav Mathematicians, Petrovac, 1995, 60–61.

[5] M. Bjelica, Extension of ratio of differences of means for equal values of variables and asymptotics, 2nd European Congress of Mathematics, Budapest, 1996, Poster 39, 1–14.

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[11] D. S. Mitrinović, Analytic inequalities, Springer-Verlag, Berlin and New York, 1970.

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[14] O. Shisha and B. Mond, Differences of means, Bull. Amer. Math. Soc. 73 (1967), 328–333.

(received 01.08.1997, in revised form 21.09.1998.)

University of Novi Sad, “M. Pupin”, 23000 Zrenjanin, Yugoslavia