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Refinements and Extensions of an Inequality
Journal of Mathematical Analysis and Applications 245, 628᎐632 Ž2000. doi:10.1006rjmaa.2000.6734, available online at http:rrwww.idealibrary.com on
NOTE Refinements and Extensions of an Inequality Young-Ho Kim Department of Applied Mathematics, Changwon National Uni¨ ersity, Changwon 641-773, Korea E-mail: [email protected] Submitted by William F. Ames Received November 30, 1999
In this article, using the properties of the power mean, the author proves the inequality n
ž
a1 n
q
a2 n
q ⭈⭈⭈ q
an n
n
Ž xqy .
/
F
n
1
Ž xqy .rx
žÝ / is1
n
a ix
F a1xqy q a2xqy q ⭈⭈⭈ qa nxqy
for a1 , a2 , . . . , a n a set of n nonnegative quantities, n G 1, and for some real numbers x, y g ⺢. 䊚 2000 Academic Press Key Words: power mean; harmonic mean; arithmetic-mean and geometric-mean inequality.
The most basic inequality is the one stating that the square of any real number is nonnegative. To make effective use of this statement, we choose as our real number the quantity b1 y b 2 , where b1 and b 2 are real. Then the inequality Ž b1 y b 2 . 2 G 0 yields, upon multiplying out, b12 q b 22 G 2 b1 b 2 . The sign of equality holds if and only if b1 s b 2 . This is the simplest version of the inequality connecting the arithmetic and geometric means. Setting b12 s a1 , b 22 s a2 in the last inequality, we obtain a1 q a2 2
G a1 a2 ,
'
valid for any two nonnegative quantities a1 and a2 . We shall begin our consideration of results which are not as immediately apparent by dis628 0022-247Xr00 $35.00 Copyright 䊚 2000 by Academic Press All rights of reproduction in any form reserved.
629
NOTE
cussing what is probably the most important inequality, and certainly a keystone of the theory of inequalities, namely the arithmetic-mean and geometric-mean inequality. The result, of singular elegance, follows: THEOREM A w1, p. 4x. Let a1 , a2 , . . . , a n be a set of n nonnegati¨ e quantities, n G 1. Then a1 q a2 q ⭈⭈⭈ qa n 1rn G Ž a1 a2 ⭈⭈⭈ a n . . Ž a. n There is strict inequality unless the a i are all equal. In this article, we give the refinements and extensions of the inequality Ža. by using the strict monotonicity of the power mean of n distinct positive numbers. For any positive values a1 , a2 , . . . , a n and any real p, we defined the power mean, or the mean of order p, by n
M p Ž a; 1rn . s M p Ž a1 , a2 , . . . , a n ; 1rn . s
1
1rp
žÝ / n
is1
a ip
.
In particular, the means of order y1, 0, 1, and 2 are the harmonic mean, the geometric mean, the arithmetic mean, and the root-mean-square. It is well known that M p Ž a1 , a2 , . . . , a n ; 1rn. is a continuous strictly increasing function of p w1, 2x. If a1 , a2 , . . . , a n ) 0, then we ha¨ e the inequalities
LEMMA.
krn
n
n
n
Fn
ž / Ł ai
ž
is1
1
Ý is1
n
k
ai
/
n
F
Ý aik
Ž 1.
is1
for 1 F k with equality holding if and only if all a i are the same; krn
n
n
F
žŁ / is1
n
ai
n
a ik
Ý
Fn
is1
1
k
žÝ / n
is1
ai
Ž 2.
for 0 F k F 1 with equality holding if and only if all a i are the same or k s 0; n
Ý
krn
n
a ik G n
is1
ž / Ł ai
n
Gn
is1
1
k
žÝ / is1
n
ai
Ž 3.
for y1 F k F 0; n
Ý is1
for k F y1.
n
a ik G n
1
žÝ / is1
nai
yk
n
Gn
ž / Ł ai
is1
krn
n
Gn
1
k
žÝ / is1
n
ai
Ž 4.
630
NOTE
The lemma is easily deduced from the fact that M p Ž a1 , a2 , . . . , a n ; 1rn. is a continuous strictly increasing function of p. THEOREM 1. Let a1 , a2 , . . . , a n be a set of n nonnegati¨ e quantities, 1 F n, 1 F x, 0 F y; then n
n
ž
1
Ý is1
n
Ž xqy .
ai
n
Fn
/
ž
Ž xqy .rx
1
Ý
n
is1
n
F
/
a ix
Ý aixqy
Ž 5.
is1
with all equalities holding if and only if all a i are the same. Let a1 , a2 , . . . , a n be a set of n positi¨ e quantities, 1 F n, 0 - x F 1, 0 F y; then Ž xqy .rn
n
n
F
žŁ / is1
ai
1
Ž xqy .rx
n
F
žÝ / is1
n
a ix
1
Ž xqy .
žÝ / is1
n
ai
Ž 6.
with equality holding if and only if all a i are the same. Proof. For 0 F a i , i s 1, 2, . . . , n, 0 - x, let a ix s A i . Then 0 F A i , s a ixqy. By Lemma Ž1. we have
AŽi xqy .r x
n
n
ž
1
Ý
n
is1
Ž xqy .rx
Ai
n
F
/
Ý AŽi xqy .r x is1
for 1 F Ž x q y .rx with all equalities holding if and only if all A i are the same; that is, n
n
1
Ž xqy .rx
žÝ / is1
n
n
F
a ix
Ý aixqy is1
for 1 F Ž x q y .rx with all equalities holding if and only if a i are the same. Observe that M p Ž a1 , a2 , . . . , a n ; 1rn. is a continuous strictly increasing function of p; we get n
n
1
Ž xqy .
žÝ / is1
n
ai
n
Fn
1
Ž xqy .rx
žÝ / is1
n
a ix
for 1 F x with equality holding if and only if x s 1. Consequently, we have the inequality Ž5. for 1 F x and 0 F y. Also, by Lemma Ž1., we have Ž xqy .rn x
n
žŁ / is1
Ai
n
F
1
Ž xqy .rx
žÝ / is1
n
Ai
631
NOTE
for 1 F Ž x q y .rx with all equalities holding if and only if all A i are the same; that is, Ž xqy .rn
n
n
F
ž / Ł ai
is1
1
Ž xqy .rx
žÝ / is1
n
a ix
for 1 F Ž x q y .rx with all equalities holding if and only if all a i are the same. Since M p Ž a1 , a2 , . . . , a n ; 1rn. is a continuous strictly increasing function of p, we get n
n
1
Ž xqy .
n
Gn
žÝ / is1
n
ai
1
Ž xqy .rx
žÝ / is1
n
a ix
for x F 1 with equality holding if and only if x s 1. Consequently, we have the inequality Ž6. for 0 - x F 1 and 0 F y. Remark. When Ž x q y .rx s 1, that is, y s 0, the inequality Ž6. is the inequality Ža. in Theorem A. By Lemmas Ž2., Ž3., and Ž4., we can also deduce the following results: THEOREM 2. Let a1 , a2 , . . . , a n be a set of n nonnegati¨ e quantities, n G 1, 0 - x F 1, yx F y F 0; then Ž xqy .rn
n
n
n
F
ž / Ł ai
is1
n
a ixqy F n
Ý is1
1
Ž xqy .rx
žÝ / is1
n
n
Fn
a ix
1
Ž xqy .
žÝ / is1
n
ai
with all equalities holding if and only if all a i are the same. THEOREM 3. Let a1 , a2 , . . . , a n be a set of n nonnegati¨ e quantities, n G 1, 1 F x, y2 x F y F yx; then n
n
ž
Ý is1
1 n
Ž xqy .
ai
n
Fn
/
ž
Ý is1
1 n
Ž xqy .rx
a ix
Fn
/
Ž xqy .rn
n
n
F
ž / Ł ai
is1
Ý aixqy . is1
THEOREM 4. Let a1 , a2 , . . . , a n be a set of n positi¨ e quantities, n G 1, 1 F x, y F y2 x; then n
n
1
Ž xqy .
žÝ / is1
n
ai
n
Fn
Ž xqy .rx
žÝ / žÝ / is1 n
Fn
1
is1
n
a ix
1
naix
n
Fn
Ž xqy .rn
ž / Ł ai
is1
y Ž xqy .rx
n
F
Ý aixqy . is1
632
NOTE
REFERENCES 1. E. F. Beckenbach and R. Bellman, ‘‘Inequalities,’’ Springer-Verlag, BerlinrNew York, 1961. 2. Z. Liu, Remark on refinements and extensions of an inequality, J. Math. Anal. Appl. 234 Ž1999., 529᎐533.
NOTE Refinements and Extensions of an Inequality Young-Ho Kim Department of Applied Mathematics, Changwon National Uni¨ ersity, Changwon 641-773, Korea E-mail: [email protected] Submitted by William F. Ames Received November 30, 1999
In this article, using the properties of the power mean, the author proves the inequality n
ž
a1 n
q
a2 n
q ⭈⭈⭈ q
an n
n
Ž xqy .
/
F
n
1
Ž xqy .rx
žÝ / is1
n
a ix
F a1xqy q a2xqy q ⭈⭈⭈ qa nxqy
for a1 , a2 , . . . , a n a set of n nonnegative quantities, n G 1, and for some real numbers x, y g ⺢. 䊚 2000 Academic Press Key Words: power mean; harmonic mean; arithmetic-mean and geometric-mean inequality.
The most basic inequality is the one stating that the square of any real number is nonnegative. To make effective use of this statement, we choose as our real number the quantity b1 y b 2 , where b1 and b 2 are real. Then the inequality Ž b1 y b 2 . 2 G 0 yields, upon multiplying out, b12 q b 22 G 2 b1 b 2 . The sign of equality holds if and only if b1 s b 2 . This is the simplest version of the inequality connecting the arithmetic and geometric means. Setting b12 s a1 , b 22 s a2 in the last inequality, we obtain a1 q a2 2
G a1 a2 ,
'
valid for any two nonnegative quantities a1 and a2 . We shall begin our consideration of results which are not as immediately apparent by dis628 0022-247Xr00 $35.00 Copyright 䊚 2000 by Academic Press All rights of reproduction in any form reserved.
629
NOTE
cussing what is probably the most important inequality, and certainly a keystone of the theory of inequalities, namely the arithmetic-mean and geometric-mean inequality. The result, of singular elegance, follows: THEOREM A w1, p. 4x. Let a1 , a2 , . . . , a n be a set of n nonnegati¨ e quantities, n G 1. Then a1 q a2 q ⭈⭈⭈ qa n 1rn G Ž a1 a2 ⭈⭈⭈ a n . . Ž a. n There is strict inequality unless the a i are all equal. In this article, we give the refinements and extensions of the inequality Ža. by using the strict monotonicity of the power mean of n distinct positive numbers. For any positive values a1 , a2 , . . . , a n and any real p, we defined the power mean, or the mean of order p, by n
M p Ž a; 1rn . s M p Ž a1 , a2 , . . . , a n ; 1rn . s
1
1rp
žÝ / n
is1
a ip
.
In particular, the means of order y1, 0, 1, and 2 are the harmonic mean, the geometric mean, the arithmetic mean, and the root-mean-square. It is well known that M p Ž a1 , a2 , . . . , a n ; 1rn. is a continuous strictly increasing function of p w1, 2x. If a1 , a2 , . . . , a n ) 0, then we ha¨ e the inequalities
LEMMA.
krn
n
n
n
Fn
ž / Ł ai
ž
is1
1
Ý is1
n
k
ai
/
n
F
Ý aik
Ž 1.
is1
for 1 F k with equality holding if and only if all a i are the same; krn
n
n
F
žŁ / is1
n
ai
n
a ik
Ý
Fn
is1
1
k
žÝ / n
is1
ai
Ž 2.
for 0 F k F 1 with equality holding if and only if all a i are the same or k s 0; n
Ý
krn
n
a ik G n
is1
ž / Ł ai
n
Gn
is1
1
k
žÝ / is1
n
ai
Ž 3.
for y1 F k F 0; n
Ý is1
for k F y1.
n
a ik G n
1
žÝ / is1
nai
yk
n
Gn
ž / Ł ai
is1
krn
n
Gn
1
k
žÝ / is1
n
ai
Ž 4.
630
NOTE
The lemma is easily deduced from the fact that M p Ž a1 , a2 , . . . , a n ; 1rn. is a continuous strictly increasing function of p. THEOREM 1. Let a1 , a2 , . . . , a n be a set of n nonnegati¨ e quantities, 1 F n, 1 F x, 0 F y; then n
n
ž
1
Ý is1
n
Ž xqy .
ai
n
Fn
/
ž
Ž xqy .rx
1
Ý
n
is1
n
F
/
a ix
Ý aixqy
Ž 5.
is1
with all equalities holding if and only if all a i are the same. Let a1 , a2 , . . . , a n be a set of n positi¨ e quantities, 1 F n, 0 - x F 1, 0 F y; then Ž xqy .rn
n
n
F
žŁ / is1
ai
1
Ž xqy .rx
n
F
žÝ / is1
n
a ix
1
Ž xqy .
žÝ / is1
n
ai
Ž 6.
with equality holding if and only if all a i are the same. Proof. For 0 F a i , i s 1, 2, . . . , n, 0 - x, let a ix s A i . Then 0 F A i , s a ixqy. By Lemma Ž1. we have
AŽi xqy .r x
n
n
ž
1
Ý
n
is1
Ž xqy .rx
Ai
n
F
/
Ý AŽi xqy .r x is1
for 1 F Ž x q y .rx with all equalities holding if and only if all A i are the same; that is, n
n
1
Ž xqy .rx
žÝ / is1
n
n
F
a ix
Ý aixqy is1
for 1 F Ž x q y .rx with all equalities holding if and only if a i are the same. Observe that M p Ž a1 , a2 , . . . , a n ; 1rn. is a continuous strictly increasing function of p; we get n
n
1
Ž xqy .
žÝ / is1
n
ai
n
Fn
1
Ž xqy .rx
žÝ / is1
n
a ix
for 1 F x with equality holding if and only if x s 1. Consequently, we have the inequality Ž5. for 1 F x and 0 F y. Also, by Lemma Ž1., we have Ž xqy .rn x
n
žŁ / is1
Ai
n
F
1
Ž xqy .rx
žÝ / is1
n
Ai
631
NOTE
for 1 F Ž x q y .rx with all equalities holding if and only if all A i are the same; that is, Ž xqy .rn
n
n
F
ž / Ł ai
is1
1
Ž xqy .rx
žÝ / is1
n
a ix
for 1 F Ž x q y .rx with all equalities holding if and only if all a i are the same. Since M p Ž a1 , a2 , . . . , a n ; 1rn. is a continuous strictly increasing function of p, we get n
n
1
Ž xqy .
n
Gn
žÝ / is1
n
ai
1
Ž xqy .rx
žÝ / is1
n
a ix
for x F 1 with equality holding if and only if x s 1. Consequently, we have the inequality Ž6. for 0 - x F 1 and 0 F y. Remark. When Ž x q y .rx s 1, that is, y s 0, the inequality Ž6. is the inequality Ža. in Theorem A. By Lemmas Ž2., Ž3., and Ž4., we can also deduce the following results: THEOREM 2. Let a1 , a2 , . . . , a n be a set of n nonnegati¨ e quantities, n G 1, 0 - x F 1, yx F y F 0; then Ž xqy .rn
n
n
n
F
ž / Ł ai
is1
n
a ixqy F n
Ý is1
1
Ž xqy .rx
žÝ / is1
n
n
Fn
a ix
1
Ž xqy .
žÝ / is1
n
ai
with all equalities holding if and only if all a i are the same. THEOREM 3. Let a1 , a2 , . . . , a n be a set of n nonnegati¨ e quantities, n G 1, 1 F x, y2 x F y F yx; then n
n
ž
Ý is1
1 n
Ž xqy .
ai
n
Fn
/
ž
Ý is1
1 n
Ž xqy .rx
a ix
Fn
/
Ž xqy .rn
n
n
F
ž / Ł ai
is1
Ý aixqy . is1
THEOREM 4. Let a1 , a2 , . . . , a n be a set of n positi¨ e quantities, n G 1, 1 F x, y F y2 x; then n
n
1
Ž xqy .
žÝ / is1
n
ai
n
Fn
Ž xqy .rx
žÝ / žÝ / is1 n
Fn
1
is1
n
a ix
1
naix
n
Fn
Ž xqy .rn
ž / Ł ai
is1
y Ž xqy .rx
n
F
Ý aixqy . is1
632
NOTE
REFERENCES 1. E. F. Beckenbach and R. Bellman, ‘‘Inequalities,’’ Springer-Verlag, BerlinrNew York, 1961. 2. Z. Liu, Remark on refinements and extensions of an inequality, J. Math. Anal. Appl. 234 Ž1999., 529᎐533.