Page 17 - MATINF Nr. 7

Means of ratios versus ratios of means 17

a 1 a 2 a n
with equality if and only if = = . . . = ;
b 1 b 2 b n
b) (Radon’s Inequality, 1913)
?
If n ∈ N , a i ≥ 0, b i > 0, ∀ i = 1, n, p ≥ 0, then
p+1 p+1 p+1 p+1
a a a (a 1 + a 2 + . . . + a n )
1 + 2 + . . . + n ≥ ,
p
b p b p b n (b 1 + b 2 + . . . + b n ) p
1 2
a 1 a 2 a n
with equality if and only if = = . . . = ;
b 1 b 2 b n
c) (Generalized Radon’s Inequality, [4])
?
If n ∈ N , a i ≥ 0, b i > 0, ∀ i = 1, n, p ≥ 0, r ≥ p + 1, then

a r 1 a r 2 a r n 1 (a 1 +a 2 + . . . +a n ) r
p + p + . . . + p ≥ · p , (10)
b 1 b 2 b n n r−p−1 (b 1 +b 2 + . . . +b n )

with equality if and only if a 1 = a 2 = . . . = a n and b 1 = b 2 = . . . = b n .

Remark 5. Obviously, Bergstr¨om’s Inequality is a particular case of Radon’s Inequality, and
both of these fractional inequalities are particular cases of Generalized Radon’s Inequality.
Also, we observe that inequality (9) is obtained from an extension of the domain of powers
in inequality (10) and for the case r = p = 1, but with the requirement that the sequences
(a 1 , a 2 , . . . , a n ) and (b 1 , b 2 , . . . , b n ) to be asynchronous.

If the sequences (a 1 , a 2 , . . . , a n ) and (b 1 , b 2 , . . . , b n ) are synchronous, from Proposition 5 we
derive a comparison of the following kind (see also [7]):

Corollary 2. 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
† 
n
P
n ! n ! a i n
1 X X 1 i=1 X a i
· a i · ≥ max n · , .
n
n b i P b i
i=1 i=1 b i i=1
i=1
It would be interesting to investigate whether for other means - besides the classic ones
considered here - can be compared the means of ratios and the ratios of means.

References

[1] B˘atinet , u-Giurgiu, D.M., M˘arghidanu D., Pop T.O., A generalization of Radon’s inequality,
Creative Math. & Inf. 20, No. 2, 2011, pp. 111–116.

[2] B˘atinet , u-Giurgiu, D.M, M˘arghidanu D., Pop T.O., A refinement of a Radon type inequality,
Creative Math. & Inf. 27, No. 2, 2018, pp. 115–122.

[3] Bullen P.S., Handbook of Means and Their Inequalities, Kluwer Academic Publishers,
Dordrecht, 2003.