Zalcman Conjecture for certain analytic and univalent functions

Lemma 1.2. [8, Lemma 2.8, p. 153] A set $\mathcal{F} \subseteq \mathcal{H}$ is locally bounded if, and only if, for each compact set $K \subset \mathbb{D}$ there is a constant M such that $$|f(z)| \le M$$ for all $f \in \mathcal{F}$ and $z \in K$ .

Theorem 1.3. [8, Montel's Theorem 2.9, p. 153] A family $\mathcal{F} \subseteq \mathcal{H}$ is normal if, and only if, is locally bounded <span id="page-1-0"></span>Corollary 1.4. [8, Corollary 2.10, p. 154] A set $\mathcal{F} \subseteq \mathcal{H}$ is compact if, and only if, it is closed and locally bounded.

Theorem 1.5. [10, Theorem 2.6 (Growth Theorem)] For each $f \in \mathcal{S}$ , $$\frac{r}{(1+r)^2} \le |f(z)| \le \frac{r}{(1-r)^2}, \quad |z| = r < 1.$$ For each $z \in \mathbb{D}$ , $z \neq 0$ , equality occurs if, and only if, f is a suitable rotation of Koebe function. For $f, g \in \mathcal{H}$ , we say that f is subordinate to g, written as $f \prec g$ or $f(z) \prec g(z)$ , if there exists an analytic function $\omega : \mathbb{D} \to \mathbb{D}$ with $\omega(0) = 0$ such that $f(z) = g(\omega(z))$ for $z \in \mathbb{D}$ . Furthermore, if g is univalent in $\mathbb{D}$ then $f \prec g$ if, and only if, f(0) = g(0) and $f(\mathbb{D}) \subseteq g(\mathbb{D})$ . If $\mathscr{G} \subseteq \mathcal{H}$ , we use the notation $s(\mathscr{G}) = \{f : f \prec g \text{ for some } g \in \mathscr{G}\}$ . If $\mathscr{G}$ is a compact subset of $\mathscr{H}$ then it is not difficult to show that $s(\mathscr{G})$ is compact subset of $\mathscr{H}$ (for instance, see [14, Lemma 5.19]). Suppose X is a linear topological vector space and $V \subseteq X$ . A point $x \in V$ is called an extreme point of V if it has no representation of the form x = ty + (1-t)z, 0 < t < 1 as a proper convex combination of two distinct points $y, z \in V$ . We denote EV the set of extreme points of V. The convex hull of a set $V \subseteq X$ is the smallest convex set containing V. The closed convex hull denoted by $\overline{co}V$ is defined as the intersection of all closed convex sets containing V. That is, the closed convex hull of V is the smallest closed convex set containing V, which is the closure of the convex hull of V. The Krein-Milman Theorem asserts that every compact subset of a locally convex topological space is contained in the closed convex hull of its extreme points (see, for instance, [9]). For a general reference and for many important results on this topic, we refer to [14]. As a first step for application of the knowledge of extreme point of these classes Brickman et al. [6] pointed out the following general results. Theorem A. Let $\mathscr{G}$ be a compact subset of $\mathcal{H}$ and J be a complex-valued continuous linear functional on $\mathcal{H}$ . Then $\max\{\operatorname{Re} J(f): f \in \overline{co}\mathscr{G}\} = \max\{\operatorname{Re} J(f): f \in \mathscr{G}\} = \max\{\operatorname{Re} J(f): f \in \overline{Eco}\mathscr{G}\}.$

Lemma 1.8. [15] $$\frac{1}{(1-z)^{\alpha+i\beta}} \in \mathcal{R}$$ if, and only if, $\alpha \geq 1$ and $\beta = 0$ . Let P denote the class of all analytic functions p in D with p(0) = 1 satisfying Re p(z) > 0 in D. Functions in the class P are called the Caratheodory ´ functions and can be expressed as (1.9) $$p(z) = 1 + \sum_{n=1}^{\infty} c_n z^n.$$ <span id="page-4-1"></span>Lemma 1.10. [\[14\]](#page-13-5) p ∈ P if, and only if, there is a probability measure µ on ∂D such that <span id="page-4-3"></span><span id="page-4-2"></span> $$p(z) = \int_{|x|=1} \frac{1+xz}{1-xz} d\mu(x).$$ Equivalently, in view of the Lemma 1.[10](#page-4-1), for p ∈ P given by (1.[9\)](#page-4-2) can be written as (1.11) $$p(z) = 1 + \sum_{n=1}^{\infty} c_n z^n = \int_0^{2\pi} \frac{1 + e^{it}z}{1 - e^{it}z} d\nu(t).$$ On comparing both the sides of (1.[11\)](#page-4-3) we obtain (1.12) $$c_n = 2 \int_0^{2\pi} e^{int} d\nu(t).$$ <span id="page-4-8"></span>Lemma 1.13. [26, Lemma 2.3, p. 507] If ^p(z) = 1 + ^P^∞ k=1 ckz ^k ∈ P, then for all n, m ∈ N, <span id="page-4-7"></span> $$|\lambda c_n c_m - c_{n+m}| \le \begin{cases} 2, & 0 \le \lambda \le 1\\ 2|2\lambda - 1|, & elsewhere \end{cases}$$ If 0 < λ < 1, the inequality is sharp for the function p(z) = (1 + z ⁿ+^m)/(1 − z ⁿ+^m). In other cases, the inequality is sharp for the function p(z) = (1 + z)/(1 − z). 2. Compactness of the set U(λ) <span id="page-4-6"></span><span id="page-4-0"></span>Theorem 2.1. For 0 < λ ≤ 1, the class U(λ) is compact.

Theorem 3.1. $\overline{co}\mathcal{U}$ consists of all functions represented by $$f(z) = \int_{|x|=1} \frac{z}{(1-xz)^2} d\mu(x),$$ where $\mu \in \wedge$ . Here $\wedge$ denotes the set of probability measure on $\partial \mathbb{D}$ . Further, $E\overline{co}\mathcal{U}$ consists functions of the form $$f(z) = \frac{z}{(1-xz)^2}, \quad |x| = 1.$$

Theorem 3.4. Let $f \in \mathcal{U}$ be given by (1.1). Then $|a_n^2 - a_{2n-1}| \leq (n-1)^2$ for $n \geq 2$ . This inequality is sharp with equality for the Koebe function and its rotations i.e., functions of the form $f(z) = z/(1-xz)^2$ where |x| = 1.

Theorem 4.2. Let $f \in \overline{co} \mathcal{U}$ be given by (1.1). Then for $n, m \geq 2$ $$|a_n a_m - a_{n+m-1}| \le \begin{cases} n+m-1, & \text{if } (n,m) \text{ is } (2,n), (m,2), (3,3), (3,4), (4,3) \\ (n-1)(m-1), & \text{otherwise.} \end{cases}$$ The second inequality is sharp and the equality holds for the Koebe function and its rotations.

Theorem 5.1. Let $f \in \mathcal{F}$ given by (1.1). Then - (i) $|a_2^2 a_3| \le 1$ (ii) $|a_2 a_3 a_4| < 2$ . These inequalities are sharp with equality for the Koebe function $k(z) = z/(1-z)^2$ and its rotations.

Definition 1.1. A set $\mathcal{F} \subseteq \mathcal{H}$ is normal if each sequence $\{f_n\}$ in $\mathcal{F}$ has a subsequence $\{f_{n_k}\}$ which converges to a function $f \in \mathcal{H}$ uniformly on every compact subset of $\mathbb{D}$ .

Definition 1.2. A set $\mathcal{F} \subseteq \mathcal{H}$ is locally bounded if for each point $a \in \mathbb{D}$ there are constants M and r > 0 such that for all $f \in \mathcal{H}$ , $$|f(z)| \le M$$ for $|z - a| < r$ . That is, $\mathcal{F}$ is locally bounded if, about each point $a \in \mathbb{D}$ there is a disk on which $\mathcal{F}$ is uniformly bounded.

Definition 1.3. If $\mathcal{F}$ is a convex subset of $\mathcal{H}$ and $J: \mathcal{H} \to \mathbb{R}$ then J is called convex on $\mathcal{F}$ provided that $J(tf + (1-t)g) \leq tJ(f) + (1-t)J(g)$ whenever $f, g \in \mathcal{F}$ and 0 < t < 1. <span id="page-2-3"></span>Theorem B. Let $\mathscr{G}$ be a compact subset of $\mathcal{H}$ and J be a real-valued, continuous and convex functional on $\overline{co}\mathscr{G}$ . Then $\max\{J(f): f \in \overline{co}\mathscr{G}\} = \max\{J(f): f \in E\overline{co}\mathscr{G}\}$ . The proof of these two results can be found in [14, Theorem 4.5, Theorem 4.6]. In order to solve such linear extremal problems over $\mathcal{G}$ , it suffices to solve them over the smaller class $E\overline{co}\mathcal{G}$ . This reduction thereby becomes an effective technique for solving various linear extremal problems. Using this technique we solve the Zalcman conjecture for the class $\mathcal{U}$ . In 1960s, L. Zalcman posed a conjecture that if a function $f \in \mathcal{S}$ is given by (1.1) then <span id="page-2-0"></span>(1.6) $$|a_n^2 - a_{2n-1}| \le (n-1)^2 \quad \text{for } n \ge 2,$$ the equality holds only for the Koebe function k(z) = z/(1 − z) ² or its rotation. It is important to note that the remarkable Zalcman conjecture implies the celebrated Bieberbach conjecture |an| ≤ n for f ∈ S (see [\[7\]](#page-13-8)). A well-known consequence of the area theorem shows that [\(1.6\)](#page-2-0) holds good for n = 2 (see [\[10\]](#page-13-4)). For f ∈ S, Krushkal has proved the Zalcman conjecture for n = 3 (see [\[17\]](#page-13-9)) and recently for n = 4, 5, 6 (see [\[18\]](#page-13-10)). For a simple and elegant proof of Zalcman conjecture for the case n = 3, we refer to [\[18\]](#page-13-10). The Zalcman conjecture for functions in the class S is still open for n > 6. However, using complex geometry and universal Teichmüller spaces Krushkal has proved it for all n ≥ 2 in his unpublished work [\[19\]](#page-13-11). The Zalcman conjecture has been proved affirmatively for certain special subclasses of S, such as starlike functions, typically real functions, close-to-convex functions [\[7,](#page-13-8) [20\]](#page-13-12) and an observation also demonstrates that the Zalcman conjecture is asymptotically true (see [\[11\]](#page-13-13)). Recently, Abu Muhana et al. [\[2\]](#page-13-14) solved Zalcman conjecture for the class F consists of the family of functions f ∈ A satisfying the condition Re (1 + zf′′(z)/f′ (z)) > −1/2 for z ∈ D. Functions in the class F are known to be convex in some direction (and hence close-to-convex and univalent) in D. In 1986, Brown and Tsao [\[7\]](#page-13-8) proved the Zalcman conjecture for the starlike functions and typically real functions. In 1988, Ma [\[20\]](#page-13-12) proved that the Zalcman conjecture for close-to-convex functions. For basic properties of starlike functions, typically real functions and close-to-convex functions we refer to [\[10,](#page-13-4) [30\]](#page-14-1). In 1999, Ma [\[21\]](#page-13-0) proposed a generalized Zalcman conjecture for f ∈ S that for n ≥ 2, m ≥ 2, $$|a_n a_m - a_{n+m-1}| \le (n-1)(m-1),$$ which is still an open problem. Ma [\[21\]](#page-13-0) has proved this generalized Zalcman conjecture for classes S ∗ and SR. Here S^R denote the class of all functions in S with real coefficients. In 2017, Ravichandran and Verma [27] proved it for the classes of starlike and convex functions of given order and for the class of functions with bounded turning. In the present paper, we prove the Zalcman conjecture and generalized Zalcman conjecture for the class U using extreme point theory. We also prove the Zalcman conjecture and generalized Zalcman conjecture for the class F for the initial coefficients. The organization of the paper is as follows. In Section [2](#page-4-0) we prove that the class U(λ) for 0 < λ ≤ 1 is compact. In particular the class U is compact. In Section [3,](#page-6-0) we will characterize the closed convex hull of the class U and its extreme points. Then by using extreme point theory, we prove the Zalcman conjecture in Section [3](#page-6-0) and generalized Zalcman conjecture for the class U in Section [4.](#page-9-0) We prove the Zalcman conjecture and generalized Zalcman conjecture for the class F for the initial coefficients in Section [5.](#page-11-0) Before we prove our main results we recall some important results which will play vital role in our proofs. In 2016, Obradović et al. [\[23\]](#page-14-2) prove the following interested result. <span id="page-4-4"></span>Proposition 1.7. [\[23\]](#page-14-2) If f ∈ U(λ) for 0 < λ ≤ 1, then for z ∈ D, $$\frac{f(z)}{z} \prec \frac{1}{(1-z)(1-\lambda z)}.$$ Let $$\mathcal{R} := \left\{ F \in \mathcal{H} : \overline{cos}(F) = \left\{ \int_{|x|=1} F(xz) \, d\mu(x) : \mu \in \wedge \right\} \right\},\,$$ where ∧ denote the set of probability measure on ∂D. We recall the following wellknown result of Hallenbeck et. al [\[15\]](#page-13-15).