Matti Vuorinen

INEQUALITIES FOR ZERO-BALANCED HYPERGEOMETRIC FUNCTIONS

The authors study certain monotoneity and convexity properties of the Gaussian hypergeometric function and those of the Euler gamma function.

Generalized convexity and inequalities

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.

ON CONFORMAL DILATATION IN SPACE

We study the conformality problems associated with quasiregular mappings in space. Our approach is based on the concept of the infinitesimal space and some new Grotzsch-Teichmuller type modulus estimates that are expressed in terms of the mean value of the dilatation coefficients.

On the degenerate Beltrami equation

We study the well-known Beltrami equation under the assumption that its measurable complex-valued coefficient μ(z) has the norm ∥μ∥∞ = 1. Sufficient conditions for the existence of a homeomorphic solution to the Beltrami equation on the Riemann sphere are given in terms of the directional dilatation coefficients of μ. A uniqueness theorem is also proved when the singular set Sing(μ) of μ is contained in a totally disconnected compact set with an additional thinness condition on Sing(μ).

On Jordan type inequalities for hyperbolic functions

This paper deals with some inequalities for trigonometric and hyperbolic functions such as the Jordan inequality and its generalizations. In particular, lower and upper bounds for functions such as and are proved.

Generalized Elliptic Integrals

Jacobi’s elliptic integrals and elliptic functions arise naturally from the Schwarz-Christoffel conformal transformation of the upper half plane onto a rectangle. In this paper we study generalized elliptic integrals which arise from the analogous mapping of the upper half plane onto a quadrilateral and obtain sharp monotonicity and convexity properties for certain combinations of these integrals, thus generalizing analogous well-known results for classical conformal capacity and quasiconformal distortion functions. An algorithm for the computation of the modulus of the quadrilateral is given.

On harmonic quasiconformal quasi-isometries

The purpose of this paper is to explore conditions which guarantee Lipschitz-continuity of harmonic maps with respect to quasihyperbolic metrics. For instance, we prove that harmonic quasiconformal maps are Lipschitz with respect to quasihyperbolic metrics.

Monotonicity Rules in Calculus

1. RULES FOR MONOTONICITY. In the first semester of calculus a student learns that if a function f is continuous on an interval [a, b] and has a positive (negative) derivative on (a, b), then f is increasing (decreasing) on [a, b]. This result is obtained easily by means of the Lagrange mean value theorem. The functions that the student proves monotone in this way are usually polynomials, rational functions, or other elementary functions. If one is attempting to establish the monotonicity of a quotient of two functions, one often finds that the derivative of the quotient is quite messy and the process tedious. Several authors have developed refinements of this method for proving monotonicity of quotients. The first such refinement of which we are aware is the following one by M. Gromov [11, p. 42], which appears in his work in differential geometry (Gromov’s proof uses only monotonicity and elementary properties of integrals):

Topics in Special Functions III

The authors provide a survey of recent results in special functions of classical analysis and geometric function theory, in particular, the circular and hyperbolic functions, the gamma function, the elliptic integrals, the Gaussian hypergeometric function, power series, and mean values.

Distortion functions for plane quasiconformal mappings

The authors study two well-known distortion functions, λ(K) andϕK(r), of the theory of plane quasiconformal mappings and obtain several new inequalities for them. The proofs make use of some properties of elliptic integrals.