12 research outputs found
Generalized Convexity and Inequalities
Let R+ = (0,infinity) 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 in M, we say that a function f : R+ to R+ is
(m1,m2)-convex if f(m1(x,y)) < or = m2(f(x),f(y)) for all x, y in 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.Comment: 17 page
Generalized Elliptic Integrals and the Legendre M-function
We study monotonicity and convexity properties of functions arising in the
theory of elliptic integrals, and in particular in the case of a
Schwarz-Christoffel conformal mapping from a half-plane to a trapezoid. We
obtain sharp monotonicity and convexity results for combinations of these
functions, as well as functional inequalities and a linearization property.Comment: 28 page
Modular Equations and Distortion Functions
Modular equations occur in number theory, but it is less known that such
equations also occur in the study of deformation properties of quasiconformal
mappings. The authors study two important plane quasiconformal distortion
functions, obtaining monotonicity and convexity properties, and finding sharp
bounds for them. Applications are provided that relate to the quasiconformal
Schwarz Lemma and to Schottky's Theorem. These results also yield new bounds
for singular values of complete elliptic integrals.Comment: 23 page
Inequalities for Means
AbstractA monotone form of L′Hospital′s rule is obtained and applied to derive inequalities between the arithmetic-geometric mean of Gauss, the logarithmic mean, and Stolarsky′s identric mean. Some related inequalities are given for complete elliptic integrals
Bounds for Quasiconformal Distortion Functions
AbstractSeveral new inequalities are proved for the distortion function Ď•K(r) appearing in the quasiconformal Schwarz lemma. Other related special functions are studied and applications are given to quasiconformal maps in the plane. Some open problems are solved, too