A logarithm type mean value theorem of the Riemann zeta function

For any integer K⩾2 and positive integer h, we investigate the mean value of |ζ(σ+it)|2k×logh|ζ(σ+it)| for all real number 01−1/K. In case K=2, h=1, this has been studied by Wang in [F.T. Wang, A mean value theorem of the Riemann zeta function, Quart. J. Math. Oxford Ser. 18 (1947) 1–3]. In this note, we give a new brief proof of Wang's theorem, and, with this method, generalize it to the general case naturally.