SOME APPLICATIONS OF AN INEQUALITY OF NORMAN LEVINSON ´ J. GIBERGANS-BAGUENA, J. J. EGOZCUE AND J. L. D´IAZ-BARRERO Abstract. Applying the inequality of N. Levinson several multivariate inequalities involving arithmetics progressions, random variables and binomial coefficients are obtained.
1. Introduction In [1] Norman Levinson presented the following generalization of an inequality of Ky Fan [2] Theorem 1.1. Let f : [0, 2s] → R be a function that with a nonnegative third derivative in (0, 2s). If ak ∈ (0, s], (1 ≤ k ≤ n), and pk > 0, (1 ≤ k ≤ n), then ! n n 1 X 1 X pk f (xk ) − f pk xk Pn k=1 Pn k=1 ! n n 1 X 1 X (1.1) ≤ pk f (2s − xk ) − f pk (2s − xk ) Pn k=1 Pn k=1 Pj 000 where Pj = k=1 pk for 1 ≤ j ≤ n. If f (t) > 0 in (0, 2s) then equality holds if and only if x1 = x2 = . . . = xn . An immediate consequence of the preceding result is the above mentioned inequality of Ky Fan. Namely, Corollary 1.2. Let 0 < xk ≤ 1/2 for k = 1, 2, · · · , n. Then !, n !n !, n !n n n Y X Y X (1.2) xk xk ≤ (1 − xk ) (1 − xk ) k=1
k=1
k=1
k=1
Proof. Indeed inequality (1.2) immediately follows setting s = 1/2 and f (t) = log(t) into (1.1). 2000 Mathematics Subject Classification. 26D15, 30A10. Key words and phrases. Multivariate Inequalities, Levinson’s inequality.
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´ J. GIBERGANS-BAGUENA, J. J. EGOZCUE AND J. L. D´IAZ-BARRERO
2. The Inequalities In the sequel some applications of the preceding inequality are given and several inequalities similar to the ones presented in ([3], [4]) are obtained. We begin with Theorem 2.1. Let x1 , x2 , . . . , xn be positive numbers such that x1 + x2 + · · · + xn = s and let pk > 0, (1 ≤ k ≤ n) and q ∈ N∗ . Then holds !q n n 1 X 1 X pk (2s − xk ) + pk xqk Pn k=1 Pn k=1 ≤
n 1 X pk xk Pn k=1
!q +
n 1 X pk (2s − xk )q Pn k=1
Proof. Since the function f : (0, +∞) → R defined by f (t) = tq , q ∈ N∗ , has third derivative f 000 (t) ≥ 0, then applying (1.1) the statement immediately follows after rearranging terms and this completes the proof. Applying the preceding result the following corollaries are obtained. Corollary 2.2. Let x1 , x2 , . . . , xn be positive numbers in arithmetic progression and let pk > 0, (1 ≤ k ≤ n) such that p1 +p2 +· · ·+pn = 1. Then, for all q ∈ N∗ , holds !q n n X X x k q xk − pk x1 + xn − pk x1 + xn − n n k=1 k=1 1 ≤ q n
"
n X
!q pk xk
−
k=1
n X
# pk xqk
k=1
Corollary 2.3. Let X be a Binomial random variable. That is, X ∼ B(n, p). Then, for all q ∈ N∗ , holds !q n n X X pk (n(n + 1) − k) pk k q + 1 − (1 − p)n 1 − (1 − p)n k=1 k=1 ≤ where pk =
n k
np 1 − (1 − p)n
q
n X pk n(n + 1) − k + 1 − (1 − p)n k=1
pk (1 − p)n−k , 0 ≤ k ≤ n.
q
APPLICATIONS OF LEVINSON’S INEQUALITY
Proof. Setting xk = k, 1 ≤ k ≤ n and noting that and
n X
n X
3
pk = 1 − (1 − p)n
k=1
pk xk = E(X) = np, then the statement immediately follows
k=1
from Theorem 2.1. Another application of Levinson’s inequality is the following result. Theorem 2.4. Let x1 , x2 , . . . , xn be positive numbers such that x1 + x2 +· · ·+xn = s and let pk > 0, (1 ≤ k ≤ n) such that p1 +p2 +· · ·+pn = 1. Then holds n X
(2.1)
! pk xk
k=1
n Y 2s
xk
k=1
pk X n −1 ≥ pk (2s − xk ) k=1
Proof. To proof the preceding statement we will consider the function f (t) = log(t) and applying (1.1), we have n X
pk log(xk ) − log
k=1
n X
! pk xk
k=1
! n 1 X pk (2s − xk ) ≤ pk log(2s − xk ) − log P n k=1 k=1 ! n n X Y Taking into account that log(xk )pk = log xk pk we have n X
k=1
log
n Y
xk pk −log
k=1
n X
! pk xk
≤ log
k=1 n Y
(2s−xk )pk −log
k=1
k=1
n X k=1
or equivalently,
n Y
pk
n Y
pk
xk (2s − xk ) k=1 k=1 ≤ log log n n X X pk xk pk (2s − xk ) k=1
k=1
! pk (2s − xk )
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´ J. GIBERGANS-BAGUENA, J. J. EGOZCUE AND J. L. D´IAZ-BARRERO
Since f (x) = log(x) is injective, we obtain n X pk (2s − xk ) p n Y 2s − xk k k=1 ≤ n X xk k=1 pk xk k=1
and the proof is complete.
Finally, we present two new multivariate inequalities involving binomial coefficients. Corollary 2.5. Let x0 , x1 , · · · , xn be positive numbers such that x0 + x1 + · · · + xn = s. Then holds #2n " n #2n n " n n X n X n Y 2s − xk (k ) (2s − xk ) ≤ xk k k xk k=0 k=0 k=0 1 n Proof. Setting pk = n , 0 ≤ k ≤ n into the preceding result and 2 k n X n taking into account the well known identity = 2n , we have k k=0 n X n n 1 (2s − x ) k k n 2n k=0 k Y 2s − xk 2n ≤ n xk 1 X n k=1 x k 2n k=0 k or equivalently, n 2n X n (2s − xk ) (nk) n Y k=0 k 2s − x k ≤ n X xk n k=0 xk k k=0 from which the statement immediately follows and this completes the proof. Corollary 2.6. If 0 ≤ xk ≤ 12 , 0 ≤ k ≤ n and x0 + x1 + · · · + xn = s. Then !2n n n n Y 1 X n 2s − xk (k ) (2s − xk ) ≤ 2n−1 k=0 k xk k=0
APPLICATIONS OF LEVINSON’S INEQUALITY
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holds. Proof. We have
n X n
n 1X n xk ≤ = 2n−1 and applying (2.1) 2 k=0 k k
k=0
yields
n X n
k=0
2n
(2s − xk ) 2n−1
n
k
≤
n Y 2s − xk (k ) k=0
and the statement follows.
xk
References 1. N. Levinson. “Generalization of an inequality of Ky Fan,” Journal of Mathematical Analysis and Applications, Vol. 8 (1964) 133–134. 2. E. F. Beckencback and R. Bellman, Inequalities, Cambridge, 1961. 3. J. Gibergans-B´ aguena and J. L. D´ıaz-Barrero. “Some Elementary Inequalities Involving Convex Functions”. Octogon Mathematics Magazine, Vol. 13, No. 2, (2005) 984–988. 4. J. L. D´ıaz-Barrero and P. G. Popescu, “Some Elemntary Inequalities for Convex Functions”, J. Ineq. Pure and Appl. Math., (2006) (Submitted). `cnica de Catalunya, Applied Mathematics III, Universitat Polite Jordi Girona 1-3, C2, 08034 Barcelona. Spain E-mail address:
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