A Note on a Theorem of Hardy

The object of this note is to give a proof by direct application of the Schwarzian inequality without any intermediary analysis, and different from Hardy's proof of a theorem given in Mathematische Zeitshrift (1920, Band 6, page 314), which states that, if k > 1,^∞∑ a^k/_n 

and ^∞∑(Sⁿ/_n )ⁿ ≤ λ ^∞∑ o, convergent.