**Rev. Anal. Num´****er. Th´****eor. Approx., vol. 37 (2008) no. 2, pp. 127–132**
**ictp.acad.ro/jnaat**

A SIMPLE PROOF OF POPOVICIU’S INEQUALITY

MIHALY BENCZE^{∗}and FLORIN POPOVICI^{†}

**Abstract.** T. Popoviciu [5] has proved in 1965 the following inequality relating
the values of a convex function*f* :*I* →Rat the weighted arithmetic means of
the subfamilies of a given family of points*x*1*, ..., x**n*∈*I:*

X

1≤i_{1}*<···<i** _{p}*≤n

(λ*i*_{1}+· · ·+*λ**i** _{p}*)

*f*

_{λ}

_{i}1*x**i*1+···+λ_{ip}*x*_{ip}*λ*_{i}_{1}+···+λ_{ip}

≤ ^{n−2}_{p−2}

"

*n−p*
*p−1*

*n*

X

*i=1*

*λ**i**f(x**i*) +

*n*

X

*i=1*

*λ**i*

!

*f* ^{λ}^{1}^{x}_{λ}^{1}^{+···+λ}^{n}^{x}^{n}

1+···+λ_{n}

#
*.*

Here*n*≥3, p∈ {2, ..., n−1}and*λ*1*, ..., λ**n* are positive numbers (representing
weights). The aim of this paper is to give a simple argument based on mathe-
matical induction and a majorization lemma.

**MSC 2000.** Primary 26A51, 26D15; Secondary 26B25.

**Keywords.** Popoviciu’s inequality, convex function, convex combination.

T. Popoviciu [5] has proved in 1965 the following inequality relating the
values of a convex function*f* :*I* →Rat the weighted arithmetic means of the
different subfamilies of a given family of points *x*_{1}*, ..., x** _{n}*∈

*I*:

X

1≤i_{1}*<···<i** _{p}*≤n

(λ*i*1 +· · ·+*λ**i**p*)*f*^{}^{λ}^{i}^{1}^{x}_{λ}^{i}^{1}^{+···+λ}^{ip}^{x}^{ip}

*i*1+···+λ_{ip}

≤ ^{n−2}_{p−2}^{}

"

*n−p*
*p−1*

*n*

X

*i=1*

*λ**i**f*(x*i*) +

*n*

X

*i=1*

*λ**i*

!

*f*^{}^{λ}^{1}^{x}_{λ}^{1}^{+···+λ}^{n}^{x}^{n}

1+···+λ*n*

#
*.*

Here*n*≥3, p∈ {2, ..., n−1}and*λ*1*, ..., λ**n*are positive numbers (representing
weights);*I* is a nonempty interval.

The inequality above (denoted (P*n,p*) in what follows) is nontrivial even
in the case of triplets (that is, when *n* = 3 and *p* = 2). Several alternative
approaches of (P_{3,2}) are discussed in the recent book of C. P. Niculescu and
L.-E. Persson [2]. See [4] and [3] for additional information.

∗Aprily Lajos College, Bra¸sov, Romania, e-mail: [email protected]

†Grigore Moisil College, Bra¸sov, Romania, e-mail: [email protected]

Theoretically, Popoviciu’s inequality is a refinement of Jensen’s inequality since it yields

*f*

*n*

P

*i=1*

*λ**i**x**i*
*n*

P

*i=1*

*λ**i*

≤ ^{1}

*n−1*
*p−1*

_{n}

P

*i=1*

*λ**i*

X

1≤i1*<···<i**p*≤n

(λ_{i}_{1}+· · ·+*λ*_{i}* _{p}*)

*f*

^{}

^{λ}

^{i}^{1}

_{λ}

^{x}

^{i}^{1}

^{+···+}

^{λ}

^{ip}

^{x}

^{ip}*i*1+···+λ_{ip}

≤ ^{n−p}_{n−1}

*n*

P

*i=1*

*λ**i**f*(x*i*)

*n*

P

*i=1*

*λ**i*

+_{n−1}^{p−1}*f*

*n*

P

*i=1*

*λ**i**x**i*
*n*

P

*i=1*

*λ**i*

≤

*n*

P

*i=1*

*λ**i**f*(x*i*)

*n*

P

*i=1*

*λ**i*

*.*

The aim of the present paper is to offer a simple argument of (P*n,p*) based
on mathematical induction and the following variant of the majorization in-
equality:

Lemma 1. *Let* *f* : [a, b]→R *be a convex function. If* *x*_{1}*, ..., x** _{n}*∈[a, b]

*and*

*a convex combination*

^{P}

^{n}*k=1*

*µ*_{k}*x*_{k}*of these points equals a convex combination*
*λ*1*a*+*λ*2*b* *of the endpoints, then*

*n*

X

*k=1*

*µ*_{k}*f*(x* _{k}*)≤

*λ*

_{1}

*f(a) +λ*

_{2}

*f*(b).

*Proof.* This can be established easily by using the barycentric coordinates
(in our case the fact that every point*x** _{k}*∈[a, b] can be expressed uniquely as
a convex combination of

*a*and

*b).*

A second argument is based on the geometric meaning of convexity. Denot-
ing by*A(x) the affine function joining (a, f*(a)) with (b, f(b)),we have

*n*

X

*k=1*

*µ*_{k}*f*(x* _{k}*) ≤

*n*

X

*k=1*

*µ*_{k}*A*(x* _{k}*) =

*A*

*n*

X

*k=1*

*µ*_{k}*x*_{k}

!

= *A*(λ1*a*+*λ*2*b) =λ*1*A(a) +λ*2*A(b)*

= *λ*1*f(a) +λ*2*f*(b).

It is worth to mention that Lemma 1 still works (with obvious changes) within the framework of convex functions on simplices.

We pass now to the proof of Popoviciu’s inequality, by considering first the
case where *n*∈N, n≥3 and *p*=*n*−1 :

(P*n,n−1*) ^{X}

1≤i≤n

*λ**i**f*(x*i*) + (n−2)

X

1≤i≤n

*λ**i*

*f*

P

1≤i≤n

*λ**i**x**i*

P

1≤i≤n

*λ**i*

≥ ^{X}

1≤j≤n

X

1≤i≤n, i6=j

*λ*_{i}

*f*

P

1≤i≤n, i6=j

*λ**i**x**i*

P

1≤i≤n, i6=j

*λ**i*

*.*

Clearly, we may assume

*x*1 ≤*x*2 ≤ · · · ≤*x**n**.*
Choose *k*∈ {1, ..., n−1}such that

*x** _{k}*≤

*n*

P

*i=1*

*λ**i**x**i*
*n*

P

*i=1*

*λ**i*

≤*x*_{k+1}

and put

*y** _{j}* =
P

1≤i≤n, i6=j

*λ**i**x**i*

P

1≤i≤n, i6=j

*λ**i*

for*j*= 1, ..., n.Then it is clear that

(1)

*n*

P

*i=1*

*λ**i**x**i*
*n*

P

*i=1*

*λ**i*

≤*y**j* ≤

*n*

P

*i=k+1*

*λ**i**x**i*
*n*

P

*i=k+1*

*λ**i*

for all *j*∈ {1, ..., k}.

We have

*k*

P

*j=1*

P

1≤i≤n, i6=j

*λ**i*

*y**j*

*k*

P

*j=1*

P

1≤i≤n, i6=j

*λ**i*

=

*k*

P

*j=1*

P

1≤i≤n, i6=j

*λ**i*

P

1≤i≤n, i6=j

*λ**i**x**i*

P

1≤i≤n, i6=j

*λ**i*

*k*

P

*j=1*

P

1≤i≤n, i6=j

*λ**i*

=

(k−1)

*n*

P

*i=1*

*λ**i**x**i*+

*n*

P

*i=k+1*

*λ**i**x**i*

(k−1)

*n*

P

*i=1*

*λ**i*+

*n*

P

*i=k+1*

*λ**i*

=

(k−1)

* _{n}*
P

*i=1*

*λ**i*

*n*

P

*i=1*

*λ**i**x**i*
*n*

P

*i=1*

*λ**i*

+

* _{n}*
P

*i=k+1*

*λ**i*

*n*

P

*i=k+1*

*λ**i**x**i*
*n*

P

*i=k+1*

*λ**i*

(k−1)

*n*

P

*i=1*

*λ**i*+

*n*

P

*i=k+1*

*λ**i*

so that by (1) and Lemma 1 we infer the inequality

*k*

X

*j=1*

X

1≤i≤n, i6=j

*λ*_{i}

*f*

P

1≤i≤n, i6=j

*λ**i**x**i*

P

1≤i≤n, i6=j

*λ**i*

≤

≤(k−1)

*n*

X

*i=1*

*λ**i*

!
*f*

*n*

P

*i=1*

*λ**i**x**i*
*n*

P

*i=1*

*λ**i*

+

*n*

X

*i=k+1*

*λ**i*

*f*

*n*

P

*i=k+1*

*λ**i**x**i*
*n*

P

*i=k+1*

*λ**i*

*.*

Or, by Jensen’s inequality,

*k*

X

*j=1*

X

1≤i≤n, i6=j

*λ*_{i}

*f*

P

1≤i≤n, i6=j

*λ**i**x**i*

P

1≤i≤n, i6=j

*λ**i*

≤

≤(k−1)

*n*

X

*i=1*

*λ*_{i}

!
*f*

*n*

P

*i=1*

*λ**i**x**i*
*n*

P

*i=1*

*λ**i*

+

*n*

X

*i=k+1*

*λ*_{i}*f*(x* _{i}*)

and

*n*

X

*j=k+1*

X

1≤i≤n, i6=j

*λ*_{i}

*f*

P

1≤i≤n, i6=j

*λ**i**x**i*

P

1≤i≤n, i6=j

*λ**i*

≤

≤(n−*k*−1)

*n*

X

*i=1*

*λ*_{i}

!
*f*

*n*

P

*i=1*

*λ**i**x**i*
*n*

P

*i=1*

*λ**i*

+

*k*

X

*i=1*

*λ*_{i}*f*(x* _{i}*)

whence we may conclude (P*n,n−1*).

Consider now the case where *n*∈N*, p*≥3.We will prove that
(P* _{n,p}*)⇒(P

*n,p−1*)

that is, if Popoviciu’s inequality works for families of *n* weighted points by
grouping them into subfamilies of size *p*∈ {3, ..., n−1}then it also works by
grouping them into subfamilies of size*p*−1.

By Lemma 1,

*λ**i*1*f(x**i*1) +· · ·+*λ**i**p**f*(x*i**p*) + (k−2)(λ*i*1+· · ·+*λ**i**p*)*f*^{}^{λ}^{i}^{1}_{λ}^{x}^{i}^{1}^{+···+λ}^{ip}^{x}^{ip}

*i*1+···+λ_{ip}

≥

≥

*p*

X

*j=1*

X

1≤k≤p, k6=j

*λ*_{i}_{k}

*f*

P

1≤k≤p, k6=j

*λ*_{ik}*x*_{ik}

P

1≤k≤p, k6=j

*λ*_{ik}

whence X

1≤i_{1}*<*···*<i**p*≤n

(λ*i*1+· · ·+*λ**i**p*)*f*^{}^{λ}^{i}^{1}_{λ}^{x}^{i}^{1}^{+···+λ}^{ip}^{x}^{ip}

*i*1+···+λ_{ip}

≥ _{p−2}^{1} − ^{n−1}_{p−1}^{}

*n*

X

*i=1*

*λ**i**f*(x*i*)

+(n−*p*+ 1) ^{X}

1≤i1*<···<i**p−1*≤n

(λ_{i}_{1}+· · ·+*λ*_{i}* _{p−1}*)

*f*

*λ**i*1*x**i*1+···+λ_{ip−1}*x*_{ip−1}*λ**i*1+···+λ_{ip−1}

*.*

By our hypothesis we get

*n−2*
*p−1*

*n−p*
*p−1*

*n*

X

*i=1*

*λ*_{i}*f*(x* _{i}*) +

*n*

X

*i=1*

*λ*_{i}

!
*f*

*n*

P

*i=1*

*λ**i**x**i*
*n*

P

*i=1*

*λ**i*

≥

≥ _{p−2}^{1} − ^{n−1}_{p−1}^{}

*n*

X

*i=1*

*λ**i**f*(x*i*) + (n−*p*+ 1)

X

1≤i1*<···<i**p−1*≤n

(λ*i*1 +· · ·+*λ**i**p−1*)*f*

*λ**i*1*x**i*1+···+λ_{ip−1}*x*_{ip−1}*λ**i*1+···+λ_{ip−1}

*,*

that is,
_{n−2}

*p−2*

*n−p*

*p−1* + ^{n−1}_{p−1}^{}_{p−2}^{1} ^{}

*n*

X

*i=1*

*λ*_{i}*f(x** _{i}*) +

^{n−2}

_{p−2}^{}

*n*

X

*i=1*

*λ*_{i}

!
*f*

*n*

P

*i=1*

*λ**i**x**i*
*n*

P

*i=1*

*λ**i*

≥ ^{n−p+1}_{p−2}^{X}

1≤i1*<···<i**p−1*≤n

(λ_{i}_{1}+· · ·+*λ*_{i}* _{p−1}*)

*f*

*λ**i*1*x**i*1+···+λ_{ip−1}*x*_{ip−1}*λ**i*1+···+λ_{ip−1}

*.*
Since

*n−2*
*p−2*

*n−p*

*p−1* + ^{n−1}_{p−1}^{}_{p−2}^{1} = ^{n−2}_{p−2}^{}^{n−p+1}* _{p−2}*
and

*n−2*
*p−2*

= ^{n−2}_{p−3}^{}^{n−p+1}* _{p−2}*
we can restate the last inequality as follows:

*n−2*
(p−1)−2

*n−(p−1)*
(p−1)−1

*n*

X

*i=1*

*λ*_{i}*f*(x* _{i}*) +

*n*

X

*i=1*

*λ*_{i}

!
*f*

*n*

P

*i=1*

*λ**i**x**i*
*n*

P

*i=1*

*λ**i*

≥ ^{X}

1≤i1*<···<i**p−1*≤n

(λ_{i}_{1} +· · ·+*λ*_{i}* _{p−1}*)

*f*

_{λ}*i*1*x**i*1+···+λ_{ip−1}*x*_{ip−1}*λ**i*1+···+λ_{ip−1}

*,*
which proves to be precisely (P*n,p−1*).

The proof of Popoviciu’s inequality is now complete.

Remark2. The induction step is not necessary in deriving the unweighted
case of the inequalities (P* _{n,2}*) :

(nP* _{n,2}*) (n − 2)

^{f(x}^{1}

^{)+···+f(x}

_{n}

^{n}^{)}+

*f*

^{x}^{1}

^{+···+x}

_{n}

^{n}^{}≥

^{2}

_{n}^{X}

1≤j<k≤n

*f*^{}^{x}^{j}^{+x}_{2} ^{k}^{}

for all *x*_{1}*, ..., x** _{n}* in the domain of

*f.*

In fact, assuming that

*x*_{1} ≤ · · · ≤*x*_{n}

we will consider first the case where

*x*1+x*n*

2 ≤ ^{x}^{1}^{+···+x}_{n}^{n}*.*
Then, by Lemma 1 we get

(M) _{n}^{1} *f*(x_{1}) +*f* ^{x}^{1}^{+x}_{2} ^{2}^{}+· · ·+*f* ^{x}^{1}^{+x}_{2} ^{n}^{} ≤ ^{1}_{2} *f*(x_{1}) +*f* ^{x}^{1}^{+···+x}_{n}^{n}^{}
while from Jensen’s inequality we infer that

(J) ^{2}_{n}^{X}

2≤j<k≤n

*f*^{}^{x}^{j}^{+x}_{2} ^{k}^{}≤ ^{2}_{n}^{n−2}_{2}

*n*

X

*i=2*

*f(x**i*).

Summing up (M) and (J) we get (nP* _{n,2}*).The case where

*x*1+x*n*

2 ≥ ^{x}^{1}^{+···+x}_{n}^{n}

can be treated in a similar way (changing the role of the indices 1 and *n* in

(M)).

At first glance Popoviciu’s inequality is a one real variable result. This
impression is strongly supported by the existence of counterexamples even in
the two real variables context. For example, think at an upsidedown regular
triangular pyramid (viewed as the graph of a convex function). Besides, all
known arguments of (P* _{n,p}*) make use of the ordering of R

*.*

However, as Professor Constantin P. Niculescu called to our attention, it is
possible to develop a higher dimensional theory of convexity based on (P_{3,2}).

This makes the objective of our joint paper [1].

REFERENCES

[1] Bencze, M., Niculescu, C. P.andPopovici, F.,*Convexity according to Popoviciu’s*
*inequality, submitted.*

[2] Niculescu, C. P. and Persson, L.-E., *Convex Functions and their applications. A*
*Contemporary Approach, CMS Books in Mathematics, vol.* **23, Springer-Verlag, New**
York, 2006.

[3] Niculescu, C. P.andPopovici, F.,*A Refinement of Popoviciu’s Inequality, Bull. Soc.*

Sci. Math. Roum.,**49**(97), no. 3, pp. 285–290, 2006.

[4] Peˇcari´c, J. E., Proschan, F.andTong, Y. C.,*Convex functions, Partial Orderings*
*and Statistical Applications, Academic Press, New York, 1992.*

[5] Popoviciu, T.,*Sur certaines in´**egalit´**es qui caract´**erisent les fonctions convexes, Analele*
S

¸tiint¸ifice Univ. “Al. I. Cuza”, Ia¸si, Sect¸ia Mat.,**11, pp. 155–164, 1965.**

Received by the editors: May 8, 2008.