volume 7, issue 4, article 148, 2006.
Received 11 June, 2006;
accepted 15 October, 2006.
Communicated by:B. Yang
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Journal of Inequalities in Pure and Applied Mathematics
NEW INEQUALITIES ABOUT CONVEX FUNCTIONS
LAZHAR BOUGOFFA
Al-imam Muhammad Ibn Saud Islamic University Faculty of Computer Science
Department of Mathematics P.O. Box 84880, Riyadh 11681 Saudi Arabia.
EMail:bougoffa@hotmail.com
c
2000Victoria University ISSN (electronic): 1443-5756 166-06
New Inequalities About Convex Functions
Lazhar Bougoffa
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J. Ineq. Pure and Appl. Math. 7(4) Art. 148, 2006
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Abstract
Iffis a convex function andx1, . . . , xnora1, . . . , anlie in its domain the following inequalities are proved
n
X
i=1
f(xi)−f
x1+· · ·+xn n
≥n−1 n
f
x1+x2 2
+· · ·+f
xn−1+xn 2
+f
xn+x1 2
and
(n−1) [f(b1) +· · ·+f(bn)]≤n[f(a1) +· · ·+f(an)−f(a)], wherea=a1+···+an nandbi=na−an−1i, i= 1, . . . , n.
2000 Mathematics Subject Classification:26D15.
Key words: Jensen’s inequality, Convex functions.
Contents
1 Main Theorems . . . 3 References
New Inequalities About Convex Functions
Lazhar Bougoffa
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1. Main Theorems
The well-known Jensen’s inequality is given as follows [1]:
Theorem 1.1. Letf be a convex function on an interval I and letw1, . . . , wn be nonnegative real numbers whose sum is1. Then for allx1, . . . , xn∈I, (1.1) w1f(x1) +· · ·+wnf(xn)≥f(w1x1+· · ·+wnxn).
Recall that a functionf is said to be convex if for anyt∈[0,1]and anyx, y in the domain off,
(1.2) tf(x) + (1−t)f(y)≥f(tx+ (1−t)y).
The aim of the present note is to establish new inequalities similar to the following known inequalities:
(Via Titu Andreescu (see [2, p. 6])) f(x1) +f(x2) +f(x3) +f
x1+x2+x3 3
≥ 4 3
f
x1 +x2 2
+f
x2+x3 2
+f
x3+x1 2
, wheref is a convex function andx1, x2, x3 lie in its domain,
(Popoviciu inequality [3])
n
X
i=1
f(xi) + n n−2f
x1 +· · ·+xn n
≥ 2
n−2 X
i<j
f
xi+xj 2
,
New Inequalities About Convex Functions
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wheref is a convex function onI andx1, . . . , xn∈I,and (Generalized Popoviciu inequality)
(n−1) [f(b1) +· · ·+f(bn)]≤f(a1) +· · ·+f(an) +n(n−2)f(a), wherea= a1+···+an n andbi = na−an−1i, i = 1, . . . , n,anda1, . . . , an ∈I.
Our main results are given in the following theorems:
Theorem 1.2. Iffis a convex function andx1, x2, . . . , xnlie in its domain, then
(1.3)
n
X
i=1
f(xi)−f
x1+· · ·+xn n
≥ n−1 n
f
x1+x2 2
+· · ·+f
xn−1+xn 2
+f
xn+x1 2
.
Proof. Using (1.2) witht= 12, we obtain (1.4) f
x1+x2 2
+· · ·+f
xn−1+xn 2
+f
xn+x1 2
≤f(x1) +f(x2) +· · ·+f(xn).
In the summation on the right side of (1.4), the expressionPn
i=1f(xi)can be written as
n
X
i=1
f(xi) = n n−1
n
X
i=1
f(xi)− 1 n−1
n
X
i=1
f(xi),
n
X
i=1
f(xi) = n n−1
" n X
i=1
f(xi)−
n
X
i=1
1 nf(xi)
# .
New Inequalities About Convex Functions
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ReplacingPn
i=1f(xi)with the equivalent expression in (1.4), f
x1+x2 2
+· · ·+f
xn−1+xn 2
+f
xn+x1 2
≤ n
n−1
" n X
i=1
f(xi)−
n
X
i=1
1 nf(xi)
# . Hence, applying Jensen’s inequality (1.1) to the right hand side of the above resulting inequality we get
f
x1+x2 2
+· · ·+f
xn−1+xn 2
+f
xn+x1 2
≤ n
n−1
" n X
i=1
f(xi)−f Pn
i=1xi
n #
,
and this concludes the proof.
Remark 1. Now we consider the simplest case of Theorem 1.2 for n = 3 to obtain the following variant of via Titu Andreescu [2]:
f(x1) +f(x2) +f(x3)−f
x1+x2+x3 3
≥ 2 3
f
x1 +x2 2
+f
x2+x3 2
+f
x3+x1 2
. The variant of the generalized Popovicui inequality is given in the following theorem.
New Inequalities About Convex Functions
Lazhar Bougoffa
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Theorem 1.3. Iff is a convex function anda1, . . . , anlie in its domain, then (1.5) (n−1) [f(b1) +· · ·+f(bn)]≤n[f(a1) +· · ·+f(an)−f(a)], wherea= a1+···+an n andbi = na−an−1i, i= 1, . . . , n.
Proof. By using the Jensen inequality (1.1),
f(b1) +· · ·+f(bn)≤f(a1) +· · ·+f(an),
and so,
f(b1) +· · ·+f(bn)
≤ n
n−1[f(a1) +· · ·+f(an)]− 1
n−1[f(a1) +· · ·+f(an)], or
f(b1) +· · ·+f(bn)
≤ n
n−1[f(a1) +· · ·+f(an)]− n n−1
1
nf(a1) +· · ·+ 1 nf(an)
,
and so
(1.6) f(b1) +· · ·+f(bn)
≤ n
n−1
f(a1) +· · ·+f(an)− 1
nf(a1) +· · ·+ 1 nf(an)
.
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Hence, applying Jensen’s inequality (1.1) to the right hand side of (1.6) we get f(b1) +· · ·+f(bn)≤ n
n−1
f(a1) +· · ·+f(an)−f
a1+· · ·+an n
, and this concludes the proof.
New Inequalities About Convex Functions
Lazhar Bougoffa
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References
[1] D.S. MITRINOVI ´C, J.E. PE ˘CARI ´C ANDA.M. FINK, Classical and New Inequalities in Analysis, Kluwer Academic Publishers, Dordrecht, 1993.
[2] KIRAN KEDLAYA, A<B (A is less than B), based on notes for the Math Olympiad Program (MOP) Version 1.0, last revised August 2, 1999.
[3] T. POPOVICIU, Sur certaines inégalitées qui caractérisent les fonctions convexes, An. Sti. Univ. Al. I. Cuza Ia¸si. I-a, Mat. (N.S), 1965.