Michael Dorff

Convolutions of planar harmonic convex mappings

Ruscheweyh and Sheil-Small proved that convexity is preserved under the convolution of univalent analytic mappings in K. However, when we consider the convolution of univalent harmonic convex mappings in , this property does not hold. In fact, such convolutions may not be univalent. We establish some results concerning the convolution of univalent harmonic convex mappings provided that it is locally univalent. In particular, we show that the convolution of a right half-plane mapping in with either another right half-plane mapping or a vertical strip mapping in is convex in the direction of the real axis. Further, we give a condition under which the convolution of a vertical strip mapping in with itself will be convex in the direction of the real axis

Landau’s Theorem for Planar Harmonic Mappings

Landau gave a lower estimate for the radius of a schlicht disk centered at the origin and contained in the image of the unit disk under a bounded holomorphic function f normalized by f(0) = f′(0) − 1 = 1. Chen, Gauthier, and Hengartner established analogous versions for bounded harmonic functions. We improve upon their estimates.

Convolutions of harmonic convex mappings

The first author proved that the harmonic convolution of a normalized right half-plane mapping with either another normalized right half-plane mapping or a normalized vertical strip mapping is convex in the direction of the real axis, provided that it is locally univalent. In this article, we prove that in general the assumption of local univalency cannot be omitted. However, we are able to show that in some cases these harmonic convolutions are locally univalent. Using this we obtain interesting examples of univalent harmonic maps one of which is a map onto the plane with two parallel slits.

Harmonic univalent mappings onto asymmetric vertical strips

Let Ωα be the asymmetrical vertical strips defined by Ωα = {w : α−π 2 sin α < Re w < α 2 sin α}, where π/2 ≤ α < π, and let D be the unit disk. We characterize the class SH(D,Ωα) of univalent harmonic orientation-preserving functions f which map D onto Ωα and are normalized by f(0) = 0, fz(0) = 0, and fz(0) > 0. Then we use this characterization to demonstrate a few other results..

ON HARMONIC CONVOLUTIONS INVOLVING A VERTICAL STRIP MAPPING

Let f_\beta = h_\beta+\bar{g}_\beta and F_a = H_a +\bar{G}_a be harmonic mappings obtained by shearing of analytic mappings h_\beta +g_\beta = 1/(2i\sin\beta)log((1 + ze^{i\beta})/(1 + ze^{-i\beta})), 0<\beta<\pi and H_a+G_a = z/(1-z), respectively. Kumar et al. [5] conjectured that if \omega(z)=e^{i\theta}z^n (\theta\in R, n\in N) and \omega_a(z)=(a-z)/(1-az), a\in(-1,1) are dilatations of f_\beta and F_a, respectively, then F_a\ast f_\beta \in S_H^0 and is convex in the direction of the real axis provided a\in[(n-2)/(n + 2), 1).They claimed to have verified the result for n = 1, 2, 3 and 4 only. In the present paper, we settle the above conjecture in the affirmative for \beta =\pi/2 and for all n\in N.

Univalency of convolutions of harmonic mappings

We consider the convolution or Hadamard product of planar harmonic mappings that are the vertical shears of the canonical half-plane mapping ? ( z ) = z / ( 1 - z ) with respective dilatations - xz and - yz , where | x | = | y | = 1 . We prove that any such convolution is univalent. Furthermore, in the case that x = y = - 1 , we show the resulting convolution is convex.

Minimal graphs in R3 over convex domains

Krust established that all conjugate and associate surfaces of a minimal graph over a convex domain are also graphs. Using a convolution theorem from the theory of harmonic univalent mappings, we generalize Krusts theorem to include the family of convolution surfaces which are generated by taking the Hadamard product or convolution of mappings. Since this convolution involves convex univalent analytic mappings, this family of convolution surfaces is much larger than just the family of associated surfaces. Also, this generalization guarantees that all the resulting surfaces are over close-to-convex domains. In particular, all the associate surfaces and certain Goursat transformation surfaces of a minimal graph over a convex domain are over close-to-convex domains.

A Family of Minimal Surfaces and Univalent Planar Harmonic Mappings

We present a two-parameter family of minimal surfaces constructed by lifting a family of planar harmonic mappings. In the process, we use the Clunie and Sheil-Small shear construction for planar harmonic mappings convex in one direction. This family of minimal surfaces, through a continuous transformation, has connections with three well-known surfaces: Enneper’s surface, the wavy plane, and the helicoid. Moreover, the identification process used to recognize the surfaces provides a connection to surfaces that give tight bounds on curvature estimates first studied in a 1988 work by Hengartner and Schober.

Obtaining Funding and Support for Undergraduate Research.

Abstract Over the past decade there has been a dramatic increase in undergraduate research activities at colleges and universities nationwide. However, this comes at a time when budgets are being tightened and some institutions do not have the resources to pursue new initiatives. In this article we present some ideas for obtaining funding and support for building an undergraduate research program in mathematics.

Some harmonic n-slit mappings

The class SH consists of univalent, harmonic, and sense-preserving functions f in the unit disk, ∆, such that f = h+g where h(z) = z+ ∑∞ 2 akz k, g(z) = ∑∞ 1 bkz k. S H will denote the subclass with b1 = 0. We present a collection of n-slit mappings (n ≥ 2) and prove that the 2-slit mappings are in SH while for n ≥ 3 the mappings are in S H . Finally we show that these mappings establish the sharpness of a previous theorem by Clunie and SheilSmall while disproving a conjecture about the inner mapping radius.