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On Inequalities Complementary to Jensen's

Published online by Cambridge University Press:  20 November 2018

Paul R. Beesack
Paul R. Beesack*
Affiliation:
Carleton University, Ottawa, Ontario
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In a paper published in 1975 [1, § 3], D. S. Mitrinovič and P. M. Vasič used the so-called “centroid method” to obtain two new inequalities which are complementary to (the discrete version of) Jensen's inequality for convex functions. In this paper we shall present a very general version of such inequalities using the same geometric ideas used in [1] but not using the centroid method itself. At the same time we shall extend the domain of the inequalities (even in the discrete case), and clarify the value of the constant (λ) appearing in the inequality. We give applications of the theorems to some general means and also to the classical means. Our first result is given as

THEOREM 1. Let v be a nonnegative measure on a σ-algebra of subsets of a set D and let q, f be real v-measurable functions on D such that q(x) > 0, –∞ < x1 ≦ f(x) ≦ x2 < ∞ for all x ∊ D and ∫D qdv = 1.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 35 , Issue 2 , 01 April 1983 , pp. 324 - 338
Copyright
Copyright © Canadian Mathematical Society 1983

References

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