Useful remarks were also made by J. Ferrand and P. Jarvi. At the final stage I have had the good fortune to work with J. Kankaanpaa, who prepared the final version of the text using the 'lEX system of D. E. Knuth and improved the text in various ways. The previewer program for 'lEX written by A. Hohti was very helpful in the course of this project. The work of Kankaanpaa was supported by a grant of the Academy of Finland. Hohti and O. Kanerva have provided their generous assistance in the use of the 'lEX system. Helsinki
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 :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.
In geometric function theory, generalized elliptic integrals and functions arise from the Schwarz-Christoffel transformation of the upper half-plane onto a parallelogram and are naturally related to Gaussian hypergeometric functions. Certain combinations of these integrals also occur in analytic number theory in the study of Ramanujan's modular equations and approximations to π. The authors study the monotoneity and convexity properties of these quantities and obtain sharp inequalities for them.
Abstract. Working in doubling metric spaces, we examine the connections between different dimensions, Whitney covers, and geometrical properties of tubular neighborhoods. In the Euclidean space, we relate these concepts to the behavior of the surface area of the boundaries of parallel sets. In particular, we give characterizations for the Minkowski and the spherical dimensions by means of the Whitney ball count.
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