Abstract
AbstractThe set of points where an entire function achieves its maximum modulus is known as the maximum modulus set. In 1951, Hayman studied the structure of this set near the origin. Following work of Blumenthal, he showed that, near zero, the maximum modulus set consists of a collection of disjoint analytic curves, and provided an upper bound for the number of these curves. In this paper, we establish the exact number of these curves for all entire functions, except for a “small” set whose Taylor series coefficients satisfy a certain simple, algebraic condition. Moreover, we give new results concerning the structure of this set near the origin, and make an interesting conjecture regarding the most general case. We prove this conjecture for polynomials of degree less than four.
Publisher
Springer Science and Business Media LLC
Subject
Applied Mathematics,Computational Theory and Mathematics,Analysis
Reference11 articles.
1. Blumenthal, O.: Sur le mode de croissance des fonctions entières. Bull. Soc. Math. France 35, 213–232 (1907)
2. Csordas, G., Ortel, M., Smith, W.: The maximum modulus function of a polynomial. Complex Var. Theory Appl. 15(2), 107–114 (1990)
3. Hardy, G.H.: The maximum modulus of an integral function. Q. J. Math. 41, 1–9 (1909)
4. Hayman, W.K.: A characterization of the maximum modulus of functions regular at the origin. J. Anal. Math. 1, 135–154 (1951)
5. Hayman, W.K., Tyler, T.F., White, D.J.: The Blumenthal conjecture. In Complex analysis and dynamical systems V, vol. 591 of Contemp. Math., pp. 149–157. Amer. Math. Soc., Providence, RI (2013)
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