Goodman's conjecture explained

Goodman's conjecture on the coefficients of multivalent functions was proposed in complex analysis in 1948 by Adolph Winkler Goodman, an American mathematician.

Formulation

Let

f(z) = \sum ^{inf ty} _{n =1} {b _{n} z ^{n}} be a p -valent function. The conjecture claims the following coefficients hold:$$|b\_n|\; \backslash le\; \backslash sum\_^\; \backslash frac|b\_k|$$

Partial results

It's known that when

p =2 , 3 , the conjecture is true for functions of the form P \circ \phi where P is a polynomial and \phi is univalent.

External sources

• 10.1090/S0002-9947-1948-0023910-X. On some determinants related to -valent functions . 1948 . Goodman . A. W. . Transactions of the American Mathematical Society . 63 . 175–192 . free .
• 10.1090/S0002-9939-1978-0460619-7. Goodman's conjecture and the coefficients of univalent functions . 1978 . Lyzzaik . Abdallah . Styer . David . Proceedings of the American Mathematical Society . 69 . 111–114 . free .
• Book: 10.1016/S1874-5709(02)80012-9 . Logarithmic Geometry, Exponentiation, and Coefficient Bounds in the Theory of Univalent Functions and Nonoverlapping Domains . [{{Google books|Wd7hKzGf9E8C|page=321|plainurl=yes}} Geometric Function Theory ]. Handbook of Complex Analysis . 2002 . Grinshpan . Arcadii Z. . 1 . 273–332 . 978-0-444-82845-3.
• AGrinshpan . A.Z.. On the Goodman conjecture and related functions of several complex variables . Department of Mathematics, University of South Florida, Tampa, FL . 1997 . 9 . 3 . 198–204. 1466800.
• 10.1090/S0002-9939-1995-1242085-7. On an identity related to multivalent functions . 1995 . Grinshpan . A. Z. . Proceedings of the American Mathematical Society . 123 . 4 . 1199 . free .