Prove that 1/(1/a + 1/b) + 1/(1/c + 1/d) ≤ 1/(1/(a+c) + 1/(b+d) ) for positive reals a, b, c, d.
Solution
A straightforward, if inelegant, approach is to multiply out and expand everything. All terms cancel except four and we are left with 2abcd ≤ a2d2 + b2c2, which is obviously true since (ad - bc)2 ≥ 0.
© John Scholes
jscholes@kalva.demon.co.uk
8 July 2002