## Numerical solution of the omitted area problem of univalent function theory.(English)Zbl 1007.30009

Summary: The omitted area problem was posed by Goodman in 1949: what is the maximum area $${\mathcal A}^*$$ of the unit disk $$\mathbb{D}$$ that can be omitted by the image of the unit disk under a univalent function normalized by $$f(0)= 0$$ and $$f'(0)= 1$$? The previous best bounds were $$0.240005\pi<{\mathcal A}^*\leq .31\pi$$. Here the problem is addressed numerically and it is found that these estimates are slightly in error. To ten digits, the correct value appears to be $${\mathcal A}^*= 0.2385813248\pi$$.

### MSC:

 30C30 Schwarz-Christoffel-type mappings 30C75 Extremal problems for conformal and quasiconformal mappings, other methods 65E05 General theory of numerical methods in complex analysis (potential theory, etc.)

### Software:

Matlab; SC Toolbox; Schwarz-Christoffel; fminsearch
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### References:

 [1] L. V. Ahlfors, Complex Analysis: An Introduction to the Theory of Analytic Functions of One Complex Variable, Cambridge University Press, New York, 1997. [2] R. W. Barnard, The omitted area problem for univalent functions, Contemp. Math. 38 (1985), 53–60. · Zbl 0562.30011 [3] .L. Lewis, On the omitted area problem, Michigan Math. J. 34 1 (1987), 13–22. · Zbl 0616.30017 [4] R. W. Barnard and K. Pearce, Rounding corners of gearlike domains and the omitted area problem, J. Comput. Appl. Math. 14 1–2 (1986), 217–226. · Zbl 0587.30002 [5] L. de Branges, A proof of the Bieberbach conjecture, Acta Math. 154 (1985), 137–152. · Zbl 0573.30014 [6] T. A. Driscoll, Algorithm 765: A Matlab toolbox for Schwarz-Christoffel mapping, ACM Trans. Math. Softw. 22 (1996), 168–186; Software available at http://www.math.udel.edu . · Zbl 0884.30005 [7] T. F. Coleman and Y. Li, An interior, trust region approach for nonlinear minimization subject to bounds, SIAM J. Opt. 6 (1996), 418–445. · Zbl 0855.65063 [8] A. W. Goodman, Note on regions omitted by univalent functions, Bull. Amer. Math. Soc. 55 (1949), 363–369. · Zbl 0033.17603 [9] A. W. Goodman and E. Reich, On regions omitted by univalent functions II, Canad. J. Math. 7 (1955), 83–88. · Zbl 0064.07404 [10] W. K. Hayman, Multivalent Functions, Cambridge University Press, Cambridge, 1994. [11] P. Henrici, Applied and Computational Complex Analysis, vol. III, Wiley, New York, 1986. · Zbl 0578.30001 [12] J. A. Jenkins, On values omitted by univalent functions, Amer. J. Math. 75 (1953), 406–408. · Zbl 0050.08302 [13] J. C. Lagarias, J. A. Reeds, M. H. Wright, and P. E. Wright, P.E., Convergence properties of the Nelder-Mead simplex method in low dimensions, SIAM J. Opt. 9 (1998), 112–147. · Zbl 1005.90056 [14] J. L. Lewis, On the minimum area problem, Indiana University Mathematics Journal 34 3 (1985), 631–661. · Zbl 0579.30007 [15] A. I. Markushevich, Theory of Functions of a Complex Variable, Chelsea Publishing Co., New York, 1985, vol. III. [16] L. N. Trefethen, Spectral Methods in Matlab, SIAM, 2000. · Zbl 0953.68643
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