GEOMETRIC MEANS, INDEX MAPPINGS AND SUPERMULTIPLICATIVITY

By

D. COMǍNESCU^{1}, S. S. DRAGOMIR^{2} and C. E. M. PEARCE^{3}

Abstract: Some maps are defined which derive naturally from weighted geometric means. They are shown to have convenient monotonicity and supermultiplicativity properties.

Key Words and Phrases: Geometric mean, supermultiplicativity, Jensen’s inequality, convex functions.

AMS (MOS) Subject Classification: 26D15.

1 Introduction

In a companion article [1] in this volume we explored various subadditive and superadditive maps that arise naturally from weighted geometric means considered as functions of all their parameters. We now complement (and conclude) our study with an investigation of a more complicated but still fairly natural combination of weighted geometric means that exhibits supermultiplicativity properties.

We begin as before by setting

P := {I|I ⊂ IN, 0 < |I| < ∞},
J^{∗}(I) := {x |p = (x_{i})_{i∈IN}, x_{i} > 0 ∀i ∈ I} (I ∈ P),
J (I) := {p |p = (p_{i})_{i∈IN}, p_{i} ≥ 0 ∀i ∈ I, P_{I} > 0} (I ∈ P),

where P_{I} := ∑_{i∈I} p_{i}. For I ∈ P, p ∈ J (I) and x ∈ J^{∗}(I), we define the geometric mean of x with weights p to be

G_{I}(p, x) := (∏_{i∈I} x_{i}^{p_{i}})^{1/P_{I}} .

The following two lemmas will prove convenient for our analysis. The first (see [1]) expresses the basic superadditivity and monotonicity properties of the mapping x ↦ G_{I}(p, x). For p, q ∈ J (I), we write p ≥ q whenever p_{i} ≥ q_{i} for all i ∈ I.

———————————————————————————————- 1 Department of Mathematics, Timişoara University, B-dul V. Parvah No.4, RO-1900, Timişoara, Romania. 2 School of Communications & Informatics, Victoria University, Melbourne VIC. 8001, Australia. 3 Department of Applied Mathematics, The University of Adelaide, Adelaide SA 5005, Australia.

LEMMA A. Suppose I ∈ P and p ∈ J (I). Then

(i) for all x, y ∈ J^{∗}(I),

G_{I}(p, x + y) ≥ G_{I}(p, x) + G_{I}(p, y) ≥ 0;

(ii) for all x, y ∈ J^{∗}(I) with x ≥ y,

G_{I}(p, x) ≥ G_{I}(p, y) ≥ 0.

The second lemma incorporates superadditivity and monotonicity properties associated with Jensen’s inequality (see [3]).

LEMMA B. Let f : C ⊆ X → IR be a convex function on the convex set C of a linear space X, and for I ∈ P, p ∈ J, x ∈ C ∩ J^{∗}(I) define

θ(I, p, f, x) := ∑_{i∈I} p_{i}f(x_{i}) − P_{I}f (\frac{1}{P_{I}} ∑_{i∈I} p_{i}x_{i}) .

Then

(i) the mapping θ(I, ·, f, x) is superadditive, nonnegative and monotone nondecreasing; (ii) the mapping θ(·, p, f, x) is monotone nondecreasing and superadditive as an index set mapping.

For I ∈ P, p ∈ J (I), x, y ∈ J^{∗}(I) we may define

ψ(I, p, x, y) := [\frac{G_{I}(p, x + y)}{G_{I}(p, x) + G_{I}(p, y)}]^{P_{I}} .

It is immediate from Lemma A that

ψ(I, p, x, y) ≥ 1.

The function ψ is the basic object of our study and is considered in the next section. In Section 3 we address an associated function of a real variable.

2 Supermultiplicativity

Our first result is as follows.

THEOREM 2.1. Suppose I ∈ P and x, y ∈ J^{∗}(I). Then ψ(I, ·, x, y) is supermultiplicative and monotone nondecreasing.

Proof. For u ∈ R, define f_{0}(u) := ln(1 + e^{u}). Then

f′′_{0}(u) = \frac{e^{u}}{(1 + e^{u})^{2}} ,

so that f_{0} is convex. We have

θ(I, p, f_{0}, x) = ∑_{i∈I} p_{i} ln(1 + exp(x_{i})) − P_{I} ln (1 + exp (\frac{1}{P_{I}} ∑_{i∈I} p_{i}x_{i}))

= ln \{\frac{∏_{i∈I}(1 + exp(x_{i}))^{p_{i}}}{[1 + exp (\frac{1}{P_{I}} ∑_{i∈I} p_{i}x_{i})]^{P_{I}}} \}

for all x_{i} ∈ R (i ∈ I).

For x, y ∈ J^{∗}(I), define ln(x/y) by (ln(x/y))_{i} := ln(x_{i}/y_{i}) (i ∈ I). Then replacing x by ln(x/y) in the above displayed relation yields

θ(I, p, f_{0}, ln(x/y)) = ln \begin{cases}∏_{i∈I} & (1 + \frac{x_{i}}{y_{i}} ) & p_{i} \\ [1 + exp (\frac{1}{P_{I}} ∑_{i∈I} p_{i} ln (\frac{x_{i}}{y_{i}} ))] & P_{I} \end{cases} \}

= ln \{\frac{∏_{i∈I}(x_{i} + y_{i})^{p_{i}} / ∏_{i∈I} y_{i}^{p_{i}}}{[1 + ∏_{i∈I} (\frac{x_{i}}{y_{i}} )^{p_{i}/P_{I}} ]^{P_{I}}} \}

= ln \{\frac{∏_{i∈I}(x_{i} + y_{i})^{p_{i}}}{[(∏_{i∈I} x_{i}^{p_{i}})^{1/P_{I}} + (∏_{i∈I} y_{i}^{p_{i}})^{1/P_{I}} ]^{P_{I}}} ]

= ln [\frac{G_{I}(p, x + y)}{G_{I}(p, x) + G_{I}(p, y)}]^{P_{I}}
= ln ψ(I, p, x, y),

whence

ψ(I, p, x, y) = exp[θ(I, p, f_{0}, ln(x/y))].

By Lemma B, θ(I, ·, f_{0}, ln(x/y)) is superadditive and monotone nondecreasing, since f_{0} is convex. The desired result follows at once from the properties of exp(·). □

The choice I = {1, 2, ⋯, n} provides the following. COROLLARY 2.2. If p_{i} ≤ 1, P_{n} > 0 and x_{i}, y_{i} > 0 for 1 ≤ i ≤ n, we have

[\frac{∏_{i=1}^{n} (x_{i} + y_{i})^{\frac{1}{n}}}{∏_{i=1}^{n} xi^{\frac{1}{n}} + ∏_{i=1}^{n} yi^{\frac{1}{n}}} ]^{n} ≥ [\frac{∏_{i=1}^{n} (x_{i} + y_{i})^{\frac{p_{i}}{Pn}}}{∏_{i=1}^{n} xi^{\frac{p_{i}}{Pn}} +∏_{i=1}^{n} yi^{\frac{1−p_{i}}{Pn}}} ]^{P_{n}} [\frac{∏_{i=1}^{n} (x_{i} + y_{i})^{\frac{1−p_{i}}{n−Pn}}}{∏_{i=1}^{n} x^{\frac{1−p_{i}}{n−Pn}} +∏_{i=1}^{n} yi^{\frac{1−p_{i}}{n−Pn}}} ]^{n−P_{n}} ≥ 1.

This leads to the following bound. COROLLARY 2.3. Set I = {1, 2, …, n} and suppose x_{i}, y_{i} > 0 for i ∈ I. Then

sup_{p∈J (I)} [\frac{∏_{i=1}^{n} (x_{i} + y_{i})^{\frac{p_{i}}{Pn}}}{∏_{i=1}^{n} xi^{\frac{p_{i}}{Pn}} + ∏_{i=1}^{n} yi^{\frac{p_{i}}{Pn}}} ]^{P_{n}} = [\frac{∏_{i=1}^{n} (x_{i} + y_{i})^{\frac{1}{n}}}{∏_{i=1}^{n} xi^{\frac{1}{n}} + ∏_{i=1}^{n} yi^{\frac{1}{n}}} ]^{n} ≥ 1.

In the following theorem we consider ψ as an index mapping.

THEOREM 2.4. Suppose I ∈ P, p ∈ J (I) and x, y ∈ J^{∗}(I). Then ψ(·, p, x, y) is monotone nondecreasing and supermultiplicative as an index set map.

Proof. From Lemma B the mapping θ(·, p, f_{0}, ln(x/y)) is superadditive and monotone nondecreasing. The desired result follows from

ψ(I, p, f_{0}, ln(x/y)) = exp[θ(I, p, f_{0}, ln(x/y))].
□

COROLLARY 2.5. Suppose I = {1, 2, …, n}, p, x, y ∈ J^{∗}(I). Then

sup_{J⊆I} [\frac{∏_{i∈J}(x_{i} + y_{i})^{\frac{p_{i}}{P_{J}}}}{∏_{i∈J} xi^{\frac{p_{i}}{P_{J}}} + ∏_{i∈J} yi^{\frac{p_{i}}{P_{J}}}} ]^{P_{J}} = [\frac{∏_{i=1}^{n} (x_{i} + y_{i})^{\frac{p_{i}}{Pn}}}{∏_{i=1}^{n} xi^{\frac{p_{i}}{Pn}} + ∏_{i=1}^{n} yi^{\frac{p_{i}}{Pn}}} ]^{P_{n}} ≥ 1.

COROLLARY 2.6. With the above assumptions

[\frac{∏_{i=1}^{n} (x_{i} + y_{i})^{\frac{p_{i}}{Pn}}}{∏_{i=1}^{n} xi^{\frac{p_{i}}{Pn}} + ∏_{i=1}^{n} yi^{\frac{p_{i}}{Pn}}} ]^{P_{n}} max_{1≤i<j≤n} \{[\frac{[(x_{i} + y_{i})^{p_{i}}(x_{j} + y_{j})^{p_{j}}]^{\frac{1}{p_{i}+p_{j}}}}{(x_{i}^{p_{i}}x_{j}^{p_{j}})^{\frac{1}{p_{i}+p_{j}}} + (y_{i}^{p_{i}}y_{j}^{p_{j}})^{\frac{1}{p_{i}+p_{j}}}} ]^{p_{i}+p_{j}} \} ≥ 1.

3 A function of a real variable

Suppose that I ∈ P, x, y ∈ J^{∗}(I), p, q ∈ J (I). We set

K(I, p, x, y) := \frac{G_{I}(p, x + y)}{G_{I}(p, x) + G_{I}(p, y)}

and for t ≥ 0 define

ϕ(t) := [\frac{K(I, p + tq, x, y)}{K(I, q, x, y)}]^{P_{I}+tQ_{I}} .

The properties of this mapping are embodied in the following theorem, wherein we set

H(I, p, q, x) := \frac{G_{I}(p, x)}{G_{I}(q, x)}.

THEOREM 3.1. We have that on [0, ∞)

(i) ϕ is monotone nondecreasing; (ii) ϕ is logarithmically concave; (iii) the inequality

[\frac{K(I, p, x, y)}{K(I, q, x, y)}]^{P_{I}} ≤ ϕ(t)

≤ \{\frac{[H(I, p, q, x + y)]^{G_{I}(q,x)+G_{I}(q,y))}}{[H(I, p, q, x)]^{G_{I}(q,x)} [H(I, p, q, y)]^{G_{I}(q,y)}} \}^{\frac{P_{I}}{G_{I} (q,x)+G_{I} (q,y)}}

is satisfied.

Proof. Let t_{2} > t_{1} ≥ 0. Then by Theorem 2.1,

ϕ(t_{2}) = ϕ(t_{1} + (t_{2} − t_{1}))
= [\frac{K(I, (p + t_{1}q) + (t_{2} − t_{1})q, x, y)}{K(I, q, x, y)}]^{P_{I}+t_{2}Q_{I}}
= \frac{[K(I, (p + t_{1}q) + (t_{2} − t_{1})q, x, y)]^{P_{I}+t_{1}Q_{I}+(t_{2}−t_{1})Q_{I}}}{[K(I, q, x, y)]^{P_{I}+t_{1}Q_{I}} [K(I, q, x, y)]^{(t_{2}−t_{1})Q_{I}}}
≤ \frac{[K(I, p + t_{1}q, x, y)]^{P_{I}+t_{1}Q_{I}} [K(I, (t_{2} − t_{1})q, x, y)]^{(t_{2}−t_{1})Q_{I}}}{[K(I, q, x, y)]^{P_{I}+t_{1}Q_{I}} [K(I, q, x, y)]^{(t_{2}−t_{1})Q_{I}}} .

Now G_{I}(αp, x) = G_{I}(p, x) for all α > 0 and so

K(I, (t_{2} − t_{1})q, x, y) = K(I, q, x, y) for all t_{2} > t_{1} ≥ 0.

Hence

ϕ(t_{2}) ≤ \frac{[K(I, p + t_{1}q, x, y)]^{P_{I}+t_{1}Q_{I}}}{[K(I, q, x, y)]^{P_{I}+t_{1}Q_{I}}} = ϕ(t_{1})

and the monotonicity of ϕ is proved.

Let α, β ≥ 0 with α + β = 1 and t_{1}, t_{2} ≥ 0. Then by Theorem 2.1

ϕ(αt_{1} + βt_{2}) = [\frac{K(I, α(p + t_{1}q) + β(p + t_{2}q), x, y)}{K(I, q, x, y)}]^{α(P_{I}+t_{1}Q_{I})+β(P_{I}+t_{2}Q_{I})}
≥ \frac{[K(I, α(p + t_{1}q), x, y)]^{α(P_{I}+t_{1}Q_{I})}}{[K(I, q, x, y)]^{α(P_{I}+t_{1}Q_{I})}} · \frac{[K(I, β(p + t_{2}q), x, y)]^{β(P_{I}+t_{2}Q_{I})}}{[K(I, q, x, y)]^{β(P_{I}+t_{2}Q_{I})}}
= \{[\frac{K(I, p + t_{1}q, x, y)}{K(I, q, x, y)}]^{P_{I}+t_{1}Q_{I}} \}^{α} \{[\frac{K(I, p + t_{2}q, x, y)}{K(I, q, x, y)}]^{P_{I}+t_{2}Q_{I}} \}^{β}
= [ϕ(t_{1})]^{α}[ϕ(t_{2})]^{β},

which proves the logarithmic concavity of ϕ. The first inequality in (iii) follows from the monotonicity of ϕ and

ϕ(0) = [\frac{K(I, p, x, y)}{K(I, q, x, y)}]^{P_{I}} .

Observe that for u > 0,

G_{I} (p + \frac{1}{u} q, x) = [∏_{i∈I} x_{i}^{(p_{i}+ \frac{1}{u} q_{i})}]^{1/(P_{I}+ \frac{1}{u} Q_{I})}
= ([∏_{i∈I} x_{i}^{(up_{i}+q_{i})}]^{1/u})^{u/(uP_{I}+Q_{I})}
= G_{I}(up + q, x).

Hence

lim_{t→∞} K(I, p + tq, x, y) = lim_{t→∞} [\frac{G_{I}(p + tq, x + y)}{G_{I}(p + tq, x) + G_{I}(p + tq, y)}]

= lim_{u→0+} \frac{G_{I} (p + \frac{1}{u} q, x + y)}{G_{I} (p + \frac{1}{u} q, x) + G_{I} (p + \frac{1}{u} q, y)}

= lim_{u→0+} \frac{G_{I}(up + q, x + y)}{G_{I}(up + q, x) + G_{I}(up + q, y)}
= \frac{G_{I}(q, x + y)}{G_{I}(q, x) + G_{I}(p, y)}
= K(I, q, x, y).

Put

p(t) := \frac{K(I, p + tq, x, y) − K(I, q, x, y)}{K(I, q, x, y)}.

Then lim_{t→∞} p(t) = 0 and thus

lim_{t→∞} ϕ(t) = lim_{t→∞}[(1 + p(t))^{1/p(t)}]^{p(t)·(P_{I}+tQ_{I})} = exp [ lim_{t→∞}[p(t)(P_{I} + tQ_{I})]].

To determine lim_{t→∞} ϕ(t), we have to compute

lim_{t→∞}[p(t)(P_{I} + tQ_{I})] = lim_{u→0+} [\frac{K(I, up + q, x, y) − K(I, q, x, y)}{K(I, q, x, y)} · \frac{uP_{I} + E_{I}}{u} ]
= \frac{Q_{I}}{K(I, q, x, y)} · lim_{u→0+} [\frac{K(I, up + q, x, y) − K(I, q, x, y)}{u}]
= \frac{Q_{I}}{K(I, q, x, y)} \frac{dg}{du}|_{u=0} ,

where

g(u) := K(I, up + q, x, y) = \frac{G_{I}(up + q, x + y)}{G_{I}(up + q, x) + G_{I}(up + q, y)}.

The calculation is conveniently performed in terms of

h(u, z) := G_{I}(up + q, z)
= exp [ln (∏_{i∈I} z_{i}^{up_{i}+q_{i}})^{1/(uP_{I}+E_{I})}]

= exp [\frac{1}{uP_{I} + E_{I}} ∑_{i∈I}(up_{i} + e_{i}) ln z_{i}]

We have

\frac{∂h(u, z)}{∂u} = exp [\frac{1}{uP_{I} + Q_{I}} ∑_{i∈I}(up_{i} + q_{i}) ln z_{i}] \frac{∂}{∂u} [\frac{1}{uP_{I} + Q_{I}} ∑_{i∈I}(up_{i} + q_{i}) ln z_{i}]

= G_{I}(up + q, z) \frac{( ∑_{i∈I} p_{i} ln z_{i})(uP_{I} + Q_{I}) − P_{I} ∑_{i∈I}(up_{i} + q_{i}) ln z_{i}}{(uP_{I} + Q_{I})^{2}}

= G_{I}(up + q, z) \frac{ln (∏_{i∈I} z_{i}^{p_{i}})^{uP_{I}+Q_{I}} − ln (∏_{i∈I} z_{i}^{up_{i}+q_{i}})^{P_{I}}}{(uP_{I} + Q_{I})^{2}}

= G_{I}(up + q, z) · ln [\begin{matrix}(∏_{i∈I} z_{i}^{p_{i}}) & uP_{I}+Q_{I} \\ (∏_{i∈I} z_{i}^{up_{i}+q_{i}}) & P_{I} \end{matrix} ]^{1/(uP_{I}+Q_{I})^{2}} .

Hence

\frac{∂h(0, z)}{∂u} = G_{I}(q, z) · ln [\begin{matrix}(∏_{i∈I} z_{i}^{p_{i}}) & 1/Q_{I} \\ (∏_{i∈I} z_{i}^{q_{i}}) & P_{I}/Q_{I}^{2} \end{matrix} ] = G_{I}(q, z) \frac{P_{I}}{Q_{I}} ln [H(I, p, q, z)] .

We now have

\frac{dg(0)}{du} = \frac{\frac{∂h(0, x + y)}{∂u} · [h(0, x) + h(0, y)] − h(0, x + y) [\frac{∂h(0, x)}{∂u} + \frac{∂h(0, y)}{∂u}]}{(h(0, x) + h(0, y))^{2}}
= \frac{G_{I}(q, x + y) \frac{P_{I}}{Q_{I}} ln [H(I, p, q, x + y)] (G_{I}(q, x) + G_{I}(q, y)) − G_{I}(q, x + y)A}{(G_{I}(q, x) + G_{I}(q, y))^{2}},

where

A = G_{I}(q, x) \frac{P_{I}}{Q_{I}} ln [H(I, p, q.x)] + G_{I}(q, y) \frac{P_{I}}{Q_{I}} ln [H(I, p, q, y)]
= \frac{P_{I}}{Q_{I}} [ln \{[H(I, p, q, x)]^{G_{I}(q,x)} [H(I, p, q, y)]^{G_{I}(q,y)}\}] .

Hence we derive

\frac{dg(0)}{du} = \frac{P_{I}}{Q_{I}} · \frac{G_{I}(q, x + y)}{(G_{I}(q, x) + G_{I}(q, y))^{2}} ln \begin{cases}[H(I, p, q, x + y)] & G_{I}(q,x)+G_{I}(q,y) \\ ( & [H(I, p, q, x)] & G_{I}(q,x) & [H(I, p, q, y)] & G_{I}(q,y) & ) \end{cases}\} .

Accordingly

lim_{t→∞}[p(t)(P_{I} + tQ_{I})]

= ln \{\frac{H(I, p, q, x + y)}{[H(I, p, q, x)]^{(G_{I}(q,x))/(G_{I}(q,x)+G_{I}(q,y))} [H(I, p, q, y)]^{(G_{I}(q,y))/(G_{I}(q,x)+G_{I}(q,y))}} \}^{P_{I}}

Finally, we obtain

lim_{t→∞} ϕ(t) = \begin{cases}[H(I, p, q, x + y)] & G_{I}(q,x)+G_{I}(q,y) \\ [H(I, p, q, x)] & G_{I}(q,x) & [H(I, p, q, y)] & G_{I}(q,y) \end{cases} \}^{P_{I}/(G_{I}(q,x)+G_{I}(q,y))}

and the theorem is proved. □

References

[1] D. Comǎnescu, S. S. Dragomir & C. E. M. Pearce, Geometric means, index mappings and entropy, this volume.

[2] S. S. Dragomir, D. Comǎnescu & C. E. M. Pearce, On some mappings associated with geometric and arithmetic means, Bull. Austral. Math. Soc. 55 (1997), 299–309.

[3] S. S. Dragomir, J. E. Pečarić & L. E. Persson, Properties of some functionals to Jensen’s inequality, Acta Math. Hungarica 70 (1996), 129–143.

[4] D.S. Mitrinović, J. E. Pečarić & A. M. Fink, Classical and New Inequalities in Analysis, Kluwer Academic Publishers, Dordrecht/Boston/London (1993).