G. D. Anderson

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.

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.

Modular equations and distortion functions

Abstract 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.

Hypergeometric Functions and Hyperbolic Metric

We obtain new inequalities for certain hypergeometric functions. Using these inequalities, we deduce estimates for the hyperbolic metric and the induced distance function on a certain canonical hyperbolic plane domain.

Functional inequalities for the Teichmüller ring capacity

The geometric study of the action of quasiconformal mappings by means of conformal invariants often leads to inequalities for special functions. Examples of such special functions are the capacities of the Grötzsch and Teichmüller rings. Several new inequalities for these and related functions are given in the multidimensional case.

Chordal lipschitz conditions and quasiconformal maps in n-space

The authors solve an extremal problem on distortion of the chordal metric under a Mobius transformation and prove a distortion theorem for quasiconformal maps in n-space.