Solution of the logarithmic coefficients conjecture in some families of univalent functions

Lemma 3.1. [8, Theorem 2.2d, p.24] Let $q \in \mathcal{Q}$ with q(0) = 1 and let $p(z) = 1 + p_n z^n + \cdots$ be analytic in $\mathbb{D}$ with $p(z) \neq 1$ . If $p \not\prec q$ in $\mathbb{D}$ , then there exits points $z_0 \in \mathbb{D}$ and $\zeta \in \partial \mathbb{D} \setminus \mathbf{E}(q)$ and there exits a real number $m \geq n \geq 1$ for which $$p(|z| < |z_0|) \subset q(\mathbb{D}), \quad p(z_0) = q(\zeta), \quad z_0 p'(z_0) = m\zeta q'(\zeta).$$

Lemma 3.2. [8, Theorem 3.1b, p.71] Let h(z) be convex in $\mathbb{D}$ with h(0) = a. If p(z) is analytic in $\mathbb{D}$ , with p(0) = a and $p(z) + zp'(z) \prec h(z)$ , then $$p(z) \prec \frac{1}{z} \int_0^z h(t) dt.$$

Lemma 3.3. [8, Corollary 3.1d.1, p.76] Let h(z) be starlike in $\mathbb{D}$ with h(0) = 0 and $a \neq 0$ . If $p(z) = 1 + p_n z^n + p_{n+1} z^{n+1} + \cdots$ is analytic in $\mathbb{D}$ satisfies the subordination relation $$\frac{zp'(z)}{p(z)} \prec h(z) \implies p(z) \prec a \exp\left(\frac{1}{n} \int_0^z \frac{h(t)}{t} dt\right).$$

Lemma 3.4. ([4]). Let $f \in \mathcal{CV}$ . If $g(z) = (1/z) \int_0^z f(t) dt$ , then g is also univalent and convex in $\mathbb{D}$ .

Theorem 3.5. Let $p(z) = 1 + p_1 z + \cdots$ be analytic function in the unit disk $\mathbb{D}$ with $p(z) \not\equiv 1$ . (3.1) $$\Re\left\{\frac{zp'(z)}{p(z)}\right\} < \frac{\lambda}{2} \qquad (0 < \lambda \le 1, z \in \mathbb{D}),$$ then there exists n > 1 such that <span id="page-5-2"></span> $$p(z) \prec (1+z^n)^{\lambda/n} =: q(z) \quad (z \in \mathbb{D}).$$

Theorem 3.11. The function $\frac{\mathbf{G}_{F_{\lambda/n,n}}(z)}{z}$ is convex univalent in $\mathbb{D}$ for each $0 < \lambda \leq 1$ and $n \geq 1$ (not normalized in the usual sense). Especially $\frac{\mathbf{G}_{F_{\lambda}}(z)}{z}$ is convex univalent in $\mathbb{D}$ , and $$\frac{2}{\lambda} \left( \frac{\mathbf{G}_{F_{\lambda}}(z)}{z} - 1 \right) \in \mathcal{CV}.$$

Theorem 3.12. Let $f \in \mathcal{N}(\lambda)$ given by (1.1). Then for |z| = r < 1 the following inequalities hold true (i) $$-\mathbf{N}_{F_{\lambda}}(-r) = r(1-r)^{\lambda} \le |f(z)| \le r(1+r)^{\lambda} = \mathbf{N}_{F_{\lambda}}(r),$$ (ii) $\left| \arg \left\{ \frac{f(z)}{z} \right\} \right| \le \lambda \sin^{-1} r.$ (ii) $$\left| \arg \left\{ \frac{f(z)}{z} \right\} \right| \le \lambda \sin^{-1} r$$ Equalities hold for the function $f(z) = \mathbf{N}_{F_{\lambda}}(z)$ given by (2.11) or its rotation.

Theorem 3.13. Let $f \in \mathcal{G}(\lambda)$ given by (1.1). Then for |z| = r < 1 it holds (i) $$\mathfrak{q}_{\lambda}(-r) = (1-r)^{\lambda} < |f'(z)| < (1+r)^{\lambda} = \mathfrak{q}_{\lambda}(r)$$ . (ii) $\left| \arg\{f'(z)\} \right| \le \lambda \sin^{-1} r$ (iii) $$-\mathbf{G}_{F_{\lambda}}(-r) = \frac{1 - (1 - r)^{\lambda + 1}}{1 + \lambda} \le |f(z)| \le \frac{(1 + r)^{\lambda + 1} - 1}{1 + \lambda} = \mathbf{G}_{F_{\lambda}}(r).$$ The above inequalities are sharp; equalities hold for $f(z) = \mathbf{G}_{F_{\lambda}}(z)$ given by (2.10) or its rotation.

Theorem 4.1. Let $f \in \mathcal{ST}_{ss}(\lambda)$ . Then the logarithmic coefficients of f satisfy $$(4.1) |\gamma_n(f)| \leq \frac{\lambda}{2n} \quad (n \geq 1),$$ (4.2) $$\sum_{n=1}^{\infty} n^2 |\gamma_n(f)|^2 \leq \frac{1}{4} \sum_{n=1}^{\infty} |B_n|^2,$$ (4.3) $$\sum_{n=1}^{\infty} |\gamma_n(f)|^2 \leq \frac{1}{4} \sum_{n=1}^{\infty} \frac{|B_n|^2}{n^2} \leq \frac{\lambda^2 \pi^2}{24}.$$ All the inequalities are sharp.

Corollary 4.4. The logarithmic coefficients of f ∈ G(λ) satisfy $$\sum_{n=1}^{\infty} |\gamma_n(f)| \leq \frac{\lambda}{2},$$ $$\sum_{n=1}^{\infty} n^2 |\gamma_n(f)|^2 \leq \frac{\lambda^2}{24} (\pi^2 - 6),$$ $$\sum_{n=1}^{\infty} (n+1)^2 |\gamma_n(f)|^2 \leq \frac{\lambda^2}{24} \pi^2,$$ $$\sum_{n=1}^{\infty} |\gamma_n(f)|^2 \leq \frac{\lambda^2}{12} (\pi^2 - 9).$$ Below, we present the another way to obtain estimate of the coefficient bounds of the functions in families G(λ) and N (λ).

Theorem 4.5. Let the function f ∈ G(λ) be given by [\(1.1\)](#page-0-1). Then |an| ≤ λ/(n(n−1)) (n ≥ 2). Equality holds for GFλ/(n−1),n−¹ defined by [\(2.9\)](#page-4-1). Proof. Let f ∈ G(λ) be given by [\(1.1\)](#page-0-1). Then, by Corollary [3.8,](#page-6-1) there exists n ≥ 1 and h ∈ ST ss(λ/n) such that $$f(z) = \mathbf{G}_g(z) \quad (z \in \mathbb{D}),$$ that is equivalent to $$\sum_{n=2}^{\infty} a_n z^n = 2 \sum_{n=2}^{\infty} \frac{n-1}{n} \gamma_{n-1}(g) z^n,$$ where γn(g) is a logarithmic coefficients of g. By Corollary [4.2](#page-9-2) we have |γn−1(g)| ≤ λ/(n−1)² , and then we conclude |an| ≤ λ/(n(n − 1)). The inequality is sharp for g = Fλ/(n−1),n−¹ or alternatively f = GFλ/(n−1),n−¹ . Note that for function G^λ given by [\(2.10\)](#page-4-2), we have $$a_n = \frac{\lambda(\lambda - 1)\cdots(\lambda - n + 2)}{n!} \quad (n \ge 2),$$ and so |an| ≤ λ/(n(n − 1)) is satisfied.

Theorem 4.6. If a function f ∈ N (λ) is of the form [\(1.1\)](#page-0-1) then |an| ≤ λ/(n − 1) for n ≥ 2. Equality holds for NFλ/(n−1),n−¹ defined by [\(2.11\)](#page-4-3). Proof. Let f ∈ N (λ) be given by [\(1.1\)](#page-0-1). Then the Corollary [3.9](#page-6-2) implies that there exists n ≥ 1 and h ∈ ST ss(λ/n) such that $$f(z) = \mathbf{N}_g(z) \quad (z \in \mathbb{D}).$$ From (2.8), we obtain $$\sum_{n=2}^{\infty} a_n z^n = 2 \sum_{n=2}^{\infty} (n-1) \gamma_{n-1}(g) z^n,$$ and applying Corollary 4.2 we get $|\gamma_{n-1}(g)| \leq \lambda/(n-1)^2$ . Thus, we conclude $|a_n| \leq \lambda/(n-1)$ and inequality is sharp for $g = F_{\lambda/(n-1),n-1}$ . On the other hand $f = \mathbf{N}_{F_{\lambda/(n-1),n-1}}$ . Note that for function $\mathbf{N}_{\lambda}(z) = z(1+z)^{\lambda}$ , we have $$a_n = \frac{\lambda(\lambda - 1)\cdots(\lambda - n + 2)}{(n-1)!} \quad (n \ge 2)$$ and so $|a_n| \leq \lambda/(n-1)$ holds.

Lemma 5.1. [5] Let $\omega$ be the Schwarz function with the power series $\omega(z) = \sum_{n=1}^{\infty} w_n z^n$ . Then $$w_2 = \xi (1 - w_1^2),$$ $$w_3 = (1 - w_1^2) (1 - |\xi|^2) \zeta - w_1 (1 - w_1^2) \xi^2,$$ for some complex number $\xi$ , $\zeta$ with $|\xi| \leq 1$ and $|\zeta| \leq 1$ .

Theorem 5.2. Let $f \in \mathcal{ST}_{ss}(\lambda)$ given by (1.1). Then <span id="page-11-0"></span> $$\left|a_2a_4 - a_3^2\right| \le \frac{\lambda^2}{4}.$$ The inequality is sharp.

Theorem 5.3. Let the function f of the form (1.1) belongs to the class $ST_{ss}(\lambda)$ . Then $|a_n| = O(1/n)$ for $n = 1, 2, 3, \ldots$

Theorem 5.4. Let $f \in \mathcal{ST}_{ss}(\lambda)$ given by (1.1). Then for real number $\delta$ , we have sharp inequalities <span id="page-12-0"></span> $$|a_{3} - \delta a_{2}^{2}| \leq \begin{cases} -\lambda^{2} \left(\delta + \frac{1 - 3\lambda}{4\lambda}\right) & \text{for } \delta < \frac{3(\lambda - 1)}{4\lambda}, \\ \frac{\lambda}{2} & \text{for } \frac{3(\lambda - 1)}{4\lambda} \leq \delta \leq \frac{1 + 3\lambda}{4\lambda}, \\ \lambda^{2} \left(\delta + \frac{1 - 3\lambda}{4\lambda}\right) & \text{for } \delta > \frac{1 + 3\lambda}{4\lambda}. \end{cases}$$

Corollary 5.5. If $f \in \mathcal{ST}_{ss}(\lambda)$ is of the form (1.1), then $$|a_2| \le \lambda, \quad |a_3| \le \frac{\lambda}{2}.$$ The first inequality is sharp for function $f(z) = \overline{\mu}F_{\lambda}(\mu z)$ and the second for $f(z) = \overline{\mu}F_{\lambda,2}(\mu z)$ , where $\mu$ is a unimodular complex number.

Corollary 5.6. Let the function $f \in \mathcal{ST}_{ss}(\lambda)$ be given by (1.1). Then $$\left|a_3 - a_2^2\right| \le \frac{\lambda}{2}.$$ The inequality is sharp for function $f(z) = \overline{\mu} F_{\lambda,2}(\mu z)$ , where $\mu$ is a unimodular complex number. Let $f \in \mathcal{S}$ given by (1.1). Then F(z) = z/f(z) is a non-vanishing analytic function in $\mathbb{D}$ and (5.5) $$F(z) = \frac{z}{f(z)} = 1 + \sum_{n=1}^{\infty} c_n z^n = 1 - a_2 z + (a_2^2 - a_3) z^3 + \cdots$$

Theorem 5.7. Let $f \in \mathcal{ST}_{ss}(\lambda)$ and F(z) = z/f(z) given by (1.1) and (5.5), respectively. Then for real number $\delta$ , we have sharp inequalities for function F(z) = z/f(z) <span id="page-13-0"></span> $$|c_2 - \delta c_1^2| \le \begin{cases} -\lambda^2 \left(\delta - \frac{\lambda + 1}{4\lambda}\right) & \text{for } \delta < \frac{\lambda - 1}{4\lambda}, \\ \frac{\lambda}{2} & \text{for } \frac{\lambda - 1}{4\lambda} \le \delta \le \frac{\lambda + 3}{4\lambda}, \\ \lambda^2 \left(\delta - \frac{\lambda + 1}{4\lambda}\right) & \text{for } \delta > \frac{\lambda + 3}{4\lambda}. \end{cases}$$

Definition 2.1. ([7]). By $\mathcal{ST}_{ss}(\lambda)$ we denote the subfamily of $\mathcal{ST}$ consisting of the functions f and satisfying the condition (2.1) $$\frac{zf'(z)}{f(z)} \prec \mathfrak{q}_{\lambda}(z) \quad (z \in \mathbb{D}),$$ where $\prec$ denotes the subordination. The condition (2.1) also means that the quantity zf'(z)/f(z) lies in a domain bounded by the spiral $SS(\lambda)$ . The geometric properties of $\mathfrak{q}_{\lambda}$ imply recurrence inclusions, below. (2.2) $$\mathcal{ST}_{ss}\left(\frac{\lambda}{n+1}\right) \subset \mathcal{ST}_{ss}\left(\frac{\lambda}{n}\right) \subset \mathcal{ST}_{ss}(\lambda) \quad (0 < \lambda \le 1, \ n \ge 1).$$ Let $F_{\lambda,n} \in \mathcal{ST}_{ss}(\lambda)$ be given by <span id="page-3-5"></span> $$\frac{zF'_{\lambda,n}(z)}{F_{\lambda,n}(z)} = \mathfrak{q}_{\lambda}(z^n) = (1+z^n)^{\lambda} \quad (z \in \mathbb{D}, n = 1, 2, \dots).$$ Then, the function $F_{\lambda,n}(z)$ is of the form (2.3) $$F_{\lambda,n}(z) = z \exp\left(\int_0^z \frac{\mathfrak{q}_{\lambda}(t^n) - 1}{t} dt\right) \quad (z \in \mathbb{D}),$$ and is extremal for various problems in $\mathcal{ST}_{ss}(\lambda)$ . Especially for n=1 we have <span id="page-3-2"></span>(2.4) $$F_{\lambda}(z) := F_{\lambda,1}(z) = z \exp\left(\int_0^z \frac{\mathfrak{q}_{\lambda}(t) - 1}{t} dt\right) = z + \lambda z^2 + \frac{3\lambda^2 - \lambda}{4} z^3 + \cdots,$$ and setting $\lambda/m$ instead of $\lambda$ with m, n = 1, 2, ..., we obtain the following form (2.5) <span id="page-3-1"></span> $$F_{\lambda/m,n}(z) = z \exp\left(\int_0^z \frac{\mathfrak{q}_{\lambda}(t^n)^{\lambda/m} - 1}{t} dt\right) = z + \frac{\lambda}{nm} z^{n+1} + \frac{\lambda^2(n+1) - nm\lambda}{4n^2 m^2} z^{2n+1} + \cdots$$ A special case (n = 1) of $F_{\lambda/m,n}$ gives (2.6) $$F_{\lambda/m}(z) = F_{\lambda/m,1}(z) = z \exp\left(\int_0^z \frac{\mathfrak{q}_{\lambda}(t)^{\lambda/m} - 1}{t} dt\right) = z + \frac{\lambda}{m} z^2 + \frac{2\lambda^2 - m\lambda}{4m^2} z^3 + \cdots \quad (z \in \mathbb{D}).$$ For $f \in \mathcal{S}$ , zf'(z)/f(z) is a non-vanishing analytic function in $\mathbb{D}$ . Then, the transform $\mathbf{G}_f$ of $f \in \mathcal{S}$ is well defined, and is of the form [9] <span id="page-3-3"></span>(2.7) $$\mathbf{G}_f(z) := \int_0^z \frac{tf'(t)}{f(t)} dt = \int_0^z \left(1 + t\left(\log\frac{f(t)}{t}\right)'\right) dt = z + 2\sum_{n=2}^\infty \frac{n-1}{n} \gamma_{n-1}(f) z^n.$$ Also, let $\mathbf{N}_f$ be given by <span id="page-3-4"></span>(2.8) $$\mathbf{N}_f(z) := z\mathbf{G}_f'(z) = z + 2\sum_{n=2}^{\infty} (n-1)\gamma_{n-1}(f)z^n.$$ We observe that $\mathbf{G}_f$ and $\mathbf{N}_f$ of $f \in \mathcal{S}$ are the functions with the power series expressed in terms of logarithmic coefficients that has the consequences in the next part of our study. For functions $F_{\lambda/m,n}$ and $F_{\lambda}$ given by (2.5) and (2.4) the transform G and N yield <span id="page-4-1"></span>(2.9) $$\mathbf{G}_{F_{\lambda/m,n}}(z) = \int_0^z \mathfrak{q}_{\lambda/m}(t^n) \, \mathrm{d}t = z + \frac{\lambda}{m(n+1)} z^{n+1} + \frac{\lambda(\lambda - m)}{2m^2(2n+1)} z^{2n+1} + \cdots$$ <span id="page-4-2"></span>(2.10) $$\mathbf{G}_{F_{\lambda}}(z) = \mathbf{G}_{F_{\lambda/1,1}}(z) = \frac{(1+z)^{1+\lambda} - 1}{1+\lambda} = z + \frac{\lambda}{2}z^2 + \frac{\lambda(\lambda-1)}{6}z^3 + \cdots$$ <span id="page-4-3"></span>(2.11) $$\mathbf{N}_{F_{\lambda/m,n}}(z) = z \mathfrak{q}_{\lambda/m}(z^n) = z + \frac{\lambda}{m} z^{n+1} + \frac{\lambda(\lambda - m)}{2m^2} z^{2n+1} + \cdots,$$ and $\mathbf{N}_{F_{\lambda}}(z) = \mathbf{N}_{F_{\lambda/1,1}}(z)$ . The sample figures of $\mathbf{N}_{F_{\lambda}}$ and $\mathbf{G}_{F_{\lambda}}$ are presented on Fig. 2 and Fig 3. Let m, n = 1, 2, ... and $0 < \lambda \le 1$ . Then, the logarithmic coefficients of $\mathbf{G}_{F_{\lambda/n,n}}$ and $F_{\lambda,n}$ are the following $$\gamma_n\left(\mathbf{G}_{F_{\lambda/n,n}}\right) = \frac{\lambda}{2n(n+1)}, \qquad \gamma_n(F_{\lambda,n}) = \frac{\lambda}{2n}, \qquad \gamma_n(F_{\lambda}) = \frac{B_n}{2n}.$$ <span id="page-4-0"></span>Also, for $m \geq n$ , we have $$\mathbf{G}_{F_{\lambda/m,n}} \in \mathcal{G}(\lambda), \qquad \mathbf{N}_{F_{\lambda/m,n}} \in \mathcal{N}(\lambda).$$  FIGURE 2. The image of $\mathbf{N}_{F_{\lambda}}(z)$ , $\mathbf{G}_{F_{\lambda}}(z)$ , $(\lambda = 1/2)$ .