arXiv subjects

Michael Dorff

Publications and source records attributed to Michael Dorff.

7 recordsLinked to original sources

arXiv2013

On harmonic convolutions involving a vertical strip mapping

Let f_β= h_β+\bar{g}_βand F_a = H_a +\bar{G}_a be harmonic mappings obtained by shearing of analytic mappings h_β+g_β= 1/(2i\sinβ)log((1 + ze^{iβ})/(1 + ze^{-iβ})), 0<β<πand H_a+G_a = z/(1-z), respectively. Kumar et al. [5] conjectured that if ω(z)=e^{iθ}z^n (θ\in R, n\in N) and ω_a(z)=(a-z)/(1-az), a\in(-1,1) are dilatations of f_βand F_a, respectively, then F_a\ast f_β\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 β=π/2 and for all n\in N.

math.CV↗

arXiv2013

An application of Cohn's rule to convolutions of univalent harmonic mappings

Dorff et al. [4], proved that the harmonic convolution of right half-plane mapping with dilatation -z and mapping f_β= h_β+ \bar{g}_β, where f_βis obtained by shearing of analytic vertical strip mapping, with dilatation e^{iθ}z^n; n = 1,2,θ\in R, is in S_H^0 and is convex in the direction of the real axis. In this paper, by using Cohn's rule, we generalize this result by considering dilatations (a-z)/(1-az), a\in (-1,1) and e^{iθ} z^n (n\in N;θ\in R) of right half-plane mapping and f_β, respectively.

math.CV↗

arXiv2013

Convolution properties of some harmonic mappings in the right-half plane

Dorff, proved in [2] that the convolution of two harmonic right-half plane mappings is convex in the direction of real axis provided that the convolution is locally univalent and sense preserving. Later, it was shown in [3] that the condition of locally univalent and sense preserving can be dropped in some special cases. In this paper, we generalize the main result from [3].

math.CV↗

arXiv2009

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 paper, 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.

math.CV↗

arXiv2009

Typically Real Harmonic Functions

We consider a class $\THO$ of typically real harmonic functions on the unit disk that contains the class of normalized analytic and typically real functions. We also obtain some partial results about the region of univalence for this class.

math.CV↗

arXiv2007

Derivative relationships between volume and surface area of compact regions in R^d

We explore the idea that the derivative of the volume, V, of a region in R^d with respect to r equals its surface area, A, where r = d V/A. We show that the families of regions for which this formula for r is valid, which we call homogeneous families, include all the families of similar regions. We determine equivalent conditions for a family to be homogeneous, provide examples of homogeneous families made up of non-similar regions, and offer a geometric interpretation of r in a few cases.

math.MG↗

arXiv2006

Minimal Surface Linear Combinatoin Theorem

Given two univalent harmonic mappings $f_1$ and $f_2$ on $\mathbb{D}$, which lift to minimal surfaces via the Weierstrass-Enneper representation theorem, we give necessary and sufficient conditions for $f_3=(1-s)f_1+sf_2$ to lift to a minimal surface for $s\in[0,1]$. We then construct such mappings from Enneper's surface to Scherk's singularly periodic surface, Sckerk's doubly periodic surface to the catenoid, and the 4-Enneper surface to the 4-noid.

math.DG↗