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Jensen's Inequality in Detail and S-Convex Functions

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  • IICSE University
Article

Jensen's Inequality in Detail and S-Convex Functions

Abstract

We here study the inequality by Jensen for the case of S- convexity.
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, Sconvex,s
1
convex, s
2
-convex, Jensen, function, sconvex
1. Introduction
2. Notations and Definitions
2.1. Notations. We use the symbology defined in [PINHEIRO2006]:
K
1
s
for the class of Sconvex functions in the first sense, some s;
K
2
s
for the class of Sconvex 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 Sconvexity, 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 Sconvexity.
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 Sconvexity,
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
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  • M R Pinheiro
M. R. Pinheiro. Convexity Secrets, Trafford. ISBN 14251-3821-7. 2008.