Harmonic mappings in the plane are univalent complex-valued harmonic functions of a complex variable. Conformal mappings are a special case where the real and imaginary parts are conjugate harmonic functions, satisfying the Cauchy-Riemann equations. Harmonic mappings were studied classically by differential geometers because they provide isothermal (or conformal) parameters for minimal surfaces. More recently they have been actively investigated by complex analysts as generalizations of univalent analytic functions, or conformal mappings. Many classical results of geometric function theory extend to harmonic mappings, but basic questions remain unresolved. This book is the first comprehensive account of the theory of planar harmonic mappings, treating both the generalizations of univalent analytic functions and the connections with minimal surfaces.  Essentially self-contained, the book contains background material in complex analysis and a full development of the classical theory of minimal surfaces, including the Weierstrass-Enneper representation. It is designed to introduce non-specialists to a beautiful area of complex analysis and geometry.

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Harmonic Mappings in the Plane

Recent advances in the field of iterative methods for solving large linear systems are reviewed. The main focus is on developments in the area of conjugate gradient-type algorithms and Krylov subspace methods for non-Hermitian matrices.

https://www.cambridge.org/core/services/aop-cambridge-core/content/view/S0962492900002245

Iterative solution of linear systems

Univalent and starlike generalized hypergeometric functions

Diagnosis, Management, and Evaluation of Chemotherapy-Induced Peripheral Neuropathy

The Convolution Transform. By Hirschmann & Widder . Pp. x + 268. 45s. 1955. (Princeton University Press. London : Cumberlege)

Convolutions in Geometric Function Theory

On Two Extremal Problems Related to Univalent Functions

Linear operators between classes of prestarlike functions

Let K(Q be the set of univalent functions in the unit disk D which are convex in the direction eip . We determine the set of analytic functions g in D which preserve K(9) under the Hadamard product, i.e., g * f e K(p) whenever f e X(d. This result contains as a special case the proof of a conjecture of Goodman and Saff about l((rp) and solves partially a multiplier problem concerning convex univalent harmonic functions in D, posed by Clunie and Sheil-Small.

/pdf/on-the-preservation-of-direction-convexity-and-the-goodman-3tbto9wqws.pdf

On the preservation of direction-convexity and the Goodman-Saff conjecture

CMFT is an international mathematics journal which publishes carefully selected original research papers in complex analysis (in a broad sense), and on applications or computational methods related to complex analysis. Survey articles of high standard and current interest can be considered for publication as well. Contributed papers should be written in English (exceptions in rare cases are tolerated), and in a lucid, expository style. Papers should not exceed 30 printed pages.

https://www.springer.com/cda/content/document/productFlyer/productFlyer_40315.pdf?SGWID=0-0-1297-174535456-0

Stephan Ruscheweyh

Papers

Convolutions in Geometric Function Theory

On Two Extremal Problems Related to Univalent Functions

Linear operators between classes of prestarlike functions

On the preservation of direction-convexity and the Goodman-Saff conjecture

Computational Methods and Function Theory