On the generalized Zalcman conjecture

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Abstract

Let $\mathcal{S}$ denote the class of analytic and univalent ({\it i.e.}, one-to-one) functions $ f(z)= z+\sum_{n=2}^{\infty}a_n z^n$ in the unit disk $\mathbb{D}=\{z\in \mathbb{C}:|z|<1\}$. For $f\in \mathcal{S}$, In 1999, Ma proposed the generalized Zalcman conjecture that $$|a_{n}a_{m}-a_{n+m-1}|\le (n-1)(m-1),\,\,\,\mbox{ for } n\ge2,\, m\ge 2,$$ with equality only for the Koebe function $k(z) = z/(1 - z)^2$ and its rotations. In the same paper, Ma \cite{Ma-1999} asked for what positive real

Results & Lemmas (2)

Each result is auto-extracted from the paper via OCR - the math is shown verbatim and may carry occasional transcription artifacts. Click to expand the full statement.

Theorem 2.1 · coeff Theorem 2.1. Let be an extremal function for the extremal problem and let be the image of |z| = 1 under 1/f(z). If and then lies either in…

Theorem 2.1. Let $f \in \mathcal{S}$ be an extremal function for the extremal problem $|2a_2a_3 - a_4|$ and let $\Gamma$ be the image of |z| = 1 under 1/f(z). If $Re \, a_2 > 0$ and $Im \, a_2 \neq 0$ then $\Gamma$ lies either in upper or lower half plane.

Theorem 2.2 · coeff Theorem 2.2. Let be given by then (2.1), with equality only for functions of the form, where is real. For the proof of Theorem 2.2, we…

Theorem 2.2. Let $$f \in S$$ be given by $f(z) = z + \sum_{n=2}^{\infty} a_n z^n$ then (2.1) $|3a_2a_3 - a_4| < 14$ , with equality only for functions of the form $$\frac{z}{(1-e^{i\theta}z)^2}$$ , where $\theta$ is real. For the proof of Theorem 2.2, we follow the technique of Ozawa [20].

Function classes studied:

Coefficient bounds & claims (3)

Machine-extracted from the paper text - useful for cross-referencing, not a verified fact.

coefficient_bound

|3*a_2*a_3 - a_4| ≤ 14 for class S (sharp) [Theorem 2.2]

function_family

Class S: class of analytic univalent functions f(z) = z + sum a_n z^n in the unit disk

function_family

Class S*: starlike functions: Re(zf'(z)/f(z)) > 0

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Results & Lemmas (2)

Coefficient bounds & claims (3)

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