## Generalized convexity and inequalities (2006)

Venue: | The University of Auckland, Report Series |

Citations: | 5 - 3 self |

### BibTeX

@ARTICLE{Anderson06generalizedconvexity,

author = {G. D. Anderson and M. K. Vamanamurthy and M. Vuorinen},

title = {Generalized convexity and inequalities},

journal = {The University of Auckland, Report Series},

year = {2006},

pages = {1--17}

}

### OpenURL

### Abstract

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 m1, m2 ∈ M, we say that a function f: R+ → R+ is (m1, m2)-convex if f(m1(x, y)) ≤ m2(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 (m1, m2)-convexity on m1 and m2 and give sufficient conditions for (m1, m2)-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. 1.

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1 |
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Citation Context ... function M : (0, ∞) × (0, ∞) → (0, ∞) is called a Mean function if (1) M(x, y) = M(y, x), (2) M(x, x) = x, (3) x < M(x, y) < y, whenever x < y, (4) M(ax, ay) = aM(x, y) for all a > 0. 2.2. Examples. =-=[10]-=-,[11] (1) M(x, y) = A(x, y) = (x + y)/2 is the Arithmetic Mean. (2) M(x, y) = G(x, y) = √ xy is the Geometric Mean. (3) M(x, y) = H(x, y) = 1/A(1/x, 1/y) is the Harmonic Mean. (4) M(x, y) = L(x, y) = ... |

1 |
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