Jensen-Type Inequalities for Invex Functions

Jensen-Type Inequalities for Invex Functions

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Craven, B. D and Dragomir, Sever S (1999) Jensen-Type Inequalities for Invex Functions. RGMIA research report collection, 2 (5).

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B.D. CRAVEN AND S.S. DRAGOMIR

Abstract. Jensen’s inequality for a real convex functionf on a convex do- main is generalised in several ways to vector functions with a cone inequality, and to invex functions generalizing convex functions.

1. Introduction

Jensen’s inequality for a real convex function f on a convex domain C states that, wheneverx1, x2, ...∈Candα1, α2, ...≥0 withP

αi= 1,

fX

α_ix_i

≤X

α_if(x_i).

This paper presents several generalisations of this inequality, to vector functionsF with a cone inequality, and to invex functions generalizing convex functions. This introduces additional gradient terms into various inequalities.

2. Basic Concepts and Definitions

Definition 1. Let X and Z be normed spaces, Q⊂Z a closed convex cone, and E⊂X a convex open set. A functionF :E→Z isQ−convex on E if

(∀x, y∈E) (∀α∈(0,1)) αF(x) + (1−α)F(y)−F(αx+ (1−α)y)∈Q.

A (Fr´echet or linear Gateaux) differentiable function F :E →Z is Q−invex (see [1],[2],[3]) if, for some scale function ω:E×E→X,

(∀x, y∈E) F(x)−F(y)−F⁰(y)ω(x−y, y)∈Q.

Remark 1. F⁰(y) denotes the derivative. Invex properties have been extensively used with optimization problems. It is well-known that Q−invex impliesQ−convex if (∀x, y∈E) ω(x−y, y) =x−y.If F is real-valued andQ=R⁺,then the usual convexity follows. FromQ−convex there follows readily

(∀xi∈E)

∀αi≥0, X

αi= 1 X

αiF(xi)−FX αixi

∈Q.

In this paper, invex shall require the function to be differentiable. (Extensions to nondifferentiable functions will be considered elsewhere.)

Denote by ≥^Q the ordering defined byQthusb≥^Q a⇔b−a∈Q.

Ifω(·, y) is linear, then (∀z=x−y)F(y+z)−F(z)≥^Q F⁰(y)Cz,whereC is a linear mapping (that may depend ony), hence (∀z) F⁰(y)z≥^Q F⁰(y)Cz;

ifQis pointed (thus if Q∩(−Q) ={0}) thenF is convex, since

(∀x) F(x)−F(y)≥Q F⁰(y) (x−y) +F⁰(y) (C−I) (x−y) =F⁰(y) (x−y). Thus functions whereω(·, y) is linear for eachy are equivalent to convex functions.

Date: 2nd August, 1999.

1

2 B.D. CRAVEN AND S.S. DRAGOMIR

Notation 1. Various results require a suitable notation, in order that the concept is not obscured by algebraic details. Let x = (x1, x2, ..., xn), where each xi is a vector in E.Let α= (α1, α2, ..., αn) where eachαi ≥0 and P

αi = 1. Denote by α·xthe inner productP

αixi,and similarly for other inner products. Denote byS a sequence of indices (i1, i2, ..., ir)taken from {1,2, ..., n} with repetitions allowed;

similarly denote byS⁰a sequence of indices{i1, i2, ..., ir+1}.Denote byxS the vector ofxi wheniruns throughS;and similarly defineαS. SimilarlyF⁰(xS)denotes the sequence of F⁰(xi)as i runs through S. Denote by Mx_S the mean of the sequence xS, and by Πα_S the product of the items in αS. If Φ (S) denotes a function of S, denote byAΦ (S)the sum of the items inΦ (S),whenS runs over all sequences of lengthr,and denote byAΦ (S⁰)the sum of the items inΦ (S⁰),whereS⁰ runs over the sequences of lengthr+ 1. ThusA(Π_αS) Φ (S)represents the sum of all terms

αi₁αi₂...αi_rΦ (i1, i2, ..., ir) asi1, i2, ..., ir run through 1,2, ..., n.

Proposition 1. Let F :E → Z be Q−invex; let x₁, x₂, ... ∈E (allowing possible repetitions), and letα1, α2, ...be nonnegative with sum 1;letw=P

αixi. Then XαiF(xi)−F(w)≥^QF⁰(w)X

ω(xi−w, w). Proof. Q−invex requires, for eachi,that

F(xi)−F(w)≥^QF⁰(w)ω(xi−w, w). Multiplication byαi and summing gives the result.

Proposition 2. LetF :E→Zbe differentiableQ−invex. Letx1, x2, ...∈E(allowing possible repetitions), and let α1, α2, ...be nonnegative with sum 1.Then

−X

j

αjF⁰(xj)ω(α·(x−xj), xj)

≥^Qα·F(xS)−F(α·x)

≥^QF⁰(α·x)X

αjω(α·(xj−x), α·x). Proof. Letw=α·x.From invex,

F⁰(xj)ω(x−xj, xj)≥^QF(xj)−F(w)≥^QF⁰(w)ω(xi−w, w). Multiplication byαj and summing gives the result.

Corollary 1. Assume also thatE isa linear subspace, andω(·, w)is linear. Then α·F(x_S)−F(α·x)≥Q 0.

Proof. P

jαjω(α·(x−xj), α·x) =ωP

j,iαjαi(xi−xj), α·x

= 0.

Corollary 2. Under hypothesis of Corollary 1, X

j

α_jF⁰(x_j)ω(x_j, x_j)−X

i,j

α_iα_jF⁰(x_j)ω(x_j, x_j)

≥Q

X

j

α_jF(x_j)−F



X

j

α_jx_j



≥Q0.

Proof. Sinceω(·, w) is linear,

−X

j

αjF⁰(xj)ω(α·(x−xj), xj)

= −X

j

F⁰(x_j)ω X

i

α_jα_i·(x_i−x_j), x_j

!

= X

j

αjF⁰(xj)ω(xj, xj)−X

i,j

αiαjF⁰(xj)ω(xj, xj). Then the result follows from Corollary 1.

Corollary 3. IfF isQ-convex, then X

j

α_jF⁰(x_j) X

i

α_ix_i

!

−X

j

α_jF⁰(x_j) X

i

α_ix_i

!

≥Q

X

j

α_jF(x_j)−F



X

j

α_jx_j



≥Q0.

Proof. By substitutingω(α·(x−x_j), x_j) =P

jα_j(x_i−x_j) into the left inequality of Proposition 2, then rearranging the terms.

3. Generalized Jensen Inequalities for Invex Functions

The following generalizations of Jensen’s inequalities, involving multiple summa- tions, hold forQ−invex functions.

Proposition 3. Let F:E→Z be differentiable Q−invex; let r≥1.Then

−A(Πα_S)F⁰(M_xS)ω((α·(x−M_xS), M_xS))

≥^QA(ΠαS)F(Mx_S)−F(α·x)

≥^QF⁰(α·x)A(ΠαS)ω(α·(Mx_S−x), Mx_S). Proof. By theQ−invex property,

−F⁰(M_xS)ω(α·(x−M_xS), M_xS)

≥^Q F(Mx_S)−F(α·x)

≥^Q F⁰(α·x)ω((Mx_S −α·x), α·x).

The result follows by applyingJ ≡A^rΠαS on the left, noting that cone inequalities are thus unchanged, andJ ξ=ξfor any argumentξ independent ofS.

Corollary 4. Assume the hypothesis of Proposition 3. If E is a linear subspace andω(·, Mx_S)is linear, then

A(ΠαS0)F(MxS)−F(α·x)≥^Q 0.

Proof. ¿From the linear assumption,

A(ΠαS)ω(α·(Mx_S−x), Mx_S) =ω(u, Mx_S), where, after simplification,

u= (A(ΠαS))α·(MxS −x) =α·(MxS −x) = 0.

4 B.D. CRAVEN AND S.S. DRAGOMIR

In the following Corollary, which generalizes Corollary 2, note thatξ=x_i1 in the first summation, andS hasr terms; in the second summation,Mx_S is unchanged, butS⁰ hasr+ 1 terms, andη=xi_r+1.

Corollary 5. Under the hypothesis of Corollary 4,

−A(ΠαS)F⁰(Mx_S)ω(ξ, Mx_S)−A(ΠαS⁰)F⁰(Mx_S)ω(η, Mx_S)

≥QA(Πα_S)F(M_xS)−F(α·x). Proof. Sinceω(·, M_xS) is linear,

−A(ΠαS)F⁰(Mx_S)ω(α·(ξ−Mx_S), Mx_S)

= A(ΠαS)F⁰(MxS)ω(MxS, MxS)−A(ΠαS)F⁰(MxS)α·ω(ξ, MxS)

= A(ΠαS)F⁰(Mx_S)ω(ξ, Mx_S)−A(ΠαS⁰)F⁰(Mx_S)ω(η, Mx_S). The conclusion follows by applying Proposition 3.

Corollary 6. IfF is convex, then (denoting ξ=x_i1)

−A(ΠαS)F⁰(Mx_S)ξ−A(ΠαS)F⁰(Mx_S) (Mx_S)

≥Q A(Πα_S)F(M_xS)−F(α·x)

≥^Q 0.

Proof. Substitute ω(α·(x−Mx_S), Mx_S) = α(x−Mx_S). The details are omit- ted.

4. Further Refinements

The notation of Corollary 5 is used to state Proposition 4, namely ξ = x_i, η=x_ir+1.Note that S hasr terms,S⁰ hasr+ 1 terms.

Proposition 4. Let F : E → Z be differentiable Q−invex. Then, with ε = (r+ 1)⁻¹,

−A(ΠαS⁰)F⁰(Mx_S)ω(ε(η−Mx_S), Mx_S)

≥QA(Πα_S)F⁰(M_xS)−A(Πα_S0)F M_xS0

≥QA(Πα_S0)F⁰ M_xS0

ω ε(M_xS −η), M_xS0 . Proof. ¿FromQ−invex,

F⁰(Mx_S)ω(ε(η−Mx_S), Mx_S)

≥Q F(M_xS)−F M_x

S0

≥Q F⁰ M_x

S0

ω ε(M_xS−η), M_x

S0

. The result follows on applying (ΠαS⁰) and averaging withA.

Corollary 7. IfF is as in Proposition 4, and ω ·, Mx_S0

is linear, then, for each r= 1,2, ...,

A(Πα_S)F(M_xS)≥Q A(Πα_S0)F M_xS0 .

Proof. This follows from the second inequality of Proposition 4, noting that, when ω ·, M_x

S0

is linear,

ω ε(M_xS−η), M_x

S0

=ε r

Xⁿ

1

ω x_ij, M_x

S0

=εω x_ir+1, M_x

S0

,

and application of A(Πα_S0) gives an equal value for each summand, so the result is 0.

Corollary 8. Under the same hypotheses as Corollary 7,

A(Πα_S)F⁰(M_xS)ξ−A(Πα_S)F⁰(M_xS(M_xS))

≥^QA(ΠαS)F(Mx_S)−A(ΠxS⁰)F Mx_s0

≥^Q0.

Proof. ¿From the first inequality of Proposition 4, using the assumed linearity to expandω(·, Mx_S) in its first argument.

For other recent results connected with Jensen’s discrete inequality, see the pa- pers [4]-[9], where further references are given

References

[1] B.D. Craven, Duality for generalized convex fractional programs, pages 473-489 of “Generalized Concavity in Optimization and Economics”, S. Schaible and W.T. Ziemba (eds.), Academic Press (1981).

[2] B.D. Craven, Invex functions and constrained local minima,Bull. Austral. Math. Soc.25(9) (1981), 37-46.

[3] B.D. Craven, Quasimin and quasisaddlepoint for vector optimization, Numer. Funct. Anal.

Optim.11(1990), 45-54.

[4] S.S. Dragomir, Some refinements of Jensen’s inequality, J. Math. Anal. Appl. 168 (1992), 518-522.

[5] S.S. Dragomir, On some refinements of Jensen’s inequality and applications,Utilitas Math.43 (1993), 235-243.

[6] S.S. Dragomir, Two mappings associated with Jensen’s inequality,Extracta Math. (Spain)8 (1993), 102-105.

[7] S.S. Dragomir, A further refinement of Jensen’s inequality,Tamkang J. Math. (Taipei, Taiwan) 25(1) (1994), 29-36.

[8] S.S. Dragomir and N.M. Ionescu, On some converse of Jensen’s inequality for convex mappings, L’Anal. Num. et Th´eor. de l’Approx.23(1994), 71-78.

[9] J.E. Peˇcari´c and S.S. Dragomir, A refinement of Jensen’s inequality and applications,Studia Univ. “Babes-Bolyai” Math.34(1) (19889), 15-19.

Department of Mathematics and Statistics, University of Melbourne, Parkville, Vic- toria 3052,, Australia

E-mail address:B.Craven@ms.unimelb.edu.au

School of Communications and Informatics, Victoria University of Technology, PO Box 14428, Melbourne City MC 8001, Victoria, Australia

E-mail address:sever@matilda.vu.edu.au