On one filtration of holomorphic functions

Theorem 1.1. If $f \in \text{Hol}(\mathbb{D}, \mathbb{C})$ and Re f'(z) > 0, then f is univalent in $\mathbb{D}$ . Thanks to this theorem, it begs to consider the class $$\mathcal{R} := \{ f \in \mathcal{A} : \operatorname{Re} f' \in \mathcal{P} \},$$ which is usually called the Noshiro-Warschawski class. More detailed information on this class can be found in [6, 8, 21, 23]. Recently, in several works [5, 14, 19] it was noticed that every function $f \in \mathcal{R}$ is a semigroup generator (see Definition 1.3 below). Moreover, elements of $\mathcal{R}$ have very Date: December 29, 2025. ²⁰²⁰ Mathematics Subject Classification. Primary 47H20, 30C45; Secondary 30C80, 58D07. $Key\ words\ and\ phrases.$ infinitesimal generators; filtrations; semigroups; starlike functions; analytic extension. specified geometric and dynamical properties. These facts among others served a base to develop theory of so-called filtrations of semigroup generators (see Definition 1.9). It can be easily seen that if we replace the Noshiro-Warschawski condition by a weaker one, namely, $$f \in \mathcal{A}, \quad \operatorname{Re} f'(z) > -\alpha$$ (1.1) for some $\alpha > -1$ , then there are non-univalent functions satisfying the last inequality. At the same time, we will see that for some positive $\alpha$ each function that satisfies (1.1) forced to be a generator. Assume f satisfies (1.1). In this paper the following questions are studied. Question 1. What geometric properties does the function f exibit? Question 2. What analytical features characterize the function f? Question 3. For which $\alpha$ does f serve as a semigroup generator? Question 4. If f is a semigroup generator, what are dynamical properties of this semi-group? Note in passing, that the inequality (1.1) is closely connected the work [9], where the set-theoretic properties of the two-parameter family of function classes defined by $$\operatorname{Re}\left((s-1)\frac{f(z)}{z} + f'(z)\right) \ge st, \quad s > 0, \ t \in [0,1),$$ were studied. Observe that (1.1) extends the last formula for s = 1 to classes of functions whose derivatives may have negative real parts. In that work the Gauß hypergeometric function ${}_{2}F_{1}(1, s; s + 1; z)$ plays a crucial role. Substituting s = 1 we get $$_{2}F_{1}(1,1;2;z) = -\frac{\log(1-z)}{z}$$ . This function will also prove to be important in the present study. In what follows, we need an additional notion. Let $f, g \in \text{Hol}(\mathbb{D}, \mathbb{C})$ and $$\Omega = \{ \omega \in \operatorname{Hol}(\mathbb{D}) : \ \omega(0) = 0 \}.$$ One says that f is subordinate g and writes $f \prec g$ , if there exists a function $\omega \in \Omega$ such that $f(z) = g(\omega(z))$ for all $z \in \mathbb{D}$ . In the case where g is univalent, conditions $f \prec g$ and $f(\mathbb{D}) \subseteq g(\mathbb{D})$ are equivalent. We will also make use of the main notions and facts from semigroup theory. For more details the reader can be referred to [4, 18, 33].

Theorem 1.4. Let $f \in \operatorname{Hol}(\mathbb{D}, \mathbb{C})$ . Then $f \in \mathcal{G}$ if and only if there are a point $\tau \in \overline{\mathbb{D}}$ and a function $q \in \operatorname{Hol}(\mathbb{D}, \mathbb{C})$ with $\operatorname{Re} q(z) \geq 0$ , $z \in \mathbb{D}$ , such that $$f(z) = (z - \tau) (1 - z\overline{\tau}) q(z), \quad z \in \mathbb{D}.$$ (1.3) Moreover, this representation is unique. We notice that formula (1.3) was obtained by Berkson and Porta in [3] and is called the Berkson-Porta representation. We further focus on the case $\tau \in \mathbb{D}$ . Up to the Möbius transformation $M_{\tau}$ $\left(M_{\tau}(z) = \frac{\tau - z}{1 - z\overline{\tau}}\right)$ of the unit disk, one can always consider generators such that f(0) = 0, or, what is the same, $\phi_t(0) = 0$ for all $t \geq 0$ . It follows from Theorem 1.4 that $f \in \mathcal{A}$ is an infinitesimal generator if and only if $\operatorname{Re} \frac{f(z)}{z} > 0$ for all $z \in \mathbb{D} \setminus \{0\}$ . We denote $$\mathcal{G}_0 := \mathcal{A} \cap \mathcal{G} = \left\{ f \in \operatorname{Hol}(\mathbb{D}, \mathbb{C}) : \frac{f(z)}{z} \in \mathcal{P} \right\}.$$ The class $\mathcal{G}_0$ is very important for the study of non-autonomous problems and geometric function theory (see, for example, [4, 6, 16, 18, 29]). Over the years, the study of the asymptotic behavior of semigroups was mainly focused on the local/global rate of convergence and the growth estimates of a semigroup with respect to its parameter. Different estimates of the rate of convergence of semigroups were obtained, see, for example, the books [4, 18, 33], the survey [25] and references therein. In view of subsequent results, we recall the following notion.

Theorem 1.6. Let $f \in \mathcal{G}_0$ and $\{\phi_t\}_{t\geq 0}$ be the semigroup generated by f. Then $\{\phi_t\}_{t\geq 0}$ is exponentially squeezing with squeezing ratio k>0 if and only if $\operatorname{Re}\frac{f(z)}{z}\geq k$ on $\mathbb{D}$ . Another direction of research on semigroup properties is focuses on the possibility of analytic extension with respect to the semigroup parameter into a complex domain. The problem is to identify conditions that allow such an extension, along with estimates of the sector in $\mathbb{C}$ to which this extension can be performed. For recent results in this direction see [2, 20, 10], see also [17]. More specifically, fix $\theta \in (0, \frac{\pi}{2}]$ and denote $$\Lambda(\theta) = \{ \zeta \in \mathbb{C} : |\arg \zeta| < \theta \}. \tag{1.4}$$

Theorem 1.7. (Theorem 2.12 in [20]). Let $\{\phi_t\}_{t\geq 0}$ be a semigroup of holomorphic self-mappings of $\mathbb D$ generated by $f\in\mathcal G_0$ , and let $\alpha\in[0,1)$ . Then $\{\phi_t\}_{t\geq 0}$ extends analytically to the sector $\Lambda\left(\frac{\pi(1-\alpha)}{2}\right)$ in $\mathbb C$ if and only if $\left|\arg\frac{f(z)}{z}\right|\leq \frac{\pi\alpha}{2}$ on $\mathbb D\setminus\{0\}$ . In the same work [20] a criterion for analytic extension was established.

Theorem 1.8. Let $f \in \mathcal{G}$ and $\theta \in [0,1)$ . Then the semigroup generated by f extends analytically to the sector $\Lambda\left(\frac{\pi(1-\theta)}{2}\right)$ in $\mathbb{C}$ if and only if $$\left|\arg\frac{f(z)}{z}\right| \le \frac{\pi\theta}{2} \text{ for all } z \in \mathbb{D} \setminus \{0\}.$$ These directions in study of dynamical systems highlight the importance of classifying generators according to the dynamical properties of the semigroups they generate. To this end, we introduce a filtration (or parametric embedding) of infinitesimal generators.

Lemma 2.4. The functions $\varphi_{\alpha}$ have the following properties: - (i) Taylor's coefficients of $\varphi_{\alpha}(z) = \sum_{n=0}^{\infty} \rho_n z^n$ are $\rho_0 = 1$ and $\rho_n = \frac{2(1+\alpha)}{n+1}$ , $n \ge 1$ ; - (ii) The Fekete-Szegö functional is $|\rho_3 \lambda \rho_2^2| = \frac{1+\alpha}{18} \cdot |9-8\lambda|, \ \lambda \in \mathbb{C};$ - (iii) The function $\varphi_{\alpha}$ , $\alpha > -1$ , is a univalent convex function; - (iv) Let $-1 < \alpha < \beta$ , then $\varphi_{\alpha} \prec \varphi_{\beta}$ .

Corollary 2.5. If $$f \in \mathcal{L}_{\alpha}$$ then $\frac{f(z)}{z} \prec \varphi_{\alpha}(z)$ . Estimates on coefficient operators remain a topic of ongoing interest in the geometric function theory community.

Proposition 2.6. Let $f \in \mathcal{L}_{\alpha}$ and $1 + \sum_{n=1}^{\infty} \phi_n z^n$ is its Taylor series. Then (i) $$|\phi_n| \le \frac{2(1+\alpha)}{n}, \ n \ge 1;$$ (ii) $$|\phi_3 - \lambda \phi_2^2| \le \frac{2(1+\alpha)}{3} \max\{1, |6\lambda - 1|\}$$ for all $\lambda \in \mathbb{C}$ and this inequality is sharp.

Theorem 2.7. Let $\alpha \geq -1$ , then - (i) $K(\mathcal{L}_{\alpha}) = -(1+2\alpha) + 2(1+\alpha)\log(2);$ - (ii) The following are equivalent: - $(1) \ \alpha \leq \alpha_*;$ - (2) The image of $\frac{f(z)}{z}$ does not contain the origin; - (3) $B(\mathcal{L}_{\alpha}) = \frac{2}{\pi} \cdot \max_{0 < \theta < \pi} \arg \varphi_{\alpha}(e^{i\theta});$ - $(4) \mathcal{L}_{\alpha} \subset \mathcal{G}$

Corollary 2.8. Let $-1 \le \alpha \le \alpha_* = \frac{2 \log 2 - 1}{2 - 2 \log 2}$ , then each element of $\mathcal{L}_{\alpha}$ is generator of exponentially squeezing semigroup with sharp squeezing ratio $K(\mathcal{L}_{\alpha}) = -(1 + 2\alpha) + 2(1 + \alpha) \log(2)$ . If $B(\mathcal{G}_1) < 1$ , then by Theorem 1.8 semigroups generated by elements of $\mathcal{G}_1$ admit analytic extension to the sector $\Lambda\left(\frac{\pi}{2}(1-B(\mathcal{G}_1))\right)$ and the angle of opening of this sector is the maximal one.

Corollary 2.9. Let $-1 \le \alpha \le \alpha_*$ and $B(\mathcal{L}_{\alpha}) = \frac{2}{\pi} \cdot \max_{0 < \theta < \pi} \arg \varphi_{\alpha}(e^{i\theta})$ . Then semigroups generated by any $f \in \mathcal{L}_{\alpha}$ admit analytic extension to the sector $\Lambda\left(\frac{\pi}{2}(1 - B(\mathcal{L}_{\alpha}))\right)$ and this sector is the maximal one. 2.3. The family $\mathcal{L}$ . Consider the family $\mathcal{L} = \{\mathcal{L}_{\alpha}\}_{{\alpha \geq -1}}$ . The following result characterizes this family from a set-theoretic perspective.

Theorem 2.10. The following assertions hold: - (a) The family $\mathcal{L} = \{\mathcal{L}_{\alpha}\}_{\alpha \in [-1,\alpha_]}$ forms a filtration with $\mathcal{L}_{-1} = \{\mathrm{Id}\}$ , $\mathcal{L}_0 = \mathcal{R}$ and $\mathcal{L}_{\alpha} = \left\{ f \in \mathcal{A} : \frac{f(z)}{z} = \frac{-\log 2 + \int_{\partial \mathbb{D}^2} F_1(1,1;2;z\overline{\zeta}) d\mu(\zeta)}{1 \log 2} \right\}.$ - (b) The filtration $\mathcal{L}$ admits a net $\{f_{\alpha}\}_{{\alpha}\in[-1,\alpha_{*}]}$ of totally extremal functions $f_{\alpha}(z):=z\varphi_{\alpha}(z)$ . (c) For every $\alpha \in [-1, \alpha_*]$ we have $f_{\alpha} \in \partial \mathcal{L}_{\alpha} := \mathcal{L}_{\alpha} \setminus \bigcup_{-1 \leq s < \alpha} \mathcal{L}_s$ . Consequently, the filtration $\mathcal{L}$ is strict.

Definition 1.2. A family $\{\phi_t\}_{t\geq 0} \subset \operatorname{Hol}(\mathbb{D})$ is called a one-parameter continuous semi-group (or just semigroup) if (a) $$\phi_{t+s} = \phi_t \circ \phi_s$$ , $t, s \ge 0$ ; and (b) $\lim_{t\to 0^+} \phi_t = \operatorname{Id}$ , where $\operatorname{Id}$ is the identity map on $\mathbb D$ and the limit is taken with respect to the topology of uniform convergence on compact sets in $\mathbb D$ . Moreover, according to the fundamental result by Berkson and Porta [3] the family $\{\phi_t\}_{t>0}$ is differentiable with respect to its parameter $t \geq 0$ , and the limit $$f = \lim_{t \to 0^+} \frac{1}{t} \left( \operatorname{Id} - \phi_t \right), \tag{1.2}$$ exists, and defines a holomorphic function on $\mathbb{D}$ . Furthermore, $\phi_t$ is the solution of the Cauchy problem: $$\frac{\partial \phi_t(z)}{\partial t} + f(\phi_t(z)) = 0$$ and $\phi_0(z) = z \in \mathbb{D}$ .

Definition 1.3. The function $f \in \operatorname{Hol}(\mathbb{D}, \mathbb{C})$ defined by (1.2) is called the infinitesimal generator of the semigroup $\{\phi_t\}_{t\geq 0} \subset \operatorname{Hol}(\mathbb{D})$ . Denote by $\mathcal{G}$ the class of all holomorphic generators on $\mathbb{D}$ . The following assertion contains a parametric representation of this class.

Definition 1.5. (see [5]). Let $\{\phi_t(z)\}_{t\geq 0}$ be a semigroup. If there exists a constant k>0 such that $$|\phi_t(z)| \le e^{-kt}|z|$$ for all $z \in \mathbb{D}$ , then $\{\phi_t\}_{t>0}$ is said to be exponentially squeezing semigroup with squeezing ratio k. The following criterion for a semigroup to be exponentially squeezing holds.

Definition 1.9. (see [5, 14, 19]). Let J be a connected subset of $\mathbb{R}$ . A filtration of $\mathcal{G}_0$ is a family $\mathfrak{F} = \{\mathfrak{F}_s\}_{s \in J}$ , $\mathfrak{F}_s \subseteq \mathcal{G}_0$ , such that $\mathfrak{F}_s \subseteq \mathfrak{F}_t$ whenever $s, t \in J$ and $s \leq t$ . Moreover, if $\mathfrak{F}_s \subseteq \mathfrak{F}_t$ for s < t $s, t \in J$ , then we say that the filtration $\{\mathfrak{F}_s\}_{s \in J}$ is strict. In this case for $t \in J$ we set $$\partial \mathfrak{F}_t = \mathfrak{F}_t \setminus \bigcup_{s \in J, s < t} \mathfrak{F}_s. \tag{1.5}$$ Let a filtration $\mathfrak{F} = {\{\mathfrak{F}_s\}}_{s \in J}$ be given. We now split Question 4 into two detailed parts for clarity. - Determine the sharp squeezing ratio for all semigroups generated by elements of each filtration set Fs; - Identify the maximal sector into which all these semigroups admit an analytic extension. Note that if the boundaries defined by (1.5) are not empty for every t ∈ J, then the filtration F is strict although for a strict filtration boundaries of its elements might be empty. The study of different classes of functions involves the searching for extremals. We will use the following notion: Definition 1.10 (see [5]). Let F ⊂ A. We say that a function f^∗ ∈ F is totally extremal for F if for every λ ∈ C and r ∈ [0, 1] $$\min_{|z|=r} \operatorname{Re}\left(\lambda \frac{f(z)}{z}\right) \ge \min_{|z|=r} \operatorname{Re}\left(\lambda \frac{f_*(z)}{z}\right) \text{ for all } f \in \mathcal{F}.$$

Definition 1.11. We say that a filtration F = {Fs}s∈^J admits a net {fs}s∈^J of totally extremal functions if for every s ∈ J, the function f^s is totally extremal for the class Fs.

Definition 2.1. For α ≥ −1 define $$\mathcal{L}_{\alpha} := \{ f \in \mathcal{A} : \operatorname{Re} f'(z) \ge -\alpha \}. \tag{2.1}$$ Obviously, if α < β then L^α ⊊ Lβ. Example 2.2. Fix α < −1. The function f(z) = zφα(z) where $$\varphi_{\alpha}(z) = -(1+2\alpha) - 2(1+\alpha) \frac{\log(1-z)}{z}, \quad z \in \mathbb{D}$$ (2.2) belongs to the class Lα. Indeed, a straightforward computation shows that Re f ′ (z) = ⁻(1+2α)+2(1+α) Re ¹ 1 − z and inf z∈D Re ¹ 1 − z = 1 2 . Hence, Re f ′ (z) > −α as required. 2.1. Analytic and geometric features of $\mathcal{L}_{\alpha}$ . The next result establishes a representation of the elements in $\mathcal{L}_{\alpha}$ .