PAVIC, ZLATKO

Research Publication Journals Repository

ADV  MATH
SCI  JOURNAL
Advances in Mathematics: Scientic Journal 3 (2014), no.2, 7381
ISSN 1857-8365
INEQUALITIES ON CONVEX COMBINATIONS WITH THE
COMMON CENTER
ZLATKO PAVI
Abstract. The article oers Jensen type inequalities for a class of functions gen-
eralizing convex functions. The results are obtained by applying such functions to
convex combinations with the common center. Functions of one and several variables
are used in these considerations.
1. Introduction
Recall the concept of convexity and anity. Let X be a real linear space. Let
x1; : : : ; xn 2 X be points (vectors), and let 1; : : : ; n 2 R be coecients (scalars).
Their linear combination
x =
nX
i=1
ixi
is convex if all coecients i are nonnegative and if their sum is equal to 1. The above
combination is ane if only the coecient sum is equal to 1. The point x itself is called
the combination center.
A set S  X is convex (respectively ane) if it contains all convex (respectively ane)
combinations of its points. The convex (respectively ane) hull of the set S is the smallest
convex (respectively ane) set containing S, and it consists of all convex (respectively
ane) combinations of points of S. The convex (respectively ane) hull of the set S is
usually denoted with convS (respectively aS).
Let C  X be a convex set. A function f : C ! R is convex if the inequality
(1.1) f
 
nX
i=1
ixi
!

nX
i=1
if(xi)
holds for all convex combinations of points xi 2 C. Let A  X be an ane set. A function
f : A ! R is ane if the equality holds in equation (1.1) for all ane combinations of
points xi 2 A. If we have two ane combinations with the common center,
nX
i=1
ixi =
mX
j=1
jyj ;
2010 Mathematics Subject Classication. 26A51, 26D15, 52A20.
Key words and phrases. Convex combination, common center, simplex.
73
74 Z. PAVI
then every ane function f satises the equality
nX
i=1
if(xi) =
mX
j=1
jf(yj):
To dene a convex function we usually take n = 2 in equation (1.1). Applying the
mathematical induction Jensen has attained the famous inequality in equation (1.1), see
[2]. Numerous books and papers have been written on Jensen's inequality. Dierent
approaches can be seen in books [6] and [9], and papers [10], [3], [7] and [5]. An overview
of dierent forms of Jensen's inequality can be found in [4].
2. Main Results
2.1. Functions of One Variable. The main result of this subsection is Theorem 2.1
relying on the idea of a convex function graph and its chord line. Using a function that
is more general than convex function, and two convex combinations with the common
center, we obtained the Jensen type inequality.
Let [a; b]  R be a bounded closed interval with endpoints a < b. Then every number
x 2 R can be uniquely presented as the binomial ane combination:
(2.1) x =
b  x
b  a
a+
x  a
b  a
b ;
which is convex if, and only if, the number x belongs to the interval [a; b]. Let I  R be
an interval containing [a; b], let f : I ! R be a function, and let f linefa;bg : R ! R be the
function of the line passing through the points A(a; f(a)) and B(b; f(b)) of the graph of
f . Applying the anity of the function f linefa;bg to the combination in (2.1), we obtain its
equation
(2.2) f linefa;bg(x) =
b  x
b  a
f(a) +
x  a
b  a
f(b) :
The consequence of the representations in equations (2.1) and (2.2) is the fact that every
convex function f : I ! R satises the inequality
(2.3) f(x)  f linefa;bg(x) for x 2 [a; b] ;
and the reverse inequality
(2.4) f(x)  f linefa;bg(x) for x 2 I n (a; b) :
In the following theorem, we use the functions satisfying the inequalities in equations
(2.3) and (2.4).
Theorem 2.1. Let I  R be an interval, let [a; b]  I be a closed interval with
endpoints a < b, and let f : I ! R be a function satisfying equations (2.3) and (2.4).
Let
Pn
i=1 iai be a convex combination of points ai 2 [a; b], and let
Pm
j=1 jbj be a
convex combination of points bj 2 I n (a; b).
If the above convex combinations have the common center
(2.5)
nX
i=1
iai =
mX
j=1
jbj ;
INEQUALITIES ON CONVEX COMBINATIONS. . . 75
then we have the inequality
(2.6)
nX
i=1
if(ai) 
mX
j=1
jf(bj):
Proof. Using the properties of the function f , and applying the anity of the function
f linefa;bg, we get
nX
i=1
if(ai) 
nX
i=1
if
line
fa;bg(ai) = f
line
fa;bg
 
nX
i=1
iai
!
= f linefa;bg
0
@ mX
j=1
jbj
1
A = mX
j=1
jf
line
fa;bg(bj)

mX
j=1
jf(bj) ;
nishing the derivation of the inequality in equation (2.6). 
The function used in Theorem 2.1 is shown in Figure 1. Although the gure represents
a continuous function, it may also be discontinuous.
Figure 1. A function satisfying equations (2.3) and (2.4)
Involving the binomial convex combination a + b to the equality in equation (2.5)
by assuming that
nX
i=1
iai = a+ b =
mX
j=1
jbj ;
and following the proof of Theorem 2.1, we achieve the double inequality
nX
i=1
if(ai)  f(a) + f(b) 
mX
j=1
jf(bj):
76 Z. PAVI
The functions used in Theorem 2.1 satisfy Jensen's inequality for convex combinations
whose points are outside the open interval (a; b), and whose center is in the closed interval
[a; b].
Corollary 2.1. Let f : I ! R be a function satisfying equations (2.3)-(2.4), and let
c =
Pm
j=1 jbj be a convex combination of points bj 2 I n (a; b).
If the center c belongs to [a; b], then
(2.7) f
0
@ mX
j=1
jbj
1
A  mX
j=1
jf(bj):
Corollary 2.2. A convex function f : I ! R satises Theorem 2.1 for every closed
interval [a; b]  I with endpoints a < b.
Corollary 2.3. Let f : I ! R be a function such that it satises Theorem 2.1 for
every closed interval [a; b]  I with endpoints a < b. Then f is convex.
Proof. Let
(2.8) c = 1b1 + 2b2
be a binomial convex combination of points b1; b2 2 I such that b1 < b2. Taking a = b1
and b = b2, equation (2.8) becomes the common center of the trivial convex combination
c 2 [a; b] and the binomial convex combination 1b1 + 2b2 of points b1; b2 2 I n (a; b).
We can apply the corollary assumption, and get the inequality
(2.9) f (1b1 + 2b2) = f(c)  1b1 + 2b2
which proves the convexity of the function f . 
Bringing together the two previous corollaries, we obtain a characterization of convex
functions as follows.
Proposition 2.1. A function f : I ! R is convex if, and only if, it satises Theorem
2.1 for every closed interval [a; b]  I with endpoints a < b.
2.2. Generalizations to More Dimensions. It can be said that the main result in this
subsection is Theorem 2.2 generalizing Theorem 2.1 to more dimensions. Example 2.1
shows that we can not transfer all results of the previous subsection to more dimensions.
Let C  R2 be a convex set, and let 4 = convfA;B;Cg  C be a triangle with vertices
A, B and C. In what follows we use a function f : C ! R that satises the inequality
(2.10) f(P )  f
plane
fA;B;Cg(P ) for P 2 4;
and the reverse inequality
(2.11) f(P )  f
plane
fA;B;Cg(P ) for P 2 C n 4
o;
where f
plane
fA;B;Cg is the function of the plane passing through the corresponding points of
the graph of f , and 4o is the interior of 4.
The generalization of Theorem 2.1 to two dimensions is as follows.
INEQUALITIES ON CONVEX COMBINATIONS. . . 77
Lemma 2.1. Let C  R2 be a convex set, let 4 = convfA;B;Cg  C be a triangle
with vertices A, B, C, and let f : C ! R be a function satisfying equations (2.10)-
(2.11). Let
Pn
i=1 iAi be a convex combination of points Ai 2 4, and let
Pm
j=1 jBj
be a convex combination of points Bj 2 C n 4
o.
If the above convex combinations have the common center
(2.12)
nX
i=1
iAi =
mX
j=1
jBj ;
then we have the inequality
nX
i=1
if(Ai) 
mX
j=1
jf(Bj):
Proof. We can implement the same analytic procedure as in Theorem 2.1 using the
anity of the plane function f
plane
fA;B;Cg. 
Including the trinomial convex combination A+B+C to the equality in equation
(2.12) by assuming that
nX
i=1
iAi = A+ B + C =
mX
j=1
jBj ;
and applying Lemma 2.1 to the left and right equality of the above equation, we obtain
the double inequality
nX
i=1
if(Ai)  f(A) + f(B) + f(C) 
mX
j=1
jf(Bj):
Corollary 2.4. Let f : C ! R be a function satisfying equations (2.10)-(2.11), and
let D =
Pm
j=1 jBj be a convex combination of points Bj 2 C n 4
o.
If the center D belongs to 4, then
f
0
@ mX
j=1
iBj
1
A  mX
j=1
jf(Bj):
The next example demonstrates that a generalization of Corollary 2.2 to convex func-
tions of two variables is not possible.
Example 2.1. Take the convex function f(x; y) = x2 + y2, the triangle with vertices
A(0; 0), B(3; 0) and C(0; 3), and the outside points B1(1; 1), B2(2; 2) and B3( 1; 1).
Then we have
1
3
A+
1
3
B +
1
3
C =
1
4
B1 +
1
2
B2 +
1
4
B3
and
6 =
1
3
f(A) +
1
3
f(B) +
1
3
f(C) >
1
4
f(B1) +
1
2
f(B2) +
1
4
f(B3) = 5:
Lemma 2.1 which refers to the triangles can be generally extended to simplexes. In
short, the convex hull of points C1; : : : ; Cp+1 2 R
p is a p-simplex in Rp if the points
C1   Cp+1; : : : ; Cp   Cp+1 are linearly independent.
78 Z. PAVI
Let C  Rp be a convex set, and let S = convfC1; : : : ; Cp+1g  C be a p-simplex with
vertices C1; : : : ; Cp+1. In this observation we use a function f : C ! R that satises the
inequality
(2.13) f(P )  f
hyperplane
fC1;:::;Cp+1g
(P ) for P 2 S;
and the reverse inequality
(2.14) f(P )  f
hyperplane
fC1;:::;Cp+1g
(P ) for P 2 C n So;
where f
hyperplane
fC1;:::;Cp+1g
is the function of the hyperplane passing through the corresponding
points of the graph of f .
Theorem 2.2. Let C  Rp be a convex set, let S = convfC1; : : : ; Cp+1g  C be a
p-simplex with vertices C1; : : : ; Cp+1, and let f : C ! R be a function satisfying
equations (2.13)-(2.14). Let
Pn
i=1 iAi be a convex combination of points Ai 2 S,
and let
Pm
j=1 jBj be a convex combination of points Bj 2 C n S
o.
If the above convex combinations have the common center
(2.15)
nX
i=1
iAi =
mX
j=1
jBj ;
then we have the inequality
(2.16)
nX
i=1
if(Ai) 
mX
j=1
jf(Bj):
Proof. To derive the inequality in equation (2.16), we may also apply the proof of The-
orem 2.1 using the anity of the hyperplane function f
hyperplane
fC1;:::;Cp+1g
. 
Including the (p + 1)-membered convex combination
Pp+1
k=1 kCk to the equality in
equation (2.15) in a way that
nX
i=1
iAi =
p+1X
k=1
kCk =
mX
j=1
jBj ;
and respecting Theorem 2.2, we get the double inequality
nX
i=1
if(Ai) 
p+1X
k=1
kf(Ck) 
mX
j=1
jf(Bj):
Corollary 2.5. Let f : C ! R be a function satisfying equations (2.13)-(2.14), and
let D =
Pm
j=1 jBj be a convex combination of points Bj 2 C n S
o.
If the center D belongs to S, then
f
0
@ mX
j=1
iBj
1
A  mX
j=1
jf(Bj):
INEQUALITIES ON CONVEX COMBINATIONS. . . 79
3. Applications to Quasi-Arithmetic Means
Functions investigated in Subsection 2.1 can be included to quasi-arithmetic means by
applying methods such as those for convex functions. More details on quasi-arithmetic
and power means can be found in [1] and [4].
Let I  R be an interval, and let
Pn
i=1 ixi be a convex combination of points xi 2 I.
The discrete '-quasi-arithmetic mean of the points xi with the coecients i is the point
(3.1) M'(xi; i) = '
 1
 
nX
i=1
i'(xi)
!
which belongs to I because the convex combination
Pn
i=1 i '(xi) belongs to '(I).
In order to apply the convexity, we use strictly monotone continuous functions '; :
I ! R such that  is convex with respect to ' ( is '-convex), which is to say that the
function f =   ' 1 is convex on the interval '(I). A similar notation is used for the
concavity.
We want to apply Theorem 2.1 to quasi-arithmetic means.
Theorem 3.1. Let I  R be an interval, and let [a; b]  I be a closed interval with
endpoints a < b. Let
Pn
i=1 iai be a convex combination of points ai 2 [a; b], and
let
Pm
j=1 jbj be a convex combination of points bj 2 I n (a; b). Let '; : I ! R
be strictly monotone continuous functions, and let f =   ' 1 be the composite
function.
If f satises equations (2.3)-(2.4) and  is increasing, and if the equality
(3.2) M'(ai; i) =M'(bj ; j)
is valid, then we have the inequality
(3.3) M (ai; i) M (bj ; j):
Proof. Take J = '(I), and [c; d] = '([a; b]) where c < d. We will apply Theorem 2.1
to the function f : J ! R, the points ui = '(ai) 2 [c; d], and the points vj = '(bj) 2
J n (c; d).
Starting with the equality '
 
M'(ai; i)

= '
 
M'(bj ; j)

, that is,
nX
i=1
iui =
mX
j=1
jvj ;
and relying on Theorem 2.1, we get
nX
i=1
if(ui) 
mX
j=1
jf(vj):
Applying the increasing function   1 to the above inequality, it follows that
  1
 
nX
i=1
if(ui)
!
   1
0
@ mX
j=1
jf(vj)
1
A ;
which is actually
M (ai; i) M (bj ; j)
because f(ui) =  (ai) and f(vj) =  (bj). 
80 Z. PAVI
All the possibilities in the above theorem are as follows.
Corollary 3.1. Let f =  ' 1 be the composition of functions  and ' 1 satisfying
the conditions of Theorem 3.1.
If either f satises equations (2.3)-(2.4) and  is increasing or  f satises equa-
tions (2.3)-(2.4) and  is decreasing, and if the equality in equation (3.2) is valid,
then the inequality in equation (3.3) holds.
If either f satises equations (2.3)-(2.4) and  is decreasing or  f satises equa-
tions (2.3)-(2.4) and  is increasing, and if the equality in equation (3.2) is valid,
then the reverse inequality in equation (3.3) holds.
Let
Pn
i=1 ixi be a convex combination of points xi 2 I. As a special case of the
quasi-arithmetic means in equation (3.1) using the functions 'r(x) = x
r for r 6= 0 and
'0(x) = lnx on the interval I = (0;+1), we can observe the discrete power means
Mr(xi; i) =
8>>>>><
>>>>>:
 
nX
i=1
i x
r
i
! 1
r
for r 6= 0
exp
 
nX
i=1
i lnxi
!
for r = 0
:
Note that
M1(xi; i) =
nX
i=1
ixi:
Corollary 3.2. Let [a; b]  (0;+1) be a closed interval with endpoints a < b. LetPn
i=1 iai be a convex combination of points ai 2 [a; b], and let
Pm
j=1 jbj be a convex
combination of points bj 2 (0;+1) n (a; b).
If the above convex combinations have the common center
M1(ai; i) =M1(bj ; j);
then
Mr(ai; i) Mr(bj ; j) for r  1;
and
Mr(ai; i) Mr(bj ; j) for r  1:
Proof. The proof follows from Theorem 3.1 and Corollary 3.1 by using convex and concave
functions such as '(x) = x,  (x) = xr for r 6= 0, and  (x) = lnx for r = 0. 
4. Applications to Integrals
The aim of the nal section is just to indicate how the above results can be transferred
to integrals. Something similar was done in [8], but in the reverse direction.
The integral analogy of the concept of convex combination is the concept of barycenter.
Let  be a positive measure on Rp, and let A  Rp be a -measurable set with (A) > 0.
INEQUALITIES ON CONVEX COMBINATIONS. . . 81
Given the positive integer n, let A = [ni=1Ani be the partition of pairwise disjoint -
measurable sets Ani. Taking points Ani 2 Ani we determine the convex combination
An =
nX
i=1
(Ani)
(A)
Ani
whose center An belongs to convA. If the sequence (An)n converges, then the -
barycenter of the set A can be dened by the point
M(A; ) = lim
n!1
 
nX
i=1
(Ani)
(A)
Ani
!
=
1
(A)
Z
A
x1 d; : : : ;
Z
A
xr d

:
Let C  Rp be a convex set, and let S  C be a p-simplex. To apply the results of
Section 2 to integrals, one can use -measurable sets A and B such that A  S and
B  C n So, and that have the common -barycenter
M(A; ) =M(B; ):
References
[1] P. S. Bullen, D. S. Mitrinovi¢, P. M. Vasi¢: Means and Their Inequalities, Reidel, Dor-
drecht, NL, 1988.
[2] J. L. W. V. Jensen: Om konvekse Funktioner og Uligheder mellem Middelværdier, Nyt tidsskrift
for matematik. B., 16 (1905) 4968.
[3] A. McD. Mercer: A variant of Jensen's inequality, Journal of Inequalities in Pure and Applied
Mathematics, 4(4), (2003) Article 73, pages 2.
[4] J. Mi¢i¢, Z. Pavi¢, J. Pe£ari¢: The inequalities for quasiarithmetic means, Abstract and
Applied Analysis, 2012, Article ID 203145, pages 25.
[5] M. S. Moslehian, M. Kian: Jensen type inequalities for Q-class functions, Bulletin of the
Australian Mathematical Society, 85 (2012) 128142.
[6] C. P. Niculescu, L. E. Persson: Convex Functions and Their Applications, Canadian Math-
ematical Society, Springer, New York, USA, 2006.
[7] Z. Pavi¢: Half convex functions, Journal of Inequalities and Applications, 2014, Article 13, pages
10.
[8] Z. Pavi¢, J. Pe£ari¢, I. Peri¢: Integral, discrete and functional variants of Jensen's inequality,
Journal of Mathematical Inequalities, 5 (2011), 253264.
[9] J. E. Pe£ari¢, F. Proschan, Y. L. Tong: Convex Functions, Partial Orderings, and Statis-
tical Applications, Academic Press, New York, USA, 1992.
[10] E. J. McShane: Jensen's inequality, Bulletin of the American Mathematical Society, 43 (1937),
521527.
Mechanical Engineering Faculty in Slavonski Brod
University of Osijek
Slavonski Brod 35000, Croatia
E-mail address: Zlatko.Pavic@sfsb.hr


INEQUALITIES ON CONVEX COMBINATIONS WITH THE COMMON CENTER

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INEQUALITIES ON CONVEX COMBINATIONS WITH THE COMMON CENTER

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