M. K. Vamanamurthy

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.

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.

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):

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.

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.

On normal subspaces

In this paper the anti-normal property is studied. A space is anti-normal if its only normal subspaces are those whose cardinalities require them to be normal. It is shown that every topological space of at least four elements contains a normal three point subspace from which it follows that there is only one non-trivial anti-normal space.

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.

Cardinality of discrete subsets of a topological space: Corrigendum

Theorem 1 and Corollary 1 of the original paper [ Bull. Austral. Math. Soc. 25 (1982), 99–101] are false unless, for example, we assume the Generalised Continuum Hypothesis. The proof of Theorem 1 correctly shows that exp(| c |) ≤ exp(| D |) but one cannot then deduce that the cardinality of C is at most that of D . All one can deduce is that the cardinality of C is less than exp(| D |).