[2] BO-YING WANG, Foundations of Majorization Inequalities (Kong zhi bu deng shi ji chu) ... SU AND FANG-UAN WANG, Theory and method of inequalities (Bu.
J ournal of Mathematical I nequalities Volume 2, Number 1 (2008), 101–107
GENERALIZATIONS OF BERNOULLI’S INEQUALITY WITH APPLICATIONS HUAN-NAN SHI (communicated by P. Bullen)
Abstract. By using methods on the theory of majorization, some new generalizations of Bernoulli’s inequality are established and some applications of the generalizations are given.
1. Introduction Let x > −1 and n is a positive integer. Then (1 + x)n 1 + nx.
(1.1)
(1.1) is known as the Bernoulli’s inequality which play an important role in analysis and its applications. So, during the past few years, many researchers obtained various generalizations, extensions of inequality (1.1). For example, the following generalizations and variants of (1.1) were recorded in [1, pp. 127–128 ]: THEOREM A. Let x > −1 . If α > 1 or α < 0 , then if 0 < α < 1 , then
(1 + x)α 1 + α x,
(1.2)
(1 + x)α 1 + α x.
(1.3)
In (1.2) and (1.3), equalities holding if and only if x = 0 . THEOREM B. Let ai 0 , xi > −1, i = 1, . . . , n , and n
a
(1 + xi ) i 1 +
i=1
n
n i=1
ai 1 . Then
a i xi ,
(1.4)
i=1
if ai 1 or ai 0 , and if xi > 0 , or −1 < xi < 0, i = 1, . . . , n , then n i=1
a
(1 + xi ) i 1 +
n
a i xi .
(1.5)
i=1
Mathematics subject classification (2000): 26D15. Key words and phrases: Bernoulli’s inequality, Schur-concavity, elementary symmetric function, majorization. The author was supported in part by the Scientific Research Common Program of Beijing Municipal Commission of Education (KM200611417009).
c , Zagreb Paper JMI-02-10
101
102
HUAN-NAN SHI
For more information on the Bernoulli’s inequality, please refer to [4, 6, 7, 8, 9, 10] and the references therein. In this paper, some new generalizations of Bernoulli’s inequality are established by the Schur-concatity of the elementary symmetric functions and the dual form of the elementary symmetric functions, and some applications of the generalizations are given. We obtain the following results. THEOREM 1. Let m, n is a positive integer, k = 1, . . . , n . (i) If m n and x > −1 , then
and
k n k i k−i Cmk 1 + x Cn Cm−n (1 + x)i , m i=0
(1.6)
k k i k−i k n Cm k Cm 1 + x (ix + k)Cn Cm−n . m
(1.7)
i=0
(ii) If m < n and x > − mn , then Cmk (1 + x)k
k i=0
and k Cm
k
kCm (1 + x)
m i k−i 1+ x , Cmi Cn−m n
k m i=0
n
ix + k
Cmi Ck−i
n−m
(1.8)
,
(1.9)
n! is the number of combinations of n elements taken k at a time, where Cnk = k!(n−k)! 0 defined Cn = 1 and Cnk = 0 for k > n . In (1.6), (1.7), (1.8) and (1.9), equalities holding if and only if x = 0 .
REMARK 1. When x = 0 , (1.6), (1.7), (1.8) and (1.9) are deduce to Vandermonde identity. k k−i Cmk = Cni Cm−n . (1.10) i=0
THEOREM 2. If ai 1 or ai 0 , and if xi > 0 or 0 xi −1 . Then Cnk
1 (1 + xi )ai n n
k
i=1
k aij 1 + xij
1i1 0 , i.e. p , q ∈ Rn++ . Thus we have Ek (pp ) Ek (qq ) and Ek∗ (pp ) Ek∗ (qq ) , i.e. (1.8)and (1.9) hold, and equality holding if and only if x = 0 . The proof of Theorem 1 is completed. Proof of Theorem 2. Set y = 1n ni=1 (1 + xi )ai , from (1.2), we have (1 + xi )ai 1 + ai xi , i = 1, . . . , n , and by lemma 1 , it follows that (y, . . . , y) ≺ ((1 + x1 )a1 , . . . , (1 + xn )an )
n
⎛ (1 + a1 x1 , . . . , 1 + an xn ) ≺ ⎝1 +
n i=1
⎞ ai xi , 1, . . . , 1⎠ .
n−1
105
GENERALIZATIONS OF BERNOULLI’S INEQUALITY WITH APPLICATIONS
And then, since Ek (xx) and Ek∗ (xx) are increasing and Schur-concave on Rn+ and are increasing and strictly Schur-concave on Rn++ for k > 1 , we have Ek (y, . . . , y) Ek ((1 + x1 )a1 , . . . , (1 + xn )an )
n
⎛ Ek (1 + a1 x1 , . . . , 1 + an xn ) Ek ⎝1 +
n i=1
⎞ ai xi , 1, . . . , 1⎠ ,
n−1
i.e. (1.11) is holds. And Ek∗ (y, . . . , y) Ek ((1 + x1 )a1 , . . . , (1 + xn )an )
n
⎛ Ek∗ (1 + a1 x1 , . . . , 1 + an xn ) Ek∗ ⎝1 +
n i=1
⎞ ai xi , 1, . . . , 1⎠ ,
n−1
i.e. (1.12) is holds. The proof of Theorem 2 is completed.
3. Applications THEOREM 3. Let ai 1 and xi 1 , i = 1, · · · , n, n ∈ N, n 2 . Then for k = 1, · · · , n , we have
k
1 + xij
aij
1i1
