On some Variants of Jensen's Inequality

On some Variants of Jensen's Inequality

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Dragomir, Sever S and Hunt, Emma (2002) On some Variants of Jensen's Inequality. RGMIA research report collection, 5 (1).

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On some variants of Jensen’s inequality

S. S. DRAGOMIR

School of Communications & Informatics, Victoria University, Vic. 8001, Australia

EMMA HUNT

Department of Mathematics, University of Adelaide, SA 5005, Adelaide, Australia

and

Surveillance Systems Division, DSTO, PO Box 1500, Edinburgh 5111, Australia

Abstract.

Some variants of Jensen’s discrete inequality are derived. These include interpolations of the basic relation for subadditive maps and of the gener- alised triangle inequality.

AMS Subject Classification: 26D15.

Key words and Phrases: Jensen’s inequality, generalised triangle inequal- ity, subadditive maps

1 Introduction

Let X be a real linear space and C ⊆ X a convex set in X, that is, a set such that

x, y ∈C and λ∈[0,1] imply λx+ (1−λ)y∈C.

If f :C →R is convex, f satisfies

f(λx+ (1−λ)y)≤λf(x) + (1−λ)f(y)

for allx, y ∈C andλ ∈[0,1]. Ifp_i ≥0 (i= 1, . . . , n) with P_n:=^Pn_i=1p_i >0 and y_i ∈C (i= 1, . . . , n), we have the Jensen inequality

f 1

P_n

n

X

i=1

p_iy_i

!

≤ 1 P_n

n

X

i=1

p_if(y_i)

(see [2] or [8, p. 6]). For some recent generalizations, refinements and appli- cations the reader is referred to [1]–[7], [9] and [8, p. 20].

In this paper we show that several new results flow from simple but judicious applications of Abel’s identity, which gives the following. Suppose X is a linear space, x_i ∈ X (i = 1, . . . , n) and s_n := ^Pn_i=1x_i. If a_i is real (i= 1, . . . , n), then

n

X

i=1

a_ix_i = a₁s₁ +

n

X

i=2

a_i(s_i−s_i−1)

=

n−1

X

i=1

(a_i−a_i+1)s_i +a_ns_n.

Consequences include an interpolation of the basic inequality for subad- ditive maps and of the generalised triangle inequality.

2 Results

We will start with the following theorem.

Theorem 2.1. Let X be a linear space and f : X → R a convex mapping, x₁, . . . , x_n ∈X and 06=a₁ ≥a₂ ≥ · · · ≥a_n≥0. Then

f a⁻¹₁

n

X

i=1

aixi

!

≤a⁻¹₁







a1f(x1) +

n

X

i=2

ai



f





i

X

j=1

xj



−f





i−1

X

j=1

xj















.

Proof. Choosep_i :=a_i−a_i+1(1≤i < n),p_n :=a_nandy_i =s_i (i= 1, . . . , n) in Jensen’s theorem. We derive

f

" Pn

i=1(a_i−a_i+1)s_i

Pn

i=1(a_i −a_i+1)

#

≤

Pn

i=1(a_i−a_i+1)f(s_i)

Pn

i=1(a_i−a_i+1) ,

where for notational simplicity we have introduced a_n+1 := 0. The desired

result now follows by Abel’s identity. 2

Corollary 2.2. Let g : X → (0,∞) be logarithmically concave, that is, let lng be concave. Under the assumptions of the theorem

g a⁻¹₁

n

X

i=1

aixi

!

≥







[g(x1)]^a1

n

Y

i=2





g^Pi_j=1x_i g^Pi−1_j=1x_j





ai





1/a1

.

The result follows from the theorem for the convex mappingf =−lng.

Suppose that the mapping ϕ : X → R is subadditive, that is, for α, β nonnegative we have

ϕ(αx+βy)≤αϕ(x) +βϕ(y).

By mathematical induction we have for all α_i ≥ 0 and y_i ∈ X (i = . . . , n) that

ϕ

n

X

i=1

α_iy_i

!

≤

n

X

i=1

α_iϕ(y_i).

This inequality may be interpolated as follows.

Corollary 2.3. Let ϕ : X → R be subadditive, y₁, . . . , y_n ∈ X and α₁ ≥α₂ ≥ · · · ≥α_n ≥0. Then

ϕ

n

X

i=1

α_iy_i

!

≤ α₁ϕ(y₁) +

n

X

i=2

α_i



ϕ





i

X

j=1

y_j



−ϕ





i−1

X

j=1

y_j









≤

n

X

i=1

α_iϕ(y_i).

Proof. Asϕis subadditive, it is convex. The first desired inequality follows from Theorem 2.1.

For the second, we observe that for 2 ≤i≤n, ϕ





i

X

j=1

y_j



−ϕ





i−1

X

j=1

y_j



≤ϕ(y_i).

Multiplying theith inequality byα_iand summing over iprovides the desired result.

Our second main result is the following.

Theorem 2.4. Letf :X →Rbe convex andx_i ∈X (i= 1, . . . , n). Suppose that m_i(i= 1, . . . , n) satisfy

i

X

j=1

m_j ≥0 (1≤i≤n)

and n

X

i=1

(n+ 1−i)m_i >0.

Then

f

Pn

i=1m_ix_i

Pn

i=1(n+ 1−i)m_i

!

≤

P_n

i=1

P_n

j=if(xj −xj+1)

Pn

i=1(n+ 1−i)m_i , where again we put x_n+1 := 0 for notational convenience.

Proof. Let s_i =^Pi_j=1m_j (1≤i≤n). Then by Abel’s identity

n

X

i=1

m_ix_i = s₁x₁+

n

X

i=2

(s_i−si−1)x_i

=

n

X

i=1

s_i(x_i−x_i+1). Applying Jensen’s inequality provides

f

" Pn

i=1s_i(x_i−x_i+1)

Pn i=1s_i

#

≤

Pn

i=1s_if(x_i−x_i+1)

Pn

i=1x_i .

The numerator on the right–hand side may be written as

n

X

i=1

m_i

n

X

j=i

f(x_j −x_j+1)

and we have the desired result. 2 Corollary 2.5. Let g : X → (0,∞) be logarithmically concave. With the above assumptions

g

Pn i=1m_ix_i

Pn

i=1(n+ 1−i)m_i

!

≥







n

Y

i=1





n

Y

j=i

g(x_j−x_j+1)





mi





1/^Pn_i=1^(n+1−i)mi

. The result follows from the theorem with the choice of convex mapping f =−lng.

3 Applications

We now derive some particular applications relating to homely choices of convex function.

1. Letx_i >0 (i= 1, . . . , n) with 0 6=a₁ ≥a₂ ≥ · · · ≥a_n≥0. Then

n

X

i=1

a_ix_i ≥a₁

"

x^a₁¹

n

Y

i=2

Pi j=1x_j

Pi−1 j=1x_j

!ai#^1/a1

.

The result follows from Corollary 2.2 with the mapping g : (0,∞)→ (0,∞) given by g(x) = x.

Suppose x₁ ≥ x₂ ≥ · · · ≥ x_n ≥ 0 and n_i ∈R with m₁ ≥ 0. In the same way we have from Corollary 2.5 that

Pn i=1m_ix_i

Pn

i=1(n+ 1−i)m_i ≥







n

Y

i=1





n

Y

j=i

(x_j−x_j+1)





mi





1/^Pn_i=1^(n+1−i)mi

.

2. Letxi >0 (i= 1, . . . , n) and 06=a1 ≥a2 ≥ · · · ≥an≥0. Then a²₁ ≤

n

X

i=1

aixi

!



a₁ x₁ −

n

X

i=2

a_ix_i

Pi−1

j=1x_j ^Pi_k=1x_k



.

This follows from Theorem 2.1 applied to the convex mapping f(x) = 1/x on the interval (0,∞).

3. Letx_i ∈R and a₁ ≥a₂ ≥ · · · ≥a_n ≥0. Then

n

X

i=1

aixi

!2

≤a1







a1x²₁+

n

X

i=2

aixi



xi+ 2

i−1

X

j=1

xj











.

This follows from Theorem 2.1 applied for the convex mapping f(x) = x² (x∈R).

4. Consider the mapping f : R → R given by f(x) = ln(1 +e^x). We have f⁰(x) =e^x/(1 +e^x) and f⁰⁰(x) =e^x/(1 +e^x)², which shows that f is convex onR.

Let 0 6=a₁ ≥a₂ ≥ · · · ≥ a_n ≥ 0 and x₁, . . . , x_n ∈ R. Then by Theorem 2.1

ln

"

1 + exp a⁻¹₁

n

X

i=1

a_ix_i

!#

≤a₁ln[1 +e^x1] +

n

X

i=2

a_i



ln







1 + exp





i

X

j=1

x_j











−ln







1 + exp





i−1

X

j=1

x_j















= ln







(1 +e^x1)^a1

n

Y

i=2





1 + exp^Pi_j=1x_j 1 + exp^Pi−1_j=1xj





ai





,

whence

1 + exp a⁻¹₁

n

X

i=1

aixi

!

≤[1 +e^x1]^a1

n

Y

i=2





1 + exp^Pi_j=1x_j 1 + exp^Pi−1_j=1x_j





ai

.

5. Let X be a real normed space and α₁ ≥ α₂ ≥ · · · ≥ α_n ≥ 0. Then for x_i ∈X (i= 1, . . . , n) we have the refinement

n

X

i=1

α_ix_i

≤ α₁ kx₁ k+

n

X

i=2

α_i





i

X

j=1

x_i

−

i−1

X

j=1

x_i





≤

n

X

i=1

α_ikx_ik

of the generalised triangle inequality. The result follows from Corollary 2.3.

References

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Appl. 163 (1992), 317–321.

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Appl. 168 (1992), 518–522.

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

[4] S. S. Dragomir, Two mappings associated with Jensen’s inequality, Ex- tracta Math. 8 (1993), 102–105.

[5] S. S. Dragomir, A further improvement of Jensen’s inequality, Tamkang J. Math. 15 (1994), 29–36.

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