A Note on the Maximum Number of Zeros of r(z) - (z)over-bar

作者：Luce Robert^*; Sete Olivier; Liesen Joerg

来源：Computational Methods and Function Theory, 2015, 15(3): 439-448.

DOI：10.1007/s40315-015-0110-6

• Complex valued harmonic function
• Rational function
• Zeros of rational harmonic functions

An important theorem of Khavinson and Neumann (Proc. Am. Math. Soc. 134: 1077-1085, 2006) states that the complex harmonic function r(z)-(z) over bar, where r is a rational function of degree n >= 2, has at most 5(n-1) zeros. In this note, we resolve a slight inaccuracy in their proof and in addition we show that for certain functions of the form r (z)-(z) over bar no more than 5(n-1)-1 zeros can occur. Moreover, we show that r (z)-(z) over bar is regular, if it has the maximal number of zeros.

• 出版日期2015-9