Let a_{1}, a_{2}, ... , a_{n} be reals in the interval (0, π) with arithmetic mean μ. Show that ∏ (sin a_{i})/a_{i} ≤ ( (sin μ)/μ )^{n}.

**Solution**

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(a_{i}) ≤ 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) = - cosec^{2}x + 1/x^{2}. But sin x ≤ x on [0, π], so f ''(x) ≤ 0 on [0, π].

© John Scholes

jscholes@kalva.demon.co.uk

30 Nov 1999