Page 15 - MATINF Nr. 7

Means of ratios versus ratios of means 15

Proposition 3. If a i > 0, b i > 0, ∀ i = 1, n are such that the sequences (a 1 , a 2 , . . . , a n ) and
(b 1 , b 2 , . . . , b n ) are asynchronous, then
 ‹
H n (b 1 , b 2 , . . . , b n ) a 1 a 2 a n
≥ H n , , . . . , . (2)
H n (a 1 , a 2 , . . . , a n ) b 1 b 2 b n

The proof results here too from the refined form of this inequality captured in the following.

Proposition 4. If a i ≥ 0, b i > 0, ∀ i = 1, n are such that the sequences (a 1 , a 2 , . . . , a n ) and
(b 1 , b 2 , . . . , b n ) are asynchronous, then
 ‹  ‹
H n (b 1 , b 2 , . . . , b n ) 1 1 1 a 1 a 2 a n
≥ H n , , . . . , · H n (b 1 , b 2 , . . . , b n ) ≥ H n , , . . . , . (3)
H n (a 1 , a 2 , . . . , a n ) a 1 a 2 a n b 1 b 2 b n
1 1
Proof. With the substitutions a i → , b i → , ∀ i = 1, n, (substitutions that keep the reverse -
b i a i
ordering of sequences), we have the following transformations:

n
P 1
A n (a 1 , a 2 , . . . , a n ) → i=1 b i = 1 , (4)
n H n (b 1 , b 2 , . . . , b n )

n
P 1
1
a i
i=1
A n (b 1 , b 2 , . . . , b n ) → = , (5)
n H n (a 1 , a 2 , . . . , a n )
n
n
P 1 P
 ‹ a i
1 1 1 i=1 b i i=1 1
A n , , . . . , = → =  1 1 1 ‹, (6)
b 1 b 2 b n n n
H n , , . . . ,
a 1 a 2 a n
 ‹  ‹
a 1 a 2 a n b 1 b 2 b n 1
A n , , . . . , → A n , , . . . , =  ‹. (7)
b 1 b 2 b n a 1 a 2 a n a 1 a 2 a n
H n , , . . . ,
b 1 b 2 b n
With the changes (4)-(7) in relation (1), it follows that

H n (a 1 , a 2 , . . . , a n ) 1 1 1
≤ ·  ‹ ≤  ‹,
H n (b 1 , b 2 , . . . , b n ) H n (b 1 , b 2 , . . . , b n ) 1 1 1 a 1 a 2 a n
H n , , . . . , H n , , . . . ,
a 1 a 2 a n b 1 b 2 b n
that is the double inequality (3) from the statement (that implies the inequality (2)).

But what happens if the sequences of numbers (a 1 , a 2 , . . . , a n ) and (b 1 , b 2 , . . . , b n ) are not
reverse - ordered (as in the statements above)? For the case in which the sequences are similarly
ordered, we have:

Proposition 5. If a i ≥ 0, b i > 0, ∀ i = 1, n are such that the sequences (a 1 , a 2 , . . . , a n ) and
(b 1 , b 2 , . . . , b n ) are similarly ordered, then
 ‹   ‹‹
1 1 1 A n (a 1 , a 2 , . . . , a n ) a 1 a 2 a n
A n (a 1 , a 2 , . . . , a n ) · A n , , . . . , ≥ max , A n , , . . . , .
b 1 b 2 b n A n (b 1 , b 2 , . . . , b n ) b 1 b 2 b n