Slice starlike functions over quaternions

Theorem 1.1. Let f(q) = q + P⁺^∞ ⁿ=2 q n ^aⁿ ∈ S^∗ . Then $$|a_n| \le n$$ , for all $n = 2, 3, \dots$ Equality |an| = n for a given n ≥ 2 holds if and only if $$f(q) = q(1 - qu)^{-*2}, \quad \forall \ q \in \mathbb{B},$$ for some ^u ^∈ ^∂B. Note that, by Theorem [1.1](#page-2-0) and Example [3.1,](#page-7-0) one can find that the Bieberbach conjecture holds for many injective slice regular functions which are not in ^V(B). Moreover, the Bieberbach conjecture over quaternions still holds for slice close-to-convex functions (see Theorem [4.3\)](#page-14-0). Related to the Bieberbach conjecture, Fekete and Szeg¨o in [\[14\]](#page-23-13) proved a striking inequality (1.1) $$|a_3 - \lambda a_2^2| \le \begin{cases} 3 - 4\lambda, & \text{if } \lambda \in (-\infty, 0), \\ 1 + 2e^{-2\lambda/(1-\lambda)}, & \text{if } \lambda \in [0, 1), \\ 4\lambda - 3, & \text{if } \lambda \in [1, +\infty), \end{cases}$$ for any injective holomorphic function F(z) = z + Σ⁺^∞ ⁿ=2anz n on D. Moreover, inequality [\(1.1\)](#page-2-1) is sharp for each ^λ ^∈ ^R. For related complex versions of Theorem [1.2,](#page-3-0) we refer the reader to the literature [\[33,](#page-24-9) [35\]](#page-24-10) and [\[13\]](#page-23-14). In this paper, we shall establish the Fekete-Szeg¨o type inequality for slice starlike functions as follows.

Theorem 1.2. Let $f(q) = q + \sum_{n=2}^{+\infty} q^n a_n \in \mathcal{S}^*$ . Then, for any $\lambda \in \mathbb{H}$ , $$|a_3 - \lambda a_2^2| \le \max\{1, |4\lambda - 3|\}.$$ This estimate is sharp for each $\lambda$ . Equality occurs if $$f(q) = q(1 - qu)^{-*2}, \quad q \in \mathbb{B},$$ for any $u \in \partial \mathbb{B}$ . Note that if $\lambda = 0$ in Theorem 1.2, then this result is already obtained in Theorem 1.1. As a very important consequence of Theorem A for the second-order coefficient, one may deduce the following well-known growth and distortion theorems. Theorem B. Let $F(z) = z + \sum_{n=2}^{+\infty} a_n z^n$ be an injective holomorphic function on $\mathbb{D}$ . Then for each $z \in \mathbb{D}$ , the following inequalities hold: (1.2) $$\frac{|z|}{(1+|z|)^2} \le |F(z)| \le \frac{|z|}{(1-|z|)^2};$$ (1.3) $$\frac{1-|z|}{(1+|z|)^3} \le |F'(z)| \le \frac{1+|z|}{(1-|z|)^3};$$ (1.4) $$\frac{1-|z|}{1+|z|} \le \left| \frac{zF'(z)}{F(z)} \right| \le \frac{1+|z|}{1-|z|}.$$ Moreover, equality holds for one of these six inequalities at some $z_0 \in \mathbb{D} \setminus \{0\}$ if and only if $$F(z) = \frac{z}{(1 - e^{i\theta}z)^2}$$ , $\forall z \in \mathbb{D}$ , for some $\theta \in \mathbb{R}$ . However, the growth and distortion theorems fail generally in $\mathbb{C}^n$ $(n \geq 2)$ and only hold for some subclass such as starlike or convex mappings (cf. [25, 28]). Based on Theorem B, the growth and distortion theorems were formulated for normalized injective slice regular functions in the special class $\mathcal{N}(\mathbb{B})$ (see [15, Theorem 3.11]). See [41] for further extensions to normalized injective (on $\mathbb{B}_I$ ) slice regular functions in $\mathcal{V}(\mathbb{B})$ . In this paper, we shall answer an open problem in [41] and show that the class $\mathcal{V}(\mathbb{B})$ is not the largest subclass of slice regular functions such that the corresponding growth, distortion, and covering theorems hold. Indeed, it is proven that the growth, distortion, and covering theorems are valid in a tighter form for $\mathcal{S}^*$ by applying a new growth theorem for the Carathéodory class in the quaternionic setting. They are essentially the first significant theorems belonging to the theory of quaternions itself for that the Taylor coefficients of slice regular functions need not belong to one slice. Notice that the close analytic relation between convex and starlike functions known as Alexander's theorem asserts that F is convex if and only if zF'(z) is starlike for the holomorphic function F on $\mathbb{D}$ normalized by F(0) = 0 and F'(0) = 1. In view of this result, we give another coefficient estimate for a subclass of slice regular functions from Theorem 1.1.

Theorem 1.3. Let $f(q) = q + \sum_{n=2}^{+\infty} q^n a_n$ be a slice regular function on $\mathbb{B}$ such that $qf'(q) \in \mathcal{S}^*$ . Then $$|a_n| \le 1$$ , for all $n = 2, 3, ...$ Equality $|a_n| = 1$ for a given $n \geq 2$ holds if and only if $$f(q) = q(1 - qu)^{-*}, \quad \forall \ q \in \mathbb{B},$$ for some $u \in \partial \mathbb{B}$ . Theorem 1.3 implies directly the following growth theorem: $$|f(q)| \le \frac{|q|}{1-|q|}, \quad \forall \ q \in \mathbb{B},$$ for any slice regular function f on $\mathbb{B}$ such that f(0) = 0 and $qf'(q) \in \mathcal{S}^*$ . For normalized convex functions F on $\mathbb{D}$ , there holds a sharper growth theorem (1.5) $$\frac{|z|}{1+|z|} \le |F(z)| \le \frac{|z|}{1-|z|}, \quad \forall \ z \in \mathbb{D},$$ which was generalized to $\mathbb{C}^n$ (see e.g., [28, 25, 37]). Usually, inequality (1.5) is just a consequence of the following distortion theorem in one and several complex variables.

Theorem 1.4. Let f be a slice regular function on $\mathbb{B}$ such that $qf'(q) \in \mathcal{S}^*$ . Then $$\frac{1}{(1+|q|)^2} \le |f'(q)| \le \frac{1}{(1-|q|)^2}, \quad \forall \ q \in \mathbb{B}.$$ Estimates are sharp. For each $q \in \mathbb{B} \setminus \{0\}$ , equality occurs if $$f(q) = q(1 - qu)^{-*}, \quad q \in \mathbb{B},$$ for some $u \in \partial \mathbb{B}$ . However, the method adopted in one and several complex variables cannot be used to our setting due to higher dimensions as well as the non-commutativity of quaternions and we do not know if the image of qf'(q) is convex for any $f \in \mathcal{S}^*$ . Fortunately, we can obtain a Koebe type theorem for slice regular functions with convex image as follows.

Theorem 1.5. Let f be a slice regular function on $\mathbb{B}$ with convex image and f'(0) = 1. Then $f(\mathbb{B})$ contains an open ball centered at f(0) of radius 1/2. Moreover, the constant 1/2 is optimal. It is worth mentioning that many methods in complex cases may fail in the quaternionic setting since the regularity does not keep under usual product and composition of two slice regular functions due to the non-commutativity of quaternions. For example, the subordination argument in [29] is not suitable to prove Theorem 5.4. In addition, the non-commutativity of quaternions forces us to formulate new auxiliary functions as in the proof of Theorem 3.11 and to explore new methods as in Theorems 5.4 and 1.5. Especially, Proposition 3.6 sets up a bridge of point-wise and regular multiplication for slice regular functions. The remaining part of this paper is organized as follows: In Sect. 2, we set up basic notation and give some preliminary results from the theory of slice regular functions over quaternions. At the end of this section, we introduce the concepts of slice starlike functions of order $\alpha$ and slice close-to-convex functions. In Sect. 3, we first offer some examples in the space $S^*$ and establish some useful lemmas to prove Theorems 1.1 and 1.2. We also recall a Schwarz lemma from [19] and establish the Rogosinski lemma for slice regular functions which is a sharpened form of the Schwarz lemma. In Sect. 4, we give a coefficient estimate for the Carathéodory class in the quaternionic setting and prove Theorems 1.1 and 1.2. In Sect. 5, we shall establish the growth and distortion theorems for some classes of slice regular functions in the quaternionic setting corresponding to Theorem B. Moreover, we formulate a result of Hayman (see Theorem 5.7) which is a precise version of the growth theorem. In Sect. 6, we shall give some results involving the radius problems for slice regular functions. Specially, we establish the quaternionic analogs of Bohr theorem and Rogosinski theorem which are of independent interest. As an application of the growth theorem, the Koebe type one-quarter theorem for the class $S^*$ is also obtained. Finally, we prove Theorem 1.5 by Proposition 3.6 which allows to establish a Bloch-Landau theorem for slice regular functions with convex image.

Theorem 2.3. Let ^f : ^B ^→ ^H be a slice regular function. Then $$f(q) = \sum_{n=0}^{\infty} q^n a_n$$ , with $a_n = \frac{f^{(n)}(0)}{n!}$ for all ^q ^∈ ^B. In fact, there are some limitations in the expansion given in Theorem [2.3.](#page-6-0) Hence, a new type of expansion occurs in [\[47\]](#page-24-16).

Proposition 2.7. Let f and g be slice regular functions on $\mathbb{B}$ . Then for all $q \in \mathbb{B} \setminus \mathcal{Z}_{f^s}$ , $$f^{-} g(q) = f(T_f(q))^{-1}g(T_f(q)),$$ where $T_f: \mathbb{B} \setminus \mathcal{Z}_{f^s} \to \mathbb{B} \setminus \mathcal{Z}_{f^s}$ is defined by $T_f(q) = f^c(q)^{-1}qf^c(q)$ . Furthermore, $T_f$ and $T_{f^c}$ are mutual inverses so that $T_f$ is a diffeomorphism. In one complex variable, the starlike function of order $\alpha \in [0, 1]$ was first introduced by Robertson [43]. Now we introduce its corresponding version for slice regular functions.

Theorem 3.2. Let $F: \mathbb{D} \to \mathbb{C}$ be a holomorphic function with F(0) = 0 and F'(0) = 1. Then F is starlike with respect to 0 if and only if $\operatorname{Re} \frac{zF'(z)}{F(z)} > 0$ on $\mathbb{D}$ . We recall also a so-called convex combination identity for slice regular functions which was used to establish sharp growth and distortion theorems for a subclass of slice regular functions (see [41] or [42]).

Lemma 3.3. Let f be a slice regular function on $\mathbb{B}$ such that $f(\mathbb{B}_I) \subseteq \mathbb{C}_I$ for some $I \in \mathbb{S}$ . Then for every $re^{J\theta} \in \mathbb{B}$ with $J \in \mathbb{S}$ , $$\left| f(re^{J\theta}) \right|^2 = \frac{1 + \langle I, J \rangle}{2} \left| f(re^{I\theta}) \right|^2 + \frac{1 - \langle I, J \rangle}{2} \left| f(re^{-I\theta}) \right|^2$$ where $\langle I, J \rangle = -\text{Re}(IJ) \in [-1, 1].$ From Theorem 3.2, Lemmas 3.3 and 3.7 below, one can show that Example 3.4. Fix $I \in \mathbb{S}$ . Let $F(z) = z + \sum_{n=2}^{+\infty} z^n a_n, a_n \in \mathbb{C}_I$ for n = 2, 3, 4... be a starlike function on $\mathbb{B}_I \simeq \mathbb{D}$ . Then $f(q) = q + \sum_{n=2}^{+\infty} q^n a_n \in \mathcal{S}^*$ . Example 3.5. Let $f \in \mathcal{S}$ be such that (3.3) $$\operatorname{Re}\left(\left(f(q) - f(-q)\right)^{-1} q f'(q)\right) > 0, \quad \forall \ q \in \mathbb{B}.$$ Then $f \in \mathcal{C}$ .

Proposition 3.6. Let f, g be two slice regular functions on $\mathbb{B}$ such that $\mathcal{Z}_f = \emptyset$ . Then we have (3.4) $$\operatorname{Re}\left(f(q)^{-1}q(q)\right) > 0 \text{ on } \mathbb{B} \Leftrightarrow \operatorname{Re}\left(f(q)^{-} q(q)\right) > 0 \text{ on } \mathbb{B},$$ and $$(3.5) |q(q)| < |f(q)| \text{ on } \mathbb{B} \Leftrightarrow |f(q)^{-} q(q)| < 1 \text{ on } \mathbb{B}.$$

Lemma 3.7. Let f be a slice regular function on $\mathbb{B}$ such that $\mathcal{Z}_f = \{0\}$ and f'(0) = 1. Then the following statements are equivalent: - (1) Re $(f(q)^{-1}qf'(q)) > \alpha$ on $\mathbb{B} \setminus \{0\}$ ; - (2) Re $(f(q)^{-} (qf'(q))) > \alpha$ on $\mathbb{B} \setminus \{0\}$ ; - (3) For any $u \in \partial \mathbb{B}$ , the function $M(r) = |f(ru)|/r^{\alpha}$ is strictly increasing on (0,1).

Corollary 3.8. Let F be holomorphic on D such that F(0) = 0 and F ′ (0) = 1. Then the following statements are equivalent. - (1) F is starlike with respect to 0; - (2) Re F(z) −1 zF′ (z)) > 0 on D; - (3) for any ^θ ^∈ [0, ²π), the function ^M(r) = ^|F(reiθ)^| is strictly increasing on [0, 1). Remark 3.9. Note that condition (2) in Corollary [3.8](#page-10-0) would imply that F is injective on D. From the formula $$\operatorname{Re}\left(\frac{zF'(z)}{F(z)}\right) = \frac{\partial}{\partial \theta} \operatorname{arg} F(re^{i\theta}), \quad z = re^{i\theta},$$ it follows that the argument of ^F(reiθ) increases monotonically for ^θ. Thus ^F is injective on ^|z^| ⁼ r < 1. By the principle of univalence on the boundary, we can conclude that F is injective on |z| ≤ r. Since r < 1 is arbitrary, F is injective on D (see e.g., [\[28,](#page-24-12) p. 38]). However, this approach is not suitable for the quaternionic setting. Now we state without proof a variant of the Schwarz lemma for slice regular functions in [\[19,](#page-23-1) Theorem 4.1] which is useful in the sequel.

Theorem 3.10. If ^f : ^B ^→ ^B be slice regular such that ^f(0) = ^f ′ (0) = . . . = f (m−1)(0) = 0 (^m ^≥ 1). Then $$|f(q)| \le |q|^m, \quad \forall \ q \in \mathbb{B},$$ and $$|f^{(m)}(0)| \le m!.$$ Both inequalities are strict (except q = 0) unless f(q) = q ^m^u for some ^u ^∈ ^∂B. In fact, we can establish a sharpened form of the Schwarz lemma for slice regular functions. Its complex version is due to Rogosinski (see e.g., [\[13,](#page-23-14) p. 200]). For $$q_0 \in \mathbb{H}$$ , denote $B(q_0, r) = \{q \in \mathbb{H} : |q - q_0| < r\}$ .

Theorem 3.11. Let $q_0, b \in \mathbb{B}$ . For the set of all slice regular functions $f : \mathbb{B} \to \mathbb{B}$ with f(0) = 0 and f'(0) = b, the range of values of $f(q_0)$ is the closed ball $\overline{B(c,r)}$ , where $$c = \frac{q_0 b(1 - |q_0|^2)}{1 - |q_0 b|^2}, \quad r = \frac{|q_0|^2 (1 - |b|^2)}{1 - |q_0 b|^2}.$$

Corollary 3.12. Let f be a slice regular function on $\mathbb{B}$ with f(0) = 0 and |f(q)| < 1 for all $q \in \mathbb{B}$ . Then $$|q| \frac{|f'(0)| - |q|}{1 - |qf'(0)|} \le |f(q)| \le |q| \frac{|q| + |f'(0)|}{1 + |qf'(0)|}, \quad \forall \ q \in \mathbb{B}.$$ Equality holds at some point $q_0 \neq 0$ if and only if $f(q) = q\varphi_a(q)u$ for some $a \in \mathbb{B}, u \in \partial \mathbb{B}$ . As an important generalization of the Schwarz lemma, the classical Schwarz-Pick lemma states that if F is a holomorphic self-mapping on $\mathbb{D}$ , then (3.10) $$|F'(z)| \le \frac{1 - |F(z)|^2}{1 - |z|^2}, \quad \forall z \in \mathbb{D}.$$ However, this classical version fails even for the slice regular automorphisms. Indeed, we take the slice regular Möbius transformation (see [46, Theorem 7.6]) $$\varphi_a(q) = (1 - q\overline{a})^{-} (a - q) = a - (1 - |a|^2) \sum_{n=1}^{+\infty} q^n \overline{a}^{n-1}.$$ Pick $a = \frac{i}{2}$ and $q_0 = \frac{i}{2}$ . By direct computation, we have $$\varphi_a(q_0) = \frac{2}{5}(i-j), \qquad \varphi_a'(q_0) = -\frac{204}{225} - \frac{96}{225}k,$$ so that $$|\varphi_a'(q_0)| = \sqrt{\frac{50832}{50625}} > \frac{1 - |\varphi_a(q_0)|^2}{1 - |q_0|^2} = \frac{68}{75}.$$ Fortunately, a variant of the Schwarz-Pick lemma for self-mapping of $\mathbb{B}$ was established in [5]. As a special case, one can obtain a coefficient estimate.

Lemma 3.13. Let f be a slice regular function on $\mathbb{B}$ such that $|f(q)| \leq 1$ for all $q \in \mathbb{B}$ . Then $$(3.11) |f'(0)| \le 1 - |f(0)|^2.$$ Equality holds in (3.11) if and only if f is of the form $$f(q) = (1 - q\overline{a})^{-} (a - q)u, \quad \forall \ q \in \mathbb{B},$$ for some $a \in \overline{\mathbb{B}}, u \in \partial \mathbb{B}$ .

Theorem 4.1. Let $f(q) = 1 + \sum_{n=1}^{\infty} q^n a_n$ be a function in $\mathcal{P}$ . Then $$\left| a_2 - \frac{a_1^2}{2} \right| \le 2 - \frac{|a_1|^2}{2}.$$ Equality holds in (4.1) if and only if $$f(q) = (q\varphi_a(q)u + 1) (1 - q\varphi_a(q)u)^{-},$$ for some $a \in \overline{\mathbb{B}}$ and $u \in \partial \mathbb{B}$ . Proof. Define $$g(q) = \begin{cases} q^{-1}(f(q)+1)^{-} (f(q)-1), & \text{if } q \in \mathbb{B} \setminus \{0\}, \\ a_1/2, & \text{if } q = 0. \end{cases}$$ From the assumption of f, we have $$|(f(q)+1)^{-1}(f(q)-1)| < 1, \forall q \in \mathbb{B},$$ then, by Proposition 3.6, the function $h(q) = (f(q) + 1)^{-} (f(q) - 1)$ is slice regular on $\mathbb{B}$ with $h(0) = 0, h'(0) = a_1/2$ and |h(q)| < 1 for all $q \in \mathbb{B}$ . By the maximum modulus principle for slice regular functions (cf. [22, Theorem 7.1]), we obtain that the function g is a slice regular function on $\mathbb{B}$ with $g'(0) = a_2/2 - a_1^2/4$ and $|g(q)| \le 1$ for all $q \in \mathbb{B}$ . From Lemma 3.13, (4.1) follows and equality holds in (4.1) if and only if $g(q) = \varphi_a(q)u$ for some $a \in \overline{\mathbb{B}}$ and $u \in \partial \mathbb{B}$ . By simple calculations, we deduce that $$f(q) = (q\varphi_a(q)u + 1) (1 - q\varphi_a(q)u)^{-}.$$ Now the proof is complete. As a direct consequence, we obtain all coefficient estimates for the Carathéodory class $\mathcal{P}$ applying the approach of finite average as in the case of one complex variable. Now let us recall this result in [39].

Theorem 4.2. Let $f(q) = 1 + \sum_{n=1}^{\infty} q^n a_n$ be a slice regular function in $\mathcal{P}$ . Then (4.2) $$\frac{1 - |q|}{1 + |q|} \le \text{Re}f(q) \le |f(q)| \le \frac{1 + |q|}{1 - |q|}, \quad \forall \ q \in \mathbb{B},$$ and $$(4.3) |a_n| \le 2, n = 1, 2, \dots.$$ Moreover, $|a_1| = 2$ or equality holds for the first or third inequality in (4.2) at some $q_0 \neq 0$ if and only if $$f(q) = (1 - qu)^{-} (1 + qu), \quad \forall \ q \in \mathbb{B},$$ for some $u \in \partial \mathbb{B}$ . Now we can prove our main coefficient estimates. Proof of Theorem 1.1. Define $$p(q) = \begin{cases} (q^{-1}f(q))^{-} f'(q) & \text{if } q \in \mathbb{B} \setminus \{0\}, \\ 1, & \text{if } q = 0, \end{cases}$$ which belongs to $\mathcal{P}$ by Lemma 3.7. Set $p(z) = 1 + qp_1 + \cdots$ . Applying Theorem 4.2 to function p, we have $|p_n| \leq 2$ for all $n \geq 1$ . Comparing the coefficients in the power series of f'(q) and $q^{-1}f(q) * p(q)$ , we have $$(4.4) na_n = p_{n-1} + a_2 p_{n-2} + \dots + a_{n-1} p_1 + a_n, \quad n = 2, 3, \dots$$ By induction, we can obtain $|a_n| \leq n$ for all $n = 2, 3, \ldots$ It is easy to see that if $|a_n| = n$ for a given n, then the above arguments imply that $|p_1| = |a_2| = 2$ , and thus, by Theorem 4.2, p is of the form $$p(q) = (1 - qu)^{-} (1 + qu) = 1 + \sum_{n=1}^{+\infty} q^n 2u^n, \quad \forall \ q \in \mathbb{B}$$ for some $u \in \partial \mathbb{B}$ . Hence, by (4.4), we have $$(n-1)a_n = 2(u^{n-1} + a_2u^{n-2} + \dots + a_{n-1}u), \quad n = 2, 3, \dots,$$ which leads to that $a_n = nu^{n-1}$ for all n = 2, 3, ..., as desired. Note that Theorem 1.1 generalizes Theorem 2.2.16 in [28] to the non-commutative algebra. Nevanlinna established the complex version of Theorem 1.1 for normalized starlike functions in 1920. In fact, the Bieberbach conjecture over quaternions still holds for slice close-to-convex functions.

Theorem 4.3. Let $f(q) = q + \sum_{n=2}^{+\infty} q^n a_n \in \mathcal{C}$ . Then $$|a_n| \leq n$$ , for all $n = 2, 3, \ldots$ Equality $|a_n| = n$ for a given $n \geq 2$ holds if and only if $$f(a) = a(1 - au)^{-*2}, \quad \forall \ a \in \mathbb{B},$$ for some $u \in \partial \mathbb{B}$ .

Theorem 5.1. Let $p(q) = 1 + \sum_{n=1}^{\infty} q^n a_n$ be a slice regular function in $\mathcal{P}$ . Then (5.1) $$\frac{1 - |q|^2}{1 + |a_1 q| + |q|^2} \le \operatorname{Re} p(q) \le |p(q)| \le \frac{1 + |a_1 q| + |q|^2}{1 - |q|^2}, \quad \forall \ q \in \mathbb{B}.$$ Moreover, equality holds for the first or third inequality in (5.1) at some $q_0 \neq 0$ if and only if $$p(q) = (q\varphi_a(q)u + 1) (1 - q\varphi_a(q)u)^{-}, \qquad \forall \ q \in \mathbb{B},$$ for some $u \in \partial \mathbb{B}$ . Proof. Consider the slice regular function $$f(q) = (p(q) + 1)^{-} (p(q) - 1), \quad \forall \ q \in \mathbb{B}.$$ Note that f(0) = 0, $f'(0) = a_1/2$ and |f(q)| < 1 for all $q \in \mathbb{B}$ by Proposition 3.6. then it follows that, by Corollary 3.12, (5.2) $$|f(q)| \le |q| \frac{|q| + |f'(0)|}{1 + |qf'(0)|} =: \mu, \quad \forall \ q \in \mathbb{B}.$$ By Proposition 2.7, it follows that $$\frac{|p(q)+1|}{|p(q)-1|} \le \mu, \quad \forall \ q \in \mathbb{B}.$$ i.e., $$\left| p(q) - \frac{1+\mu^2}{1-\mu^2} \right| \le \frac{2\mu}{1-\mu^2}, \quad \forall \ q \in \mathbb{B}.$$ Hence, (5.1) follows from $$|p(q)| \le \frac{1+\mu^2}{1-\mu^2} + \frac{2\mu}{1-\mu^2} = \frac{1+|qa_1|+|q|^2}{1-|q|^2}, \quad \forall \ q \in \mathbb{B}.$$ and $$\operatorname{Re} p(q) \ge \frac{1+\mu^2}{1-\mu^2} - \frac{2\mu}{1-\mu^2} = \frac{1-|q|^2}{1+|qa_1|+|q|^2}, \quad \forall \ q \in \mathbb{B}.$$ If one of equalities above occurs for some point $q_0 \neq 0$ , then (5.2) turns into an equality at $\widetilde{q}_0 = T_{p^c+1}(q_0) = (p(q_0)+1)^{-1}q_0(p(q_0)+1)$ . Whence Corollary 3.12 implies that $$f(q) = q\varphi_a(q)u, \quad \forall \ q \in \mathbb{B},$$ for some $u \in \partial \mathbb{B}$ . Thus $$p(q) = (q\varphi_a(q)u + 1) (1 - q\varphi_a(q)u)^{-}, \quad \forall \ q \in \mathbb{B},$$ for some $u \in \partial \mathbb{B}$ . The converse part can be verified easily by direct calculations. Now the proof is complete. Now we can establish the growth and distortion theorems for $\mathcal{S}^*$ .

Theorem 5.2. Let $f(q) = q + \sum_{n=2}^{\infty} q^n a_n \in \mathcal{S}^*$ . Then the following inequalities hold for all $q \in \mathbb{B}$ (5.3) $$\frac{|q|}{1+|a_2q|+|q|^2} \le |f(q)| \le \frac{|q|}{1-|q|^2} \left(\frac{1+|q|}{1-|q|}\right)^{\frac{|a_2|}{2}};$$ (5.4) $$\frac{1 - |q|^2}{(1 + |a_2q| + |q|^2)^2} \le |f'(q)| \le \frac{1 + |a_2q| + |q|^2}{(1 - |q|^2)^2} \left(\frac{1 + |q|}{1 - |q|}\right)^{\frac{|a_2|}{2}};$$ (5.5) $$\frac{1 - |q|^2}{1 + |a_2 q| + |q|^2} \le \frac{|qf'(q)|}{|f(q)|} \le \frac{1 + |a_2 q| + |q|^2}{1 - |q|^2}.$$

Theorem 5.3. Let $f \in \mathcal{S}^*$ . Then the following inequalities hold for all $q \in \mathbb{B}$ (5.8) $$\frac{|q|}{(1+|q|)^2} \le |f(q)| \le \frac{|q|}{(1-|q|)^2};$$ (5.9) $$\frac{1-|q|}{(1+|q|)^3} \le |f'(q)| \le \frac{1+|q|}{(1-|q|)^3};$$ (5.10) $$\frac{1-|q|}{1+|q|} \le \frac{|qf'(q)|}{|f(q)|} \le \frac{1+|q|}{1-|q|}.$$ All of these estimates are sharp. For each $q \in \mathbb{B} \setminus \{0\}$ , equality occurs if $$f(q) = q(1 - qu)^{-*2}, \quad q \in \mathbb{B},$$ for some $u \in \partial \mathbb{B}$ . As a consequence of (5.8), Theorem 1.4 follows. Now we establish a more general version.

Theorem 5.4. Let m be a positive integer and $f(q) = q + \sum_{n=m+1}^{+\infty} q^n a_n$ be a slice regular function on $\mathbb{B}$ such that $qf'(q) \in \mathcal{S}^*$ . Then $$\frac{1}{(1+|q|^m)^{2/m}} \le |f'(q)| \le \frac{1}{(1-|q|^m)^{2/m}}, \quad \forall \ q \in \mathbb{B}.$$

Theorem 5.5. Let $f(q) = q + \sum_{n=m+1}^{+\infty} q^n a_n \in \mathcal{S}^*$ for some positive integer m. Then $$\frac{|q|}{(1+|q|^m)^{2/m}} \le |f(q)| \le \frac{|q|}{(1-|q|^m)^{2/m}}, \quad \forall \ q \in \mathbb{B}.$$ In fact, estimates on the right side of [\(5.8\)](#page-17-1) and [\(5.9\)](#page-17-2) can be directly obtained from coefficient estimates in Theorem [1.1.](#page-2-0) Equality on the right side of [\(5.8\)](#page-17-1) or [\(5.9\)](#page-17-2) occurs for some ^q⁰ ^∈ ^B \ {0}, if and only if $$f(q) = q(1 - qu)^{-*2}, \quad q \in \mathbb{B},$$ for some ^u ^∈ ^∂B. Moreover, those two estimates hold in a larger class. Let f, g be two slice regular functions on ^B. Write ^f ^≺^N ^g if there exists ^w ∈ N (B) with ^w(0) = 0 and ^|w(q)^| ^&lt; 1 for all ^q ^∈ ^B such that ^f(q) = ^g(w(q)) for all ^q ^∈ ^B (see [\[16,](#page-23-12) Definition 2.10]). Note that the composition of slice regular functions, when defined, does not give generally a slice regular function. However, if ^w ∈ N (B), then the composition f ◦ w is slice regular for any slice regular function f. Since the proof of Theorem [5.6](#page-19-1) below is similar to the classical case (see e.g., [\[13,](#page-23-14) p. 202]), we omit it here.

Theorem 5.6. Let f, g be two slice regular functions on ^B. If ^f ∈ C and ^g ^≺^N ^f, then the following inequalities hold for all ^q ^∈ ^B $$|g(q)| \le \frac{|q|}{(1-|q|)^2}$$ and $|g'(q)| \le \frac{1+|q|}{(1-|q|)^3}$ . In fact, we can obtain more information related to [\(5.8\)](#page-17-1). Note that the author in [\[31\]](#page-24-24) established the Hayman's regularity theorem making use of complex version of Theorem [5.7](#page-19-0) for the p-univalent holomorphic function.

Theorem 5.7. Let ^f ∈ S^∗ and M∞(r, f) = max^|q|=^r |f(q)| for r ∈ (0, 1). Then the function φ(r) = 1 r (1− r) ²M∞(r, f) is decreasing on (0, 1) and hence tends to a limit ^α ^∈ [0, 1]. For any ^α ^∈ [0, 1], there exists ^u ^∈ ^∂^B such that $$\lim_{r \nearrow 1^{-}} (1-r)^2 |f(ru)| = \alpha.$$ Proof. From the proof in Theorem [5.3,](#page-17-3) we have (5.12) $$\frac{|f(r_2u)|}{|f(r_1u)|} \le \frac{r_2(1-r_1)^2}{r_1(1-r_2)^2}.$$ Choose ^u ^∈ ^∂^B such that ^M∞(r2, f) = ^|f(r2u)|. Whence the above inequality yields that $$\frac{1}{r_2}(1-r_2)^2 M_{\infty}(r_2, f) \le \frac{1}{r_1}(1-r_1)^2 |f(r_1 u)| \le \frac{1}{r_1}(1-r_1)^2 M_{\infty}(r_1, f),$$ that is to say φ(r) decreases on (0, 1) and hence tends to a limit α. From [\(5.8\)](#page-17-1), there holds that φ(r) ∈ [0, 1], thus α ∈ [0, 1]. Let {rn} be a sequence increasing to 1 and choose ^uⁿ ^∈ ^∂^B such that ^M∞(rn, f) = ^|f(rnun)|. From the compactness of ^∂B, there exists a cluster point ^u^∞ ^∈ ^∂B. From [\(5.12\)](#page-19-2), we have, for r < rn, $$\alpha \le \frac{1}{r_n} (1 - r_n)^2 |f(r_n u_n)| \le \frac{1}{r} (1 - r)^2 |f(r u_n)|.$$ Letting n → ∞, we obtain $$\alpha \le \frac{1}{r}(1-r)^2|f(ru_{\infty})| \le \frac{1}{r}(1-r)^2M_{\infty}(r,f),$$ as desired.

Theorem 6.1. Let $f(q) = \sum_{n=0}^{+\infty} q^n a_n$ be a slice regular function on $\mathbb{B}$ such that $f(q) \in \Pi := \{q \in \mathbb{H} : \text{Re } q \leq 1\}$ for all $q \in \mathbb{B}$ . Then (6.2) $$\sum_{n=1}^{+\infty} |q^n a_n| \le \operatorname{dist}(f(0), \partial \Pi), \quad |q| \le \frac{1}{3}.$$

Lemma 6.2. Let $g_{m,r}(z) = \sum_{n=0}^{m} r^n z^n, z \in \mathbb{D}, r \in \mathbb{R}$ . Then $$\operatorname{Re} g_{m,r}(z) \geq \frac{1}{2}$$ for all $z \in \mathbb{D}$ and $r \in (0, r_m)$ , where $r_1 = 1/2, r_2 = \sqrt{3/8}$ , and $r_m$ for $m \geq 3$ is the unique positive solution of the equation $$1 - r - 2r^{m+1} = 0.$$ Moreover, $$1 - \frac{\log m}{m} < r_m < 1 - \frac{1}{m}, \quad m \ge 3.$$ Based on Lemma 6.2, we now establish the quaternionic analogy of the Rogosinski theorem as follows.

Theorem 6.3. Let $f = g + \overline{h} : \mathbb{B} \to \mathbb{H}$ be such that the image $f(\mathbb{B})$ is convex, where $g(q) = \sum_{n=0}^{+\infty} q^n a_n$ and $h(q) = \sum_{n=0}^{+\infty} q^n b_n$ are slice regular functions on $\mathbb{B}$ . Then each partial sum $$S_m(q) = \sum_{n=0}^{m} q^n a_n + \overline{\sum_{n=0}^{m} q^n b_n}$$ maps $B(0, r_m)$ into $f(\mathbb{B})$ , where $r_m$ are given by Lemma 6.2.

Theorem 6.4. Let $f \in \mathcal{S}^*$ , then it holds that $$B(0,\frac{1}{4}) \subset f(\mathbb{B}).$$ The estimate is precise.

Theorem 6.5. Let $f(q) = q + \sum_{n=3}^{+\infty} q^n a_n \in \mathcal{S}^*$ , then it holds that $$B(0,\frac{1}{2}) \subset f(\mathbb{B}).$$ 6.4. Bloch-Landau Theorem. In complex analysis, a well-known result of Bloch-Landau theorem says that if F is a holomorphic function in $\mathbb{D}$ with the only restriction F'(0) = 1, then the image of F contains a disc of radius $r \geq b$ , where b is an absolute constant. Now we introduce its corresponding version of slice regular functions. For $a \in \mathbb{B}$ and the slice regular function f on $\mathbb{B}$ with f'(0) = 1, denote by r(a, f) the radius of the largest ball contained in the image of f centered at f(a) and denote $r(f) = \sup\{r(a, f), a \in \mathbb{B}\}$ . In [48], we have proved that r(f) > 0.23 for any slice regular function f with Bloch seminorm $\sup_{q \in \mathbb{B}} (1 - |q|^2) |f'(q)| = 1$ . In this paper, we restrict functions f in r(f) to be slice regular functions with convex image. Define (6.4) $$C = \inf\{r(f) : f \text{ is slice regular on } \mathbb{B} \text{ such that } f'(0) = 1 \text{ and } f(\mathbb{B}) \text{ convex } \}.$$ Let us prove Theorem 1.5 before giving the upper and lower bounds of C. Proof of Theorem 1.5. Let f be as described in the theorem and assume that f(0) = 0 for otherwise we can consider the slice regular function f - f(0). Let $p \in \partial f(\mathbb{B})$ be a point at minimum distance from the origin 0. From the open mapping theorem for slice regular function [17], we have |p| > 0. If $|p| = +\infty$ , the theorem has been proved. Otherwise, $0 < |p| < +\infty$ , we obtain that (6.5) $$\operatorname{Re}\left(f(q)\overline{p}\right) < |p|^2, \quad \forall \ q \in \mathbb{B},$$ since $f(\mathbb{B})$ is convex. Consider the slice regular function $$g(q) = (2|p|^2 - f(q)\overline{p})^{-} f(q)\overline{p}, \quad \forall \ q \in \mathbb{B}.$$ Then it follows that g(0) = 0 and $g'(0) = (2p)^{-1}$ . From (6.5), we have $$|f(q)\overline{p}| < |2|p|^2 - f(q)\overline{p}|, \quad \forall \ q \in \mathbb{B},$$ which implies, by (3.5) in Proposition 3.6, |g(q)| < 1 for all $q \in \mathbb{B}$ . The Schwarz lemma for slice regular functions in Theorem 3.10 implies that $|g'(0)| \le 1$ . Hence, $|p| \ge 1/2$ , which shows the image $f(\mathbb{B})$ contains the open ball B(0, 1/2). To see that the constant 1/2 is optimal, we consider the slice regular function given by $$f(q) = q(1-q)^{-1}, \quad \forall \ q \in \mathbb{B}.$$ It is easy to show that f'(0) = 1 and $f(\mathbb{B}) = \{q \in \mathbb{H} : \operatorname{Re} q > -1/2\}$ is convex and contains the open ball centered at 0 of radius 1/2 while 1/2 is optimal, as desired. As a consequence of Theorem 1.5, we give a Bloch-Landau theorem for slice regular functions with convex image.

Theorem 6.6. The constant C given in (6.4) has the following estimate $$\frac{1}{2} \le C \le \frac{\pi}{4}.$$ Proof. From Theorem [1.5,](#page-4-2) it suffices to show that C ≤ π/4. To this end, let us consider the slice regular function $$f(q) = \sum_{n=0}^{+\infty} \frac{q^{2n+1}}{2n+1}, \quad q \in \mathbb{B},$$ with f ′ (0) = 1. > Note that the image ^f(B) = {^q ^∈ ^H : ^|Im ^q^| ^&lt; π 4 } is convex, the desired result follows. We have proved the Bieberbach conjecture over quaternions for slice close-to-convex functions. It is still an open problem if the Bieberbach conjecture holds true for injective slice regular functions. Open question : Does it hold that ^|an| ≤ n, n = 2, ³, . . . , provided that ^f(q) = ^q + Σ⁺^∞ ⁿ=2q n aⁿ is an injective slice regular function in the open unit ball B?

Definition 2.1. Let $\Omega$ be a domain in $\mathbb{H}$ . A function $f:\Omega\to\mathbb{H}$ is called (left) slice regular if, for all $I\in\mathbb{S}$ , its restriction $f_I$ to $\Omega_I$ is holomorphic, i.e., it has continuous partial derivatives and satisfies $$\bar{\partial}_I f(x+yI) := \frac{1}{2} \left( \frac{\partial}{\partial x} + I \frac{\partial}{\partial y} \right) f_I(x+yI) = 0$$ for all x + yI ∈ Ω^I .

Definition 2.2. Let ^f : ^B ^→ ^H be a slice regular function. For each ^I ^∈ ^S, the ^I-derivative of ^f at q = x + yI is defined by $$\partial_I f(x+yI) := \frac{1}{2} \left( \frac{\partial}{\partial x} - I \frac{\partial}{\partial y} \right) f_I(x+yI)$$ on B^I . The slice derivative of f is the function f ′ defined by ^∂^I ^f on ^B^I for all ^I ^∈ ^S. All slice regular functions on B can be expressed as power series.

Definition 2.4. Let ^f, ^g : ^B ^→ ^H be two slice regular functions of the form $$f(q) = \sum_{n=0}^{\infty} q^n a_n, \qquad g(q) = \sum_{n=0}^{\infty} q^n b_n.$$ The regular product (or ∗-product) of f and g is the slice regular function defined by $$f * g(q) = \sum_{n=0}^{\infty} q^n \left( \sum_{k=0}^n a_k b_{n-k} \right).$$

Definition 2.5. Let ^f(q) = ^P^∞ n=0 q n aⁿ be a slice regular function on B. We define the regular conjugate of f as $$f^c(q) = \sum_{n=0}^{\infty} q^n \bar{a}_n,$$ and the symmetrization of f as $$f^{s}(q) = f f^{c}(q) = f^{c} f(q) = \sum_{n=0}^{\infty} q^{n} \left( \sum_{k=0}^{n} a_{k} \bar{a}_{n-k} \right).$$ Both f c and f s are slice regular functions on B. Let Zf^s denote the zero set of the symmetrization f s of f.

Definition 2.6. Let f be a slice regular function on B. If f does not vanish identically, its regular reciprocal is the function defined by $$f^{-*}(q) := f^{s}(q)^{-1} f^{c}(q)$$ which is slice regular on ^B \ Zf^s .

Definition 2.8. Let $\alpha < 1$ . We shall say that f is a slice starlike function of order $\alpha$ in the unit ball $\mathbb{B}$ if the slice regular function $f \in \mathcal{S}$ satisfies $$\operatorname{Re}\left(f(q)^{-1}qf'(q)\right) > \alpha, \quad q \in \mathbb{B} \setminus \{0\}.$$ Specially, we shall say that f is a slice starlike function for $\alpha = 0$ . Let $S^(\alpha)$ denote the class of slice starlike function of order $\alpha$ and write $S$ instead when $\alpha = 0$ . With the concept of slice starlike functions, we can introduce the concept of slice close-to-convex functions given by $$\mathcal{C} = \{ f : \mathbb{B} \to \mathbb{H} : \exists \ h \in \mathcal{S}^* \text{ such that } \operatorname{Re} \left( h(q)^{-1} q f'(q) \right) > 0 \text{ on } \mathbb{B} \}.$$ Note that the inclusion $\mathcal{S}^* \subset \mathcal{C}$ holds. In one complex variable case, a holomorphic function f in $\mathcal{C}$ is the so-called close-to-convex function introduced by Kaplan. In [32, Theorem 1], it was proven that every close-to-convex function is injective which satisfies naturally the Bieberbach conjecture. In fact, this result was proved early by Reade in 1955 (see e.g., [34, 38]). Note that we have omitted the word "left" for simplicity in the article. Certainly, the right slice regular functions have completely analogous theories. This observation shall be used in the proof of Theorem 3.11.