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volume 6, issue 4, article 120, 2005.

Received 12 May, 2005;

accepted 22 September, 2005.

Communicated by:C.E.M. Pearce

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Journal of Inequalities in Pure and Applied Mathematics

MULTIVARIATE VERSION OF A JENSEN-TYPE INEQUALITY

ROBERT A. AGNEW

Deerfield, IL 60015-3007, USA.

EMail:[email protected]

c

2000Victoria University

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Multivariate Version of a Jensen-Type Inequality

Robert A. Agnew

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J. Ineq. Pure and Appl. Math. 6(4) Art. 120, 2005

http://jipam.vu.edu.au

Abstract

A univariate Jensen-type inequality is generalized to a multivariate setting.

2000 Mathematics Subject Classification:Primary 26D15.

Key words: Convex functions, Tchebycheff methods, Jensen’s inequality.

1. Introduction

The following theorem was proved in [1], using Tchebycheff methods [4], [5], to extend a result obtained in [2] for the Laplace transform. It was later reproved in [3], [6], [7] using Jensen’s inequality.

Theorem 1.1. Let X be a nonnegative random variable withE(X) = µ > 0 and E(X²) = λ < ∞. Suppose that f : [0,∞) → R with f(0) = 0 and g(x) = f(x)/x convex on (0,∞). Then, E(f(X)) ≥ µg(λ/µ) = (µ²/λ) f(λ/µ)and the bound is sharp.

We next provide a natural multivariate generalization of Theorem1.1, using the same approach as [1], followed by examples to illustrate its application.

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2. Main Result

Let S = (0,∞)ⁿ and letg₁, ..., g_n be real-valued functions onS. For any column vectorx= (x1, . . . , xn)^T ∈S,letf(x) =Pn

i=1xigi(x)and letei denote thei^thunit column vector inRⁿ.

Theorem 2.1. Letg₁, ..., g_n be convex onS, and letX = (X₁, . . . , X_n)^T be a random column vector inSwithE(X) = µ= (µ1, . . . , µn)^T andE XX^T

= Σ +µµ^T for covariance matrixΣ. Then,

(2.1) E(f(X))≥

n

X

i=1

µ_ig_i Σe_i

µ_i +µ

and the bound is sharp.

Proof. By convexity, for anyξ_i ∈S, there exists ab_i(ξ_i)∈Rⁿsuch that (2.2) g_i(x)≥g_i(ξ_i) +b_i(ξ_i)^T(x−ξ_i)

for allx∈S, i.e., there exists a supporting hyperplane atξ_i. Hence,

E(f(X)) =

n

X

i=1

E(X_ig_i(X)) (2.3)

≥

n

X

i=1

E X_i

g_i(ξ_i) +b_i(ξ_i)^T (X−ξ_i)

≥

n

X

i=1

µ_i

g_i(ξ_i) +b_i(ξ_i)^T

E

XX_i µ_i

−ξ_i

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But

E(XXi) = E XX^Tei

=E XX^T

ei = Σei+µµi. Then, (2.2) and (2.3) together imply that

ξ_i =E

XX_i µi

= Σe_i µi

+µ

yields the maximum bound which is obviously attained whenXis concentrated atµ.

Theorem2.1is a true multivariate extension as the following examples illustrate. As indicated in [2] for the Laplace transform, certain extensions are only nominally multivariate and fall within the domain of Theorem 1.1 because the random variables are combined in a univariate linear combination.

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3. Examples

Example 3.1. Letg_i(x) =α_i+β_i^Txbe linear withα_i ∈Randβ_i ∈Rⁿ. Then

f(x) =

n

X

i=1

xigi(x) =

n

X

i=1

xi αi+β_i^Tx

is a general quadratic function which can also be written as f(x) = α^Tx+ x^TB xwhereα= (α₁, . . . , α_n)^T andB = [β₁, . . . , β_n]^T. Then we have

E(f(X)) =E

n

X

i=1

X_i α_i+β_i^TX

!

=

n

X

i=1

α_iµ_i+β_i^T (Σe_i+µµ_i)

=

n

X

i=1

µi

αi+β_i^T Σe_i

µ_i +µ

=α^Tµ+µ^TB µ+tr (BΣ)

so the Theorem 2.1 bound is, not surprisingly, exact in this general quadratic case.

Example 3.2. Let g_i(x) = ρ_i Qn

j=1x^−γ_j ^ij withρ_i > 0andγ_ij > 0. Here, the gi might represent Cournot-type price functions (inverse demand functions) for quasi-substitutable products wherex_iis the supply of productiandg_i(x₁, . . . , x_n) is the equilibrium price of product i, given its supply and the supplies of its al- ternates. Then, xigi(x) represents the revenue from product i and f(x) =

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Pn

i=1x_ig_i(x)represents total market revenue for the ensemble of products. In this context, we would normally expect γij ∈ (0,1)for viable products. Then, with probabilistic supplies, we have

E(f(X))≥

n

X

i=1

µ_ig_i Σe_i

µ_i +µ

=

n

X

i=1

µ_iρ_i

n

Y

j=1

σ_ij µ_i +µ_j

^−γ^ij

whereσ_ij is theij^thelement ofΣ. This example demonstrates that Theorem2.1 has an interesting application in economic oligopoly theory.

In Example3.2,g_i(x) = e^hⁱ^(x)where

h_i(x) =ln ρ_i−

n

X

j=1

γ_ijlnx_j

is convex onS. In general, ifk :R→Ris convex nondecreasing andh:S→ Ris convex, theng(x) =k(h(x))is convex onSsince

k h λ x⁽¹⁾+ (1−λ) x⁽²⁾

≤k λ h x⁽¹⁾

+ (1−λ)h x⁽²⁾

≤λ k h x⁽¹⁾

+ (1−λ) k h x⁽²⁾ for any x⁽¹⁾, x⁽²⁾ ∈ S and λ ∈ [0,1]. Other examples satisfying Theorem2.1 can be generated by composing the linear functions of Example3.1with convex nondecreasing functions likek(u) = e^u, k(u) = u+√

u²+ 1 = e^sinh⁻¹ ^u, or k(u) = max (0, u).

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References

[1] R.A. AGNEW, Inequalities with application in economic risk analysis, J.

Appl. Prob., 9 (1972), 441–444.

[2] D. BROOK, Bounds for moment generating functions and for extinction probabilities, J. Appl. Prob., 3 (1966), 171–178.

[3] B. GULJAŠ, C.E.M. PEARCE AND J. PE ˇCARI ´C, Jensen’s inequality for distributions possessing higher moments, with applications to sharp bounds for Laplace-Stieltjes transforms, J. Austral. Math. Soc. Ser. B, 40 (1998), 80–85.

[4] S. KARLINANDW.J. STUDDEN, Tchebycheff Systems: with Applications in Analysis and Statistics, Wiley Interscience, 1966.

[5] J.F.C. KINGMAN, On inequalities of the Tchebychev type, Proc. Camb.

Phil. Soc., 59 (1963), 135–146.

[6] C.E.M. PEARCE AND J.E. PE ˇCARI ´C, An integral inequality for convex functions, with application to teletraffic congestion problems, Math. of Opns. Res., 20 (1995), 526–528.

[7] A.O. PITTENGER, Sharp mean-variance bounds for Jensen-type inequali- ties, Stat. & Prob. Letters, 10 (1990), 91–94.