In this paper, we consider a family of functions that are meromorphic and univalent in the unit disk Δ = {z ∈ C: ∣z∣ < 1} and that have some rather striking geometric properties. These functions all have the form μ(z)=1/z+Σ from k=1 to n of a_kz~k with ∣a_n∣=1. It is well known that if μ given above is univalent in the disk then 0≠μ'(z)=-1/(z~2)+Σ from k=1 to n of ka_kz~(k-1)=-1/(z~2)(1-Σ from k=1 to n of ka_kz~(k+1)). Therefore, ∣ na_n ∣ ≤ 1 and equality is possible only if all zeros of - z~2μ'(z) lie on the circle {∣z∣ = 1}. In that case, a_(n-1) = 0 and (k - 1)a_(k-1) = -na_n(n - k)a_(n-k) [1, p. 166; 2, p. 10]. We will call a function given by a meromorphic polynomial of degree n.
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