Sharp bounds for the valence of certain logharmonic polynomials
Abstract
Consider a logharmonic polynomial; that is, a product of the form p ( z ) q ( z ) ¯ p(z)\overline {q(z)} , where p p , q q are holomorphic polynomials. Assume q q is linear and denote by n n the degree of p p . It was recently shown in Khavinson, Lundberg, and Perry [ On the valence of logharmonic polynomials , Recent progress in function theory and operator theory, Amer. Math. Soc., Providence, RI, 2024] that the valence of such a logharmonic polynomial is at most 3 n − 1 3n-1 ; in this paper we show that their 3 n − 1 3n-1 upper bound is sharp. Together with the work of Khavinson et al., this resolves a conjecture in Bshouty and Hengartner [Ann. Univ. Mariae Curie-Sklodowska Sect. A 54 (2000), pp. 21–25].
Fields of Study & Subjects
Analytic and geometric function theory
Meromorphic and Entire Functions
Mathematical functions and polynomials
Publication details
- Journal / source
- Proceedings of the American Mathematical Society
- Volume
- 154
- Issue
- 8
- Pages
- 3415–3426
- Publication date
- 5 March 2026
- Language
- EN