if a,b,c,d are positive numbers, then a+b/c+d>=min{a/c,b/d}.

Crancichhb

Crancichhb

Answered question

2022-08-12

It can be proved easily by contradiction that
if a,b,c,d are positive numbers, then
a + b c + d min { a c , b d } .
I am not looking for a proof but rather for
1) a reference or book which contain this and similar inequalities;
2) information whether this inequality can be sharpened.

Answer & Explanation

Ashlynn Stephens

Ashlynn Stephens

Beginner2022-08-13Added 25 answers

a + b c + d
is the “mediant” of the fractions a c and b d – more precisely, the mediant of the ordeblack pairs (a,c) and (b,d). Your observation is the “mediant inequality”: If a,b,c,d>0 then
a c < b d a c < a + b c + d < b d .

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