Abstract
Brannan showed that a normalized univalent polynomial of the form $P(z)=z+a_2 z^2+\ldots + a_{n-1}z^{n-1}+\frac{z^n}{n}$ is starlike if and only if $a_2=\ldots=a_{n-1}=0$. We give a new and simple proof of his result, showing further that it is also equivalent to the membership of $P$ in the Noshiro-Warschawski class of univalent functions whose derivative has positive real part in the disk. Both proofs are based on the Fejér lemma for trigonometric polynomials with positive real part.
Results & Lemmas (2)
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PROPOSITION 1.
PROPOSITION 1. If n ≥2 and the polynomial Q(z) = 1+c1z +c2z2 +...+cn−1zn−1 + zn has positive real part in D then c1 = c2 =... = cn−1 = 0.
PROPOSITION 1. If n ≥2 and the polynomial Q(z) = 1+c1z +c2z2 +...+cn−1zn−1 + zn has positive real part in D then c1 = c2 = ... = cn−1 = 0.
THEOREM 2.
THEOREM 2. Let P(z) = z + a2z2 +...+ an−1zn−1 + anzn be a polynomial in the class S with all critical points on the unit circle (that is,…
THEOREM 2. Let P(z) = z + a2z2 +...+ an−1zn−1 + anzn be a polynomial in the class S with all critical points on the unit circle (that is, |an| = 1 n). Then the following statements are equivalent: (a) P′ has positive real part in D. (b) a2 = ... = an−1 = 0. (c) P is starlike.
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