On the generalized Zalcman functional for some classes of univalent functions

Theorem 1. Let $f(z) = z + a_2 z^2 + a_3 z^3 + \ldots$ , $f \in S$ , and let $\alpha$ be its Hayman index. Then (1) $$\lim_{n \to \infty} \frac{|a_n^2 - a_{2n-1}|}{(n-1)^2} = \alpha^2.$$ Also, if $B_n = \sup_{f \in S} |a_n^2 - a_{2n-1}|$ then $$\lim_{n \to \infty} \frac{B_n}{(n-1)^2} = 1.$$

Corollary 2. If $f \in S$ is not a rotation of the Koebe function, then for every $\delta \in (0, 1 - \alpha^2)$ there exist $n_0 \in \mathbb{N}$ such that $$|a_n^2 - a_{2n-1}| \le (1 - \delta)(n - 1)^2$$ , for all $n \geq n_0$ . Estimates for the closed convex hull of convex functions. Denote by C the class of convex functions in S. A typical example is the half-plane function $\ell(z) = \frac{z}{1-z}$ . It is well known that the coefficient estimate can be improved a great deal for the functions in C: by a theorem of Loewner, they must satisfy $|a_n| \leq 1$ , with equality only for the function $\ell$ and its rotations (see [4, Corollary on p. 45]. It would, thus, be natural to expect a similar improvement for convex functions in the Zalcman conjecture and this is indeed the case. Denote by co(C) the convex hull of C and by $\overline{co(C)}$ its closure in the topology of uniform convergence on compact subsets of $\mathbb{D}$ . Note that this larger class no longer consists only of univalent functions. By a rotation of a function f in S, as is usual [4, Chapter 2], we mean the function $f_c(z) = \overline{c}f(cz)$ , |c| = 1, which is again in S. The estimate below is known when $\lambda = 0$ so there is no need to include that case in the result.

Theorem 3. Let $0 < \lambda \le 2$ . If $f \in \overline{co(C)}$ , then $|\lambda a_n^2 - a_{2n-1}| \le 1$ for all $n \ge 2$ . For any fixed n and $\lambda < 2$ , equality holds only for the functions of the following form (and for their rotations): $$f(z) = \sum_{k=1}^{2n-2} m_k \frac{z}{1 - e^{i\theta_k} z},$$ where $0 \le m_k \le 1$ , $\theta_k = \frac{(2k+1)\pi}{2n-2}$ , and $$\sum_{k=1}^{n-1} m_{2k} = \sum_{k=1}^{n-1} m_{2k-1} = 1/2.$$

Theorem 4. If $0 < \lambda \le 4/3$ and $f \in \mathcal{R}$ , then for all $n \ge 2$ we have $$|\lambda a_n^2 - a_{2n-1}| \le \frac{2}{2n-1}$$ . For $\lambda < 4/3$ and any fixed $n \geq 2$ , equality holds only for the functions of the following form (and for their rotations): $$f(z) = \sum_{k=1}^{2n-2} m_k \left( 2e^{-i\theta_k} \log \frac{1}{1 - e^{i\theta_k} z} - z \right) = \sum_{k=1}^{2n-2} 2m_k e^{-i\theta_k} \log \frac{1}{1 - e^{i\theta_k} z} - z ,$$ where $$0 \le m_k \le 1$$ , $\theta_k = \frac{(2k+1)\pi}{2n-2}$ , $\sum_{k=1}^{n-1} m_{2k} = \sum_{k=1}^{n-1} m_{2k-1} = 1/2$ .

Lemma 5. Let $\lambda > 0$ , $n \geq 2$ , and consider the triangle $$\Delta = \{(u, v) \in \mathbb{R}^2 : u \ge 0, v \ge 0, nu + (2n - 1)v \le 1\}$$ in the uv-plane. Then $$\max_{(u,v)\in\Delta} (\lambda u^2 + v) = \max\left\{\frac{\lambda}{n^2}, \frac{1}{2n-1}\right\},\,$$ and equality can hold only at the points $(u,v)=(0,\frac{1}{2n-1})$ and $(u,v)=(\frac{1}{n},0)$ .

Theorem 6. If $\lambda > 0$ and $f \in \mathcal{H}$ , then for each $n \geq 2$ we have (2) $$|\lambda a_n^2 - a_{2n-1}| \le \max\left\{\frac{\lambda}{n^2}, \frac{1}{2n-1}\right\}.$$ Equality holds if and only if $$f(z) = \begin{cases} z + \frac{\alpha}{2n-1} z^{2n-1}, & \text{for } \lambda \le \frac{n^2}{2n-1} \\ z + \frac{\alpha}{n} z^n, & \text{for } \lambda \ge \frac{n^2}{2n-1}, \end{cases}$$ where $\alpha$ is a complex number such that $|\alpha| = 1$ . Note that the rotations are already included in the form of extremal functions. Also, two different types of extremal functions exist in the case $\lambda = \frac{n^2}{2n-1}$ .