The positive reals v, w, x, y, z satisfy 0 < h ≤ v, w, x, y, z ≤ k. Show that (v + w + x + y + z)(1/v + 1/w + 1/x + 1/y + 1/z) ≤ 25 + 6( √(h/k) - √(k/h) )2. When do we have equality?
Solution
Fix four of the variables and allow the other to vary. Suppose, for example, we fix all but x. Then the expression on the lhs has the form (r + x)(s + 1/x) = (rs + 1) + sx + r/x, where r and s are fixed. But this is convex. That is to say, as x increases if first decreases, then increases. So its maximum must occur at x = h or x = k. This is true for each variable.
Suppose all five are h or all five are k, then the lhs is 25, so the inequality is true and strict unless h = k. If four are h and one is k, then the lhs is 17 + 4(h/k + k/h). Similarly if four are k and one is h. If three are h and two are k, then the lhs is 13 + 6(h/k + k/h). Similarly if three are k and two are h.
h/k + k/h ≥ 2 with equality iff h = k, so if h < k, then three of one and two of the other gives a larger lhs than four of one and one of the other. Finally, we note that the rhs is in fact 13 + 6(h/k + k/h), so the inequality is true with equality iff either (1) h = k or (2) three of v, w, x, y z are h and two are k or vice versa.
© John Scholes
jscholes@kalva.demon.co.uk
20 Aug 2002