About the equivalence of some classical inequalities. I. (English)

Gaz. Mat., Ser. B 120, No. 4, 171-179 (2015).

Summary: The equivalence of the classical inequalities studied in [{\it G. H. Hardy} et al., Inequalities. Cambridge: Univ. Press (1934; Zbl 0010.10703); {\it L. Maligranda}, Math. Inequal. Appl. 1, No. 1, 69‒83 (1998; Zbl 0889.26001); ibid. 4, No. 2, 203‒207 (2001; Zbl 0987.26011); {\it A. W. Marshall} and {\it I. Olkin}, Inequalities: theory of majorization and its applications. New York etc.: Academic Press (1979; Zbl 0437.26007)], follows from Jensen inequality as a property of the convex functions. Following a long way, but simple and generally, in this paper we show that the equivalence of classical inequalities in finite dimensional case can be proved without using directly Jensen inequality.