SUPERQUADRACITY OF FUNCTIONS AND
REARRANGEMENTS OF SETS
Superquadracity and
Rearrangements
SHOSHANA ABRAMOVICH
Shoshana Abramovich
Department of Mathematics
University of Haifa
Haifa, 31905, Israel
EMail: abramos@math.haifa.ac.il
vol. 8, iss. 2, art. 46, 2007
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Contents
Received:
26 December, 2006
Accepted:
29 May, 2007
Communicated by:
S.S. Dragomir
2000 AMS Sub. Class.:
26D15.
Key words:
Superquadratic functions, Convex functions, Jensen’s inequality.
Abstract:
In this paper we establish upper bounds of
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n X
|xi − xi+1 |
xi + xi+1 f
+f
,
2
2
i=1
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xn+1 = x1
when the function f is superquadratic and the set (x) = (x1 , . . . , xn ) is given
except its arrangement.
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Contents
1
Introduction
3
2
The Main Results
7
Superquadracity and
Rearrangements
Shoshana Abramovich
vol. 8, iss. 2, art. 46, 2007
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1.
Introduction
We start with the definitions and results of [1] and [5] which we use in this paper.
Definition 1.1. The sets (y− ) = y1− , . . . , yn− and (− y) = (− y1 , . . . ,− yn ) are
symmetrically decreasing rearrangements of an ordered set (y) = (y1 , . . . , yn ) of n
real numbers, if
y1−
(1.1)
≤
yn−
≤
y2−
≤ ··· ≤ y
−
Superquadracity and
Rearrangements
[ n+2
2 ]
Shoshana Abramovich
vol. 8, iss. 2, art. 46, 2007
and
(1.2)
−
yn ≤
−
y1 ≤
−
yn−1 ≤ · · · ≤
−
y[ n+1 ] .
2
A circular rearrangement of an ordered set (y) = (y1 , . . . , yn ) is a cyclic rearrangement of (y) or a cyclic rearrangement followed by inversion.
Definition 1.2. An ordered set (y) = (y1 , . . . , yn ) of n real numbers is arranged
in circular symmetric order if one of its circular rearrangements is symmetrically
decreasing.
Theorem A ([1]). Let F (u, v) be a symmetric function defined for α ≤ u, v ≤ β
2 F (u,v)
for which ∂ ∂u∂v
≥ 0.
Let the set (y) = (y1 , . . . , yn ), α ≤ yi ≤ β, i = 1, . . . , n be given except its
arrangement. Then
n
X
F (yi , yi+1 ) ,
(yn+1 = y1 )
i=1
is maximal if (y) is arranged in circular symmetrical order.
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Definition 1.3 ([5]). A function f , defined on an interval I = [0, L] or [0, ∞) is
superquadratic, if for each x in I, there exists a real number C (x) such that
f (y) − f (x) ≥ C (x) (y − x) + f (|y − x|)
for all y ∈ I.
A function is subquadratic if −f is superquadratic.
Lemma A ([5]). Let f be a superquadratic function with C (x) as in Definition 1.3.
Superquadracity and
Rearrangements
Shoshana Abramovich
(i) Then f (0) ≤ 0.
vol. 8, iss. 2, art. 46, 2007
(ii) If f (0) = f 0 (0) = 0, then C (x) = f 0 (x) whenever f is differentiable.
(iii) If f ≥ 0, then f is convex and f (0) = f 0 (0) = 0.
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The following lemma presents a Jensen’s type inequality for superquadratic functions.
Lemma B ([6, Lemma
Let xr ≥ 0, 1 ≤
P 2.3]). Suppose that f is superquadratic.
P
r ≤ n and let x = nr=1 λr xr , where λr ≥ 0, and nr=1 λr = 1. Then
n
X
r=1
λr f (xr ) ≥ f (x) +
n
X
λr f (|xr − x|) .
r=1
If f (x) is subquadratic, the reverse inequality holds.
From Lemma B we get an immediate result which we state in the following
lemma.
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Lemma C. Let f (x) be superquadratic on [0, L] and let x, y ∈ [0, L], 0 ≤ λ ≤ 1,
then
λf (x) + (1 − λ) f (y)
≥ f (λx + (1 − λ) y) + λf ((1 − λ) |y − x|) + (1 − λ) f (λ |y − x|)
t−1 X
k
f 2λ (1 − λ) |1 − 2λ| |x − y|
≥ f (λx + (1 − λ) y) +
k=0
+ λf (1 − λ) |1 − 2λ|t |x − y| + (1 − λ) f λ |1 − 2λ|t |x − y| .
Superquadracity and
Rearrangements
Shoshana Abramovich
vol. 8, iss. 2, art. 46, 2007
If f is positive superquadratic we get that:
t−1 X
λf (x)+(1 − λ) f (y) ≥ f (λx + (1 − λ) y)+
f 2λ (1 − λ) |1 − 2λ|k |x − y|
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k=0
More results related to superquadracity were discussed in [2] to [6].
In this paper we refine the results in [7] by showing that for positive superquadratic
functions we get better bounds than in [7].
Theorem B ([7, Thm. 1.2]). If f is a convex function and x1 , x2 , . . . , xn lie in its
domain, then
n
X
i=1
x1 + · · · + xn
n
n−1
x1 + x2
xn−1 + xn
xn + x1
≥
f
+ ··· + f
+f
.
n
2
2
2
f (xi ) − f
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Theorem C ([7, Thm. 1.4]). If f is a convex function and a1 , . . . , an lie in its
domain, then
(n − 1) [f (b1 ) + · · · + f (bn )] ≤ n [f (a1 ) + · · · + f (an ) − f (a)] ,
where a =
a1 +···+an
n
and bi =
na−ai
,
n−1
i = 1, . . . , n.
Superquadracity and
Rearrangements
Shoshana Abramovich
vol. 8, iss. 2, art. 46, 2007
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2.
The Main Results
Theorem 2.1. Let f (x) be a superquadratic function on [0, L] . Then for xi ∈
[0, L] , i = 1, . . . , n, where xn+1 = x1 ,
!
!!
n
n
n
X
X
n−1X
xi + xi+1
|xi − xi+1 |
(2.1)
f
+f
n i=1
2
2
i=1
i=1
!
!
!
n
n
n
n
X
X
X
1X
x
xi
j
≤
f (xi ) − f
−
f xi −
n
n
n
i=1
i=1
i=1
Superquadracity and
Rearrangements
Shoshana Abramovich
vol. 8, iss. 2, art. 46, 2007
j=1
holds. If f 000 (x) ≥ 0 too, then
n
(2.2)
n−1X
f
n i=1
n
X
xi + xi+1
!
n
X
|xi − xi+1 |
!!
+f
2
2
i=1
i=1
n
n−1X
x
bi + x
bi+1
|b
xi − x
bi+1 |
≤
f
+f
n i=1
2
2
!
!
!
n
n
n
n
X
X
X
1X
x
xi
j
−
f xi −
≤
f (xi ) − f
,
n
n
n
i=1
j=1
i=1
i=1
where (b
x) = (b
x1 , . . . , x
bn ) is a circular symmetrical rearrangement of (x) = (x1 , . . . , xn ) .
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Example 2.1. The functions
f (x) = xn ,
n ≥ 2,
x ≥ 0,
and the function
f (x) =
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x2 log x, x > 0,
0,
x=0
are superquadratic with an increasing second derivative and therefore (2.2) holds
for these functions.
Proof. Let f be a superquadratic function on [0, L] . Then by Lemma B we get for
0 ≤ α ≤ 1, 1 ≤ k ≤ n and xi ∈ [0, L] , xn+1 = x1 ,
(2.3)
n
X
f (xi )
Superquadracity and
Rearrangements
i=1
n
n
kX
n−k X
=
f (xi ) +
f (xi )
n i=1
n i=1
Shoshana Abramovich
vol. 8, iss. 2, art. 46, 2007
n
n
n−k X
kX
=
(αf (xi ) + (1 − α) f (xi+1 )) +
f (xi )
n i=1
n i=1
n
n−k X
≥
f (αxi + (1 − α) xi+1 )
n i=1
n
n−k X
(αf ((1 − α) |xi+1 − xi |) + (1 − α) f (α |xi+1 − xi |))
+
n i=1
!
Pn
X
Pn
n
x
x
1
i
i
i=1
+k f
+
f xi − i=1 .
n
n
n
i=1
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For k = 1 and α = 21 we get that (2.1) holds.
2 F (u,v)
If f 000 (x) ≥ 0, then ∂ ∂u∂v
≥ 0, where
F (u, v) = f (u + v) + f (|u − v|) ,
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u, v ∈ [0, L] .
Therefore according to Theorem A, the sum
n
X
|xi + xi+1 |
xi + xi+1
f
+f
,
2
2
i=1
xn+1 = x1 ,
is maximal for (b
x) = (b
x1 , . . . , x
bn ) , which is the circular symmetric rearrangement
of (x) . Therefore in this case (2.2) holds as well.
Remark 1. For a positive superquadratic function f , which according to Lemma A
is also a convex function, (2.1) is a refinement of Theorem B.
If f 000 (x) ≥ 0, (2.2) is a refinement of Theorem B as well.
Remark 2. Theorem B is refined by
!
Pn
n
n
X
X
n−1
x
bi + x
bi+1
i=1 xi
f (xi ) − f
≥
f
n
n
2
i=1
i=1
n
n−1X
xi + xi+1
≥
f
,
n i=1
2
because a convex function f satisfies the conditions of Theorem A for F (u, v) =
f (u + v) .
The following inequality is a refinement of Theorem C for a positive superquadratic
function f , which is therefore also convex. The inequality results easily from Lemma
B and the identity
!
n
n
n
X
X
1 X
f (ai ) =
f (aj ) (1 − δij )
n − 1 j=1
i=1
i=1
(where δij = 1 for i = j and δij = 0 for i 6= j), therefore the proof is omitted.
Superquadracity and
Rearrangements
Shoshana Abramovich
vol. 8, iss. 2, art. 46, 2007
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Theorem 2.2. Let f be a superquadratic function on [0, L] , and let xi ∈ [0, L],
i = 1, . . . , n. Then
!
!
n
n
X
X
n
f (yi )
f (xi ) − f (x) −
n−1
i=1
i=1
!
n
n
n
XX
1
1 X
≥
f (|yi − xj |) (1 − δij ) +
f (|x − xi |) ,
n − 1 i=1 j=1
n − 1 i=1
where x =
Pn
xi
i=1 n ,
yi =
nx−xi
n−1
, i = 1, . . . , n.
Superquadracity and
Rearrangements
Shoshana Abramovich
vol. 8, iss. 2, art. 46, 2007
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References
[1] S. ABRAMOVICH, The increase of sumsand products dependent on
(y1 , . . . , yn ) by rearrangement of this set, Israel J. Math., 5(3) (1967).
[2] S. ABRAMOVICH, S. BANIĆ AND M. MATIĆ, Superquadratic functions in
several variables, J. Math. Anal. Appl., 327 (2007), 1444–1460.
[3] S. ABRAMOVICH, S. BANIĆ AND M. KLARICIĆ BACULA, A variant of
Jensen-Steffensen’s inequality for convex and superquadratic functions, J. Ineq.
Pure & Appl. Math., 7(2) (2006), Art. 70. [ONLINE: http://jipam.vu.
edu.au/article.php?sid=687].
[4] S. ABRAMOVICH, S. BANIĆ, M. MATIĆ AND J. PEČARIĆ, JensenSteffensen’s and related inequalities for superquadratic functions, to appear in
Math. Ineq. Appl.
[5] S. ABRAMOVICH, G. JAMESON AND G. SINNAMON, Refining Jensen’s inequality, Bull. Sci. Math. Roum., 47 (2004), 3–14.
Superquadracity and
Rearrangements
Shoshana Abramovich
vol. 8, iss. 2, art. 46, 2007
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[6] S. ABRAMOVICH, G. JAMESON AND G. SINNAMON, Inequalities for averages of convex and superquadratic functions, J. Ineq. Pure & Appl. Math., 5(4)
(2004), Art. 91. [ONLINE: http://jipam.vu.edu.au/article.php?
sid=444].
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[7] L. BOUGOFFA, New inequalities about convex functions, J. Ineq. Pure &
Appl. Math., 7(4) (2006), Art. 148. [ONLINE: http://jipam.vu.edu.
au/article.php?sid=766].
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