On an extension of shapiro's cyclic inequality

Abstract

In 1954, Harold Seymour Shapiro proposed the inequality for a cyclic sum innvariables as follows ...In this note, by studying1.2in the casen4, we show that it is true whenp≥q,and false whenp<q. Moreover, we give a sufficient condition ofp,qunder which1.2is true in the casen5. It is worth saying that ifp<q, then 1.2is false for every even n≥4. Two open questions are discussed at the end of this paper....