Maxwell O. Reade

On a Briot-Bouquet Differential Subordination

Let p(z) be analytic in the unit disc Δ, let h(z) be convex (univalent) in Δ, and let β and γ be complex numbers. The authors show that if p(z) = zp′(z)(βp(z) + γ)−1 ≺ h(z) (where ≺ denotes subordination), then p(z) ≺ h(z). They prove, further, that if, in addition, the differential equation q(z) + zq′(z)(βq(z) + γ)−1 = h(z) has a univalent solution q(z), then the sharp subordination p(z) ≺ q(z) holds. Applications of these results in the field of univalent functions are given.

Mean-values and harmonic polynomials

lying in D. Conversely, if f(x, y) is superficially summable in the interior of a finite domain D, and if (1) holds for each point (x0, yo) and each discD(#o, yo;r) about (x0, yo) in D, then/(x, y) is harmonic in D(l). It follows that (1) may be taken as the defining equation for harmonic functions. 0.2. Similarly, if/(x, y) is superficially summable in the interior of a finite simply-connected domain D, and if/(x, y) is summable on each circle

On the starlikeness and convexity of a class of analytic functions

We extend some results concerning the univalence of rational functions recently obtained by Mitrinovi<c. Sufficient conditions for starlikeness and convexity are also obtained.

The radius of -convexity for the class of starlike univalent functions, -real

We use a result due to Gutljanski; to obtain the radius of a-convexity for the class S* of starlike univalent functions for real a.

On the Univalence of Functions Defined by Certain Integral Transforms

Abstract The integral transform F(z) = ∝ 0 z (f′(t)) α ( g(t) t ) β dt , where α and β are real, of pairs of special analytic functions f ( z ) = z + ···, g ( z ) = z + ···, univalent in the open unit disc Δ is studied. The transform and our results extend some recent results due to Shirakova.

The koebe domain of the classes N1(a) and N2(a) of nevanlinna analytic functions

In this paper we find the Koebe domain of analytic functions having the form (1) and (2). We also find the domain of the values of all such functions.

Extremal problems for certain analytic (Nevanlinna) functions

AbstractWe obtain sharp bounds on some basic functionals defined on the sets of all analytic functions having the representations

The Hardy classes for functions in the class MV[α, k]

f\left( z \right) \equiv \int\limits_{ - 1}^1 {\frac{{d\mu \left( t \right)}}{{z - t}}}