Convexity according to the geometric mean

By Constantin P. Niculescu. Mathematical Inequalities & Applications, 2000

292 citations recorded by the source corpus.

We develop a parallel theory to the classical theory of convex functions, based on a change of variable formula, by replacing the arithmetic mean by the geometric one. It is shown that many interesting functions such as exp sinh cosh sec csc arc sin Γ etc illustrate the multiplicative version of convexity when restricted to appropriate subintervals of 0 .A s a consequence, we are not only able to improve on a number of classical elementary inequalities but also to discover new ones.

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/96650c0adccf84600c2231f2caeb553a9db11b1c

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In Applied Mathematics

In Geometry and Topology

Citations in the corpus

Ostrowski type inequalities for $p$-convex functions (2016, methodology): “See [8,14] for some kinds of convexity obtained by using weighted means.”
The Schur multiplicative and harmonic convexities of the complete symmetric function (2011, background): “In particular, the multiplicatively and harmonic convex functions have been attracting the attention of many mathematicians [2], [9]–[11], [16], [19], [29], [38], [40], [41].”
Selected Topics on Hermite-Hadamard Inequalities and Applications (2003, background): “([128]) Suppose that I is a subinterval of (0,∞).”
Geometrically concave univariate distributions (2010, background): “We note that the geometrically convex functions have been studied already in 1928 by Montel [17], however, the roots of this topic can be traced long before him, as it was pointed out recently by Niculescu [21].”
The Hermite–Hadamard inequality for log-convex functions (2012, background): “See [15,16,2] for a discussion of convexity from this point of view.”
Conformable Integral Inequalities of the Hermite-Hadamard Type in terms of GG- and GA-Convexities (2019, background): “In [15, 16], Niculescu defined the GA- and GG-convex functions as follows.”
Weak majorization inequalities and convex functions (2003, background): “For more examples and properties of multiplicativily convex functions the reader is referred to [10].”
Hermite–Hadamard type integral inequalities for geometric-arithmetically s-convex functions (2013, background): “2 When s D 1, the above-defined geometric-arithmetically s-convex function becomes a GA-convex function defined in [10, 11].”