On Hardy spaces, univalent functions and the second coefficient

Theorem 1. If $f \in C$ then $f' \in H^p$ for all $p < \frac{1}{1+|a_2|}$ and $f \in H^q$ for all $q < \frac{1}{|a_2|}$ . Both estimates are sharp. The sharpness comes from the sector function, with $\alpha \in [0,1]$ , given by $$s_{\alpha}(z) = \frac{1}{2\alpha} \left[ \left( \frac{1+z}{1-z} \right)^{\alpha} - 1 \right] = z + \alpha z^2 + \frac{1+2\alpha^2}{3} z^3 + \dots$$ (1) (understood as $z \mapsto \frac{1}{2} \log \frac{1+z}{1-z}$ when $\alpha = 0$ ), which maps $\mathbb{D}$ onto a sector of opening $\alpha \pi$ . For the coefficients of functions in C we find their sharp order of growth and an explicit bound. Recall that $|a_2| \leq 1$ with equality only for rotations of the half-plane function $s_1(z) = \frac{z}{1-z}$ .

Theorem 2. If $f \in C$ then $a_n \in O(n^{|a_2|-1})$ , which is sharp, and $$|a_n| \le \exp\left(\frac{|a_2|^2 - 1}{2}\right), \qquad n \ge 2.$$ Following Cruz and Pommerenke [8] we define the $lower\ order$ of a function f in $\mathbb D$ as $$\beta = \inf_{\zeta \in \mathbb{D}} |A_f(\zeta)|,$$ where the expression $$A_f(\zeta) = \frac{1}{2} (1 - |\zeta|^2) \frac{f''(\zeta)}{f'(\zeta)} - \overline{\zeta}, \qquad \zeta \in \mathbb{D},$$ is the second coefficient of the Koebe transform of f, given by $$F_{\zeta}(z) = \frac{f\left(\frac{\zeta + z}{1 + \overline{\zeta}z}\right) - f(\zeta)}{(1 - |\zeta|^2)f'(\zeta)} = z + A_f(\zeta)z^2 + \dots, \qquad \zeta, z \in \mathbb{D}.$$ (2) Clearly, $0 \le \beta \le |a_2|$ , while we will prove in Section 3 that $\beta = |a_2|$ if and only if either $a_2 = 0$ or f is a rotation of $s_{\alpha}$ . We have the following improvement of Theorem 1.

Theorem 3. If $f \in C$ then $f' \in H^p$ for all $p < \frac{1}{1+\beta}$ and $f \in H^q$ for all $q < \frac{1}{\beta}$ . Convex functions and the angle at infinity. We define the angle at infinity of a convex domain $\Omega$ as $$\Theta(\Omega) = \inf\{\theta \in [0, \pi] : \Omega \text{ is contained in a sector of aperture } \theta \};$$ (3) see [5, §3]. An equivalent definition in terms of half-tangents is given in Section 4. We regard a sector of aperture zero as an infinite strip. Clearly, $\Theta = 0$ for bounded domains, infinite half-strips and strips. Note that the infimum in (3) is not always attained. For example, $\Theta = 0$ for a parabolic region but no parallel strip contains it. Simple modifications provide similar examples for other values of $\Theta$ in $(0, \pi)$ . On the other hand, if $\Theta = \pi$ then this value is always attained by any of the supporting half-planes of $\Omega$ . In fact, we will prove that in this case $\Omega$ itself must be a half-plane. Our main theorem relates the angle at infinity with the lower order as follows.

Theorem 4. If $$f \in C$$ then $\Theta(f(\mathbb{D})) = \beta \pi$ . From this, we deduce some corollaries. We will be needing the following theorem of Chuaqui and Osgood [6] and Fournier, Ma and Ruscheweyh [17]. We provide a new proof of this in Section 8. Theorem A ([6, 17]). If $f \in C$ has $a_2 = 0$ then f is either bounded or a rotation of $s_0(z) = \frac{1}{2} \log \frac{1+z}{1-z}$ . We now deduce our first corollary from Theorem 4. Corollary 1. If $f \in C$ then its image $f(\mathbb{D})$ lies in some sector of aperture $|a_2|\pi$ . Indeed, if $a_2 = 0$ then the corollary holds for the two cases given in Theorem A. If $a_2 \neq 0$ then, by the aforementioned characterization of the case when $\beta$ equals $|a_2|$ , we see that f is either a rotation of a sector function $s_{\alpha}$ or $\Theta < |a_2|\pi$ , again, two cases for which the corollary holds. Corollary 1 is sharp in view of the functions $s_{\alpha}$ . Moreover, the aperture $|a_2|\pi$ cannot be improved to $\beta\pi$ since, as discussed earlier, the infimum in (3) is not always attained. By subordination, if f is holomorphic in $\mathbb{D}$ and its range lies in some sector of aperture $\theta$ then $f \in H^q$ , for $q < \frac{\pi}{\theta}$ ; see Exercise 2 in [9, Chap. 1]. Hence, from Corollary 1 we deduce the following, which is the Hardy space estimate for the function f in Theorem 1, thus given an alternative geometric proof. Corollary 2. If $$f \in C$$ then $f \in H^q$ for all $q < \frac{1}{|a_2|}$ . Finally, since for functions in C we have $|a_2| = 1$ only for half-plane mappings, we obtain the following. Corollary 3. If $f \in C$ with $\Theta(f(\mathbb{D})) = \pi$ then f is a half-plane mapping. Convex functions and smoothness at the boundary. Let $\Lambda_t$ denote the class of functions $\phi$ defined on $[0, 2\pi]$ which satisfy the Lipschitz condition of order $t \in (0, 1]$ : $$|\phi(x) - \phi(y)| \le c |x - y|^t,$$ for some c > 0. Clearly, s < t implies $\Lambda_s \supset \Lambda_t$ . Moreover, we consider the wider class $\Lambda_t^p$ , for $t \in (0,1]$ and $p \geq 1$ , of functions $\phi \in L^p(0,2\pi)$ that satisfy the integral Lipschitz condition $$\left(\int_0^{2\pi} |\phi(\theta+h) - \phi(\theta)|^p d\theta\right)^{1/p} \le c h^t, \quad \text{for } h > 0,$$ where c > 0 is some constant. It is easy to see that if s < t then $\Lambda_s^p \supset \Lambda_t^p$ , while also p < q implies $\Lambda_t^p \supset \Lambda_t^q$ . Here is our main theorem.

Theorem 5. If $f \in C$ is not a rotation of the half-plane function $s_1(z) = \frac{z}{1-z}$ then $f(e^{i\theta}) \in \Lambda_t^p$ with $t = \frac{1}{p} - |a_2|$ , for every $1 \le p < \frac{1}{|a_2|}$ . This is sharp in the sense that for every $\alpha \in [0,1)$ there exists $f \in C$ with $\alpha = |a_2|$ , for which $f(e^{i\theta}) \notin \Lambda_t^p$ with $t = \frac{1}{p} - \alpha + \varepsilon$ , for every $1 \le p < \frac{1}{\alpha}$ and $\varepsilon > 0$ . In the case $a_2 = 0$ we are able to prove a stronger Lipschitz condition on the boundary.

Theorem 6. Let $f \in C$ with $a_2 = 0$ . If f is not a rotation of the parallel strip function $s_0(z) = \frac{1}{2} \log \frac{1+z}{1-z}$ then $f(e^{i\theta}) \in \Lambda_t$ for $t = \frac{1-3|a_3|}{3(1+|a_3|)}$ . Other geometric subclasses of S and Hardy spaces. Theorem 1 takes the following form in $S$ , the subclass of S consisting of functions whose range is starlike* with respect to the origin. This refines another theorem of Eenigenburg and Keogh [12]. Theorem 7. If f ∈ S ∗ then f ′ ∈ H^p for all p < ² 4+|a2| and f ∈ H^q for all q < ² 2+|a2| . Both estimates are sharp. For the subclass K of close-to-convex functions, a theorem of Leung [29] provides the sharp estimate: if f ∈ K then the derivative f ′ belongs to H^p for p < 1/3, which is sharp. The following theorem provides a slight improvement of this when a² = 0 while showing that it remains sharp when a² ̸= 0. Theorem 8. Let f ∈ K. We have that - (i) if a² = 0 then f ′ ∈ H1/³ and f ∈ H1/² , while - (ii) if a² ̸= 0 then f ′ ∈ H^p for all p < 1/3 and f ∈ H^q for all q < 1/2. All four estimates are sharp. Let R denote the subclass of S consisting of functions f whose range is convex in one direction, that is, there exists a direction θ ∈ [0, π) such that for each line L parallel to the line {teiθ : t ∈ R} the intersection L ∩ f(D) is either empty or a connected set. Domains that are convex in one direction have been of importance in the theory of harmonic mappings; see [11, Chap. 3]. Let R⁺ denote the class of functions f in S whose range is convex in the positive direction, which means that for each w ∈ f(D) the half-line {w + t : t ≥ 0} lies in f(D). This is an important class in the theory of semigroups of holomorphic functions in the disk; see [15]. It is well known that R⁺ ⊂ R ⊂ K, so that the estimates in Theorem 8 apply to functions in R⁺ and R. Moreover, these estimates are optimal when a² ̸= 0. However, for a² = 0 the optimal exponent exhibits a jump. Theorem 9. Let f ∈ S and assume that a² = 0. We have that - (i) if f ∈ R then f ′ ∈ H^p for all p < 1/2 and f ∈ H^q for all q < 1, while - (ii) if f ∈ R⁺ then f ′ ∈ H^p for all p < 1 and f ∈ H^q for all q < ∞. All four estimates are sharp. We now show that the improvement of the Hardy space estimate for closeto-convex functions given in Theorem 8 for the case a² = 0 does not persist in the class S, in fact, not even in the class SR, the subclass of S consisting of functions that have real coefficients. On the other hand, it is not difficult to see that Prawitz' Hardy space estimate for functions in S also holds in the class of typically real functions T, which also contains non-univalent functions. Theorem 10. It holds that T ⊂ H^p for all p < 1/2. For every α ∈ [0, 2] there exists f ∈ S^R with a² = α and such that f /∈ H1/² . Integrability of the derivative of univalent functions. Recall that the Nevanlinna class N consists of holomorphic functions f in D having bounded characteristic: $$\sup_{r \in (0,1)} \int_0^{2\pi} \log^+ |f(re^{i\theta})| d\theta < \infty,$$ where $\log^+ x = \max\{0, \log x\}$ . It is known that N is wider than all Hardy spaces, that is, $\bigcup_{p>0} H^p \subset N$ ; see [9, §2.1]. The Bloch-Nevanlinna conjecture, asserting that $f' \in N$ whenever $f \in N$ , has been proven false by various authors. A striking counterexample was provided by Lohwater, Piranian and Rudin [32]. They showed that for a suitably chosen increasing sequence $\{n_p\}$ of positive integers, the function $$\Phi(z) = \int_0^z \exp\left(\frac{1}{2} \sum_{p=1}^\infty w^{n_p}\right) dw$$ is holomorphic in $\mathbb{D}$ , continuous and injective in $\overline{\mathbb{D}}$ , and for almost all $\theta$ it satisfies $$0 = \liminf_{r \to 1} |\Phi'(re^{i\theta})| < \limsup_{r \to 1} |\Phi'(re^{i\theta})| = +\infty$$ and $$-\infty = \liminf_{r \to 1} \arg \Phi'(re^{i\theta}) < \limsup_{r \to 1} \arg \Phi'(re^{i\theta}) = +\infty.$$ This means that $\Phi \in H^{\infty} \cap S_{\mathbb{R}}$ but $\Phi' \notin N$ . It's easy to see that $a_2(\Phi) = \frac{1}{4}$ (it relies on the choice $n_1 = 1$ made in [32]). We mention that the construction of $\Phi$ can be modified so that the second coefficient takes any value in the interval $[0, \frac{\pi}{8})$ , but we will not supply a proof for this. Instead, composing the Koebe function with an integral transform of the function $\Phi$ , we will prove the following.

Theorem 11. For every $\alpha \in [0,2)$ there exists $f \in S_{\mathbb{R}}$ such that $a_2 = \alpha$ and $f' \notin N$ . It is interesting to note that in view of this theorem, the pathological behavior of the function $\Phi$ can be encountered arbitrary close to the Koebe function k, whose derivative $k'(z) = \frac{1+z}{(1-z)^3}$ belongs to $H^p$ for all p < 1/3, therefore to N as well. The organization of the article is as follows. We prove Theorems 1, 2 and 3 in Section 3, Theorem 4 in Section 4, Theorems 5 and 6 in Section 5, Theorems 7, 8, 9 and 10 in Section 6, and Theorem 11 in Section 7. Theorem A is given a new proof in an appendix in Section 8.

Lemma 1. If $f \in C$ then $|a_3| \leq \frac{1+2|a_2|^2}{3}$ . Equality holds only for rotations of the sector function (1).

Lemma 2. If $f \in C$ then $\beta = |a_2|$ if and only if either $a_2 = 0$ or f is a rotation

Lemma 4. Let $f \in C$ with associated measure $\mu$ and assume that $f(\lambda_0) = \infty$ for some $\lambda_0 \in \mathbb{T}$ . Then $$\Theta(f(\mathbb{D})) = (2\mu(\lambda_0) - 1)\pi.$$

Lemma 5. Let $f \in C$ with $a_2 = 0$ . Then $$|f'(z)| \le (1 - |z|)^{-\frac{2(1+3|a_3|)}{3(1+|a_3|)}}, \qquad z \in \mathbb{D}.$$