summed numerator and summed denominator

If  a1b1,,anbn  are any real fractions with positive denominators and


are the least and the greatest ( of the fractions, then

ma1++anb1++bnM. (1)

The equality signs are valid if and only if all fractions are equal; in this case one has


Proof.  Set  q1:=a1b1,  …,  qn:=anbn.  Then we have  a1++an=b1q1++bnqn,  which apparently has the lower boundMathworldPlanetmath(b1++bn)m  and the upper bound(b1++bn)M.  Dividing the three last expressions by the sum  b1++bn  yields the asserted double inequalityMathworldPlanetmath (1).

Remark.  Cf. also the mediant.

Title summed numerator and summed denominator
Canonical name SummedNumeratorAndSummedDenominator
Date of creation 2013-10-11 15:35:42
Last modified on 2013-10-11 15:35:42
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 11
Author pahio (2872)
Entry type Theorem
Classification msc 11A99
Related topic InequalityForRealNumbers