Generalized convexity and inequalities

By G. D. Anderson, Mavina Vamanamurthy, Matti Vuorinen. 2007

253 citations recorded by the source corpus.

Abstract Let R + = ( 0 , ∞ ) and let M be the family of all mean values of two numbers in R + (some examples are the arithmetic, geometric, and harmonic means). Given m 1 , m 2 ∈ M , we say that a function f : R + → R + is ( m 1 , m 2 ) -convex if f ( m 1 ( x , y ) ) ⩽ m 2 ( f ( x ) , f ( y ) ) for all x , y ∈ R + . The usual convexity is the special case when both mean values are arithmetic means. We study the dependence of ( m 1 , m 2 ) -convexity on m 1 and m 2 and give sufficient conditions for ( m 1 , m 2 ) -convexity of functions defined by Maclaurin series. The criteria involve the Maclaurin coefficients. Our results yield a class of new inequalities for several special functions such as the Gaussian hypergeometric function and a generalized Bessel function.

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/579386ffdd28a312404b777baf303d41e7dc5208

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In Applied Mathematics

In Geometry and Topology

Citations in the corpus

The Schur multiplicative and harmonic convexities of the complete symmetric function (2011, background): “[3] discussed an attractive class of inequalities, which are related to multiplicatively and harmonic convex functions.”
Generalized elliptic integrals and the Legendre M-function (2007, background): “the assertion follows from [ AVV2 , Theorem 3.2(1)].”
Mills' ratio: Monotonicity patterns and functional inequalities (2008, background): “5 from [1] the function x → x2r ′(x) is strictly decreasing, which is proved in part (d) of Theorem 2.”
Turán type inequalities for generalized complete elliptic integrals (2007, background): “2. From the first part of Theorem 3.2 due to Anderson et al. [ 6 ] it is known that r � logfa,b,c(r) = logF(a,b,c,r)isconvexon (0,1),wherea,b,c > 0andc(a+b+1) ≥ ab. Using this result, we get”
Geometrically concave univariate distributions (2010, background): “Moreover, it is easy to verify (see for example [4,21]) that a differentiable function f is geometrically convex (concave) if and only if the function x → xf ′(x)/ f (x) is increasing (decreasing).”