Stečkin proved an inequality on Fejér means of Fourier series He said, "Proving similar inequality for other summability method is an interesting problem." We generalize Stečkin's inequality and give various applications to summability methods.
Yang, G. S., & Liu, S. J. (2009). A Simple Proof of Inequalities Related to Means. Tamkang Journal of Mathematics, 40(4), 429-436. https://www.airitilibrary.com/Article/Detail?DocID=00492930-200912-201012230067-201012230067-429-436