H¨older Means, Lehmer Means, and xy1 log cosh x

The monotonicity of the two variable H¨older mean is a consequence of a certain inequality for log cosh x. This in turn is shown to be a limiting case of an inequality not involving any transcendental functions. The proof of this elementary inequality is based on the preliminary result that H2 nq1FLnwhere Hnand Ln are the H¨older and Lehmer means of order n. Some further inequalities for log cosh x are also proved.