
06515124
j
2016a.00695
Ceau\c{s}u, Traian
About the equivalence of some classical inequalities. I.
Gaz. Mat., Ser. B 120, No. 4, 171179 (2015).
2015
Romanian Mathematical Society (Societatea de \c{S}tiin\c{t}e Matematice din Rom\^ania), Bucharest
EN
H30
I30
Cauchy inequality
rootmeansquare inequality
rearrangement inequality
CauchyBunyakovskiSchwarz inequality
Bernoulli inequality
Young inequality
RadoPopoviciu inequality
Maclaurin inequality
Maclaurin inequality
RogersH\"older inequality
Rogers inequality
Lyapunov inequality
powermean inequality
Minkowski inequality
Zbl 0010.10703
Zbl 0987.26011
Zbl 0437.26007
Zbl 0889.26001
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, 6983 (1998; Zbl 0889.26001); ibid. 4, No. 2, 203207 (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.