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Int. Journal of Math. Analysis, Vol. 3, 2009, no. 2, 95 - 98

Jensen’s Inequality in Detail

and S-Convex Functions

I. M. R. Pinheiro

P.O. Box 12396, A’Beckett St.

Melbourne, Victoria, Australia, 8006

mrpprofessional@yahoo.com

Abstract. We here study the inequality by Jensen for the case of S-

convexity.

Mathematics Subject Classiﬁcation: 39A12

Keywords: convex, S−convex,s

1

−convex, s

2

-convex, Jensen, function, s−convex

1. Introduction

2. Notations and Definitions

2.1. Notations. We use the symbology deﬁned in [PINHEIRO2006]:

• K

1

s

for the class of S−convex functions in the ﬁrst sense, some s;

• K

2

s

for the class of S−convex functions in the second sense, some s;

• K

0

for the class of convex functions;

• s

1

for the variable S,0<s

1

≤ 1, used for the ﬁrst type of S-convexity;

• s

2

for the variable S,0<s

2

≤ 1, used for the second type of s-convexity.

Remark 1. The class of 1-convex functions is simply a restriction of the

class of convex functions, which is attained when X =

+

,

K

1

1

≡ K

2

1

≡ K

0

.

2.2. Deﬁnitions. We use the deﬁnitions presented in [PINHEIRO2006] in

what regards S−convexity, as well as in [PINHEIRO2008], in what regards

convexity:

Deﬁnition 1. f : I− > is considered convex iﬀ

f[λx +(1− λ)y] ≤ λf(x)+(1− λ)f(y)

96 I. M. R. Pinheiro

∀x, y ∈ I,λ ∈ [0, 1].

Deﬁnition 2. A function f : X− > is said to be s

1

-convex if the inequality

f(λx +(1− λ

s

)

1

s

y) ≤ λ

s

f(x)+(1− λ

s

)f(y)

holds ∀λ ∈ [0, 1]; ∀x, y ∈ X; X ⊂

+

.

Remark 2. If the complementary concept is veriﬁed, then f is said to be

s

1

−concave.

Deﬁnition 3. A function f : X− > is called s

2

−convex, s =1, if the graph

lies below a ‘bent chord’ (L) between any two points, that is, for every compact

interval J ⊂ I, with boundary ∂J, it is true that

sup

J

(L − f ) ≥ sup

∂J

(L − f ).

Deﬁnition 4. A function f : X− > is said to be s

2

−convex if the inequality

f(λx +(1− λ)y) ≤ λ

s

f(x)+(1− λ)

s

f(y)

holds ∀λ ∈ [0, 1]; ∀x, y ∈ X; X ⊂

+

.

Remark 3. If the complementary concept is veriﬁed, then f is said to be

s

2

−concave.

3. Preliminaries

3.1. Jensen’s Inequality. We ﬁnd Jensen’s inequality deﬁned the following

way (see [PEARCE2002], for instance, page 21):

For a real convex function Φ, numbers x

i

in its domain, and positive weights

a

i

, Jensen’s inequality can be stated as:

Φ

p

i=1

a

i

x

i

p

i=1

a

i

≤

p

i=1

a

i

Φ(x

i

)

p

i=1

a

i

,

with the inequality reversed if Φ is concave. As a particular case, if the weights

a

i

are all equal to unity, then

Φ

p

i=1

x

i

p

≤

p

i=1

Φ(x

i

)

p

.

In the vast majority of the literature, however (see [YEH1999], for instance),

Jensen’s inequality is mentioned without the denominator. As incredible as

it may seem, both [PEARCE2002] and [YEH1999] seem to think that the

statement above is the same as Jensen’s inequality, provided one states that

the sum from the denominator is one...

Once more, an equivocated assertion. Of course it is not the same. Basically,

if we tie one coeﬃcient to the other with the condition that the sum is one, we

Jensen’s Inequality in detail and S-convex functions 97

cannot state that this is the same as a generic situation, where the coeﬃcients

are allowed to hold any sum they like...

Of course there will be several cases which will not be included in Jensen’s

inequality which are included here.

Proof. Steps:

• The convexity condition would give us easily the broader inequality, once

all which is made is forcing the sum of the coeﬃcients to be one via

division by the total of the coeﬃcients accompanying the top domain

members.

• Proved.

3.2. Extension of broader inequality to S−convexity.

s

1

−convexity

For a real s

1

−convex function Φ, numbers x

i

in its domain, and positive weights

a

i

, we have:

Φ

p

i=1

a

1

s

i

x

i

(

p

i=1

a

i

)

1

s

≤

p

i=1

a

i

Φ(x

i

)

p

i=1

a

i

.

Proof. Steps:

• Suﬃces, once more, applying the deﬁnition to the forced ‘sum-one’ coef-

ﬁcients.

• Proved.

s

2

−convexity

For a real s

2

−convex function Φ, numbers x

i

in its domain, and positive weights

a

i

, we have:

Φ

p

i=1

a

i

x

i

p

i=1

a

i

≤

p

i=1

a

s

i

Φ(x

i

)

(

p

i=1

a

i

)

s

.

Proof. Steps:

• Suﬃces, once more, applying the deﬁnition to the forced ‘sum-one’ coef-

ﬁcients.

• Proved.

98 I. M. R. Pinheiro

4. References

[PEARCE2002] C. E. M. Pearce and S. S. Dragomir. Selected Topics on

Hermite-Hadamard Inequalities and Applications, RGMIA monographs, on-

line at rgmia.vu.edu.au, 2002.

[PINHEIRO2008] M. R. Pinheiro. Convexity Secrets, Traﬀord. ISBN 1-

4251-3821-7. 2008.

[PINHEIRO2006] M. R. Pinheiro. Exploring the concept of S−convexity,

Aequationes Mathematicae, Acc. 2006, V. 74, I.3, 2007.

[YEH1999] J. Yeh. Lectures on Real Analysis, World Scientiﬁc. ISBN:981023936X.

1999.

Received: September 23, 2008