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Means of ratios versus ratios of means

Dorin M˘arghidanu 1

For two pairs of numerical sequences, the classical means of ratios and the ratios of classical
means associated with those sequences are compared. The conditions when these comparisons
can be made are also analyzed.

Given the positive real numbers x 1 , x 2 , . . . , x n the following classical means are well known
(see for example [3], [8]):

n
1 X
A n (x 1 , x 2 , . . . , x n ) := · x k (arithmetic mean of numbers x 1 , x 2 , . . . , x n ),
n
k=1
Ì
n
Y
G n (x 1 , x 2 , . . . , x n ) := n x k (geometric mean of numbers x 1 , x 2 , . . . , x n ),
k=1
n
H n (x 1 , x 2 , . . . , x n ) := (harmonic mean of numbers x 1 , x 2 , . . . , x n ),
n
P 1
k=1 x k
as well as the inequality between them,

A n (x) ≥ G n (x) ≥ H n (x)

(and the equalities hold if and only if x 1 = x 2 = . . . = x n ).

Relative to two sequences of numbers (a 1 , a 2 , . . . , a n ) and (b 1 , b 2 , . . . , b n ), a i ≥ 0, b i > 0,
a i
∀i = 1, n, we are interested in comparing the means of the ratios with the ratios of the means
b i
associated with the given sequences, under possible additional conditions that must be fulfilled
by the given numbers.

Remark 1. To this purpose, from the three classical means the most convenient is the geometric
mean, for which we have (without other conditions imposed - apart from the positivity of the
numbers a i ≥ 0, b i > 0, ∀i = 1, n) the next obvious equality:

 ‹
a 1 a 2 a n G n (a 1 , a 2 , . . . , a n )
G n , , . . . , = .
b 1 b 2 b n G n (b 1 , b 2 , . . . , b n )

For the other two types of means, some additional conditions are required for the given
number sequences, such as those related to the monotony of sequences. Let us mention here the
following one.

1
Profesor dr., Colegiul Nat , ional ,,Al. I. Cuza”, Corabia, d.marghidanu@gmail.com

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