An inequality relating means

Extract

1. Let a, b be positive constants; and let y₁, y₂, …, y_n be real exponents, not all equal, having arithmetic mean y defined by

(here, and in what follows, the summation ∑ extends over the values i = 1, 2, …, n). Then it is clear that

since the right-hand sides are the geometric means of the positive numbers whose arithmetic means stand on the left-hand sides. I know of no results, however, which relate the ratios and and I have had occasion recently to require such results. This note gives an inequality between these ratios, subject to certain restrictions on a and b.