Let a1, a2, ... , an be reals in the interval (0, π) with arithmetic mean μ. Show that ∏ (sin ai)/ai ≤ ( (sin μ)/μ )n.
Products are intractable, whereas sums are easy, so we take logs. Let f(x) = ln ( (sin x)/x) = ln sin x - ln x. The required relation is now that 1/n ∑ f(ai) ≤ f(μ). This is true if the curve is concave, in other words if f ''(x) ≤ 0.
Differentiating, we have: f '(x) = cot x - 1/x, f ''(x) = - cosec2x + 1/x2. But sin x ≤ x on [0, π], so f ''(x) ≤ 0 on [0, π].
39th Putnam 1978
© John Scholes
30 Nov 1999