Uniformly locally univalent harmonic mappings associated with the pre-Schwarzian norm

Theorem 2.1. Let $f = h + \overline{g}$ be a sense-preserving harmonic mapping in $\mathbb{D}$ . Then either $||P_{h+\varepsilon g}|| = ||P_f|| = \infty$ or both $||P_{h+\varepsilon g}||$ and $||P_f||$ are finite for each $\varepsilon \in \overline{\mathbb{D}}$ . If $||P_f|| < \infty$ , then the inequality $$(2.1)$$ holds for each $\varepsilon \in \overline{\mathbb{D}}$ . In particular, $$|||P_h|| - ||P_f||| \le 1.$$ The constant 1 is sharp in the two estimates.

Corollary 2.1. Let $f = h + \overline{g}$ be a sense-preserving harmonic mapping in $\mathbb{D}$ . If $||P_h|| < \infty$ , then for any $\varepsilon_1$ , $\varepsilon_2 \in \overline{\mathbb{D}}$ , we have the following inequalities. - (1) The sharp inequality $|||P_{h+\varepsilon_1g}|| ||P_{h+\varepsilon_2g}||| \le 2$ holds. - (2) If $|\varepsilon_1| = |\varepsilon_2|$ , then $||P_{h+\varepsilon_1\overline{g}}|| = ||P_{h+\varepsilon_2\overline{g}}||$ . If $|\varepsilon_1| \neq |\varepsilon_2|$ , then $$||P_{h+\varepsilon_1\overline{g}}|| - ||P_{h+\varepsilon_2\overline{g}}|| \le |\varepsilon_1| + |\varepsilon_2| < 2.$$ (3) If $|\varepsilon_1| \leq |\varepsilon_2|$ , then we have the sharp inequality $|||P_{h+\varepsilon_1g}|| - ||P_{h+\varepsilon_2\overline{g}}||| \leq 1$ . If $|\varepsilon_1| > |\varepsilon_2|$ , then $$\left| ||P_{h+\varepsilon_1 q}|| - ||P_{h+\varepsilon_2 q}|| \right| < 1 + |\varepsilon_1| + |\varepsilon_2| < 3.$$

Corollary 2.2. Let $f = h + \overline{g}$ be a sense-preserving harmonic mapping in $\mathbb{D}$ . If $h + \varepsilon_1 g$ is univalent (resp. convex) in $\mathbb{D}$ for some $\varepsilon_1 \in \overline{\mathbb{D}}$ , then $||P_{h+\varepsilon\overline{g}}|| < 9$ (resp. 7) and $||P_{h+\varepsilon g}|| \le 8$ (resp. 6) for each $\varepsilon \in \overline{\mathbb{D}}$ . Furthermore, the constants 8 and 6 are sharp. Conversely, if either $||P_{h+\varepsilon_1\overline{g}}|| \ge 9$ (resp. 7) or $||P_{h+\varepsilon_2g}|| > 8$ (resp. 6) for some $\varepsilon_1$ , $\varepsilon_2 \in \overline{\mathbb{D}}$ , then $h + \varepsilon g$ is not univalent (resp. convex) in $\mathbb{D}$ for any $\varepsilon \in \overline{\mathbb{D}}$ . The harmonic Koebe function $K = h_K + \overline{g_K}$ and the harmonic half-plane mapping $L = h_L + \overline{g_L}$ still show its sharpness in the corresponding cases because $h_K(z) - g_K(z) = \frac{z}{(1-z)^2}$ is univalent in $\mathbb D$ and $h_L(z) + g_L(z) = \frac{z}{1-z}$ is univalent and convex in $\mathbb D$ , respectively.

Theorem 3.1. Let $f = h + \overline{g}$ be a sense-preserving and ULU harmonic mapping in $\mathbb{D}$ . Then we have $$(3.1) (1-|z|^2)|P_h(z)| \le 2(\alpha/t+|z|) and (1-|z|^2)|P_f(z)| \le 2(\alpha_0/t+|z|)$$ for every $z \in \mathbb{D}$ , where $$t = \begin{cases} \frac{e^{\rho(f)} - 1}{e^{\rho(f)} + 1} & \text{if } \rho(f) < \infty, \\ 1 & \text{if } \rho(f) = \infty. \end{cases}$$ In particular, if f is univalent in $\mathbb{D}$ , then $$||P_h|| \le 2(\alpha + 1)$$ and $||P_f|| \le 2(\alpha_0 + 1)$ . Proof. Suppose that $f = h + \overline{g} \in \text{ULU}$ . Then f is univalent in each hyperbolic disk $d_h(z, \rho(f))$ for every $z \in \mathbb{D}$ . Fix $z \in \mathbb{D}$ and let $\phi(\zeta) = \frac{t\zeta + z}{1 + t\overline{z}\zeta}$ ( $\zeta \in \mathbb{D}$ ), where t is defined as above. Using the Koebe transformation, we get that $$F_{1}(\zeta) = \frac{(f \circ \phi)(\zeta) - (f \circ \phi)(0)}{(f \circ \phi)_{\zeta}(0)}$$ $$= \frac{h(\phi(\zeta)) - h(z)}{th'(z)(1 - |z|^{2})} + \frac{g(\phi(\zeta)) - g(z)}{th'(z)(1 - |z|^{2})}$$ $$= H_{1}(\zeta) + \overline{G_{1}(\zeta)}$$ and $F_1 \in \mathcal{S}_H$ . A simple computation yields that $$|H_1''(0)| = t \left| (1 - |z|^2) \frac{h''(z)}{h'(z)} - 2\overline{z} \right| \le 2\alpha,$$ which implies the first inequality in (3.1). Using the affine change, we have that $$F_{2}(\zeta) = \frac{F_{1}(\zeta) - \overline{b_{1}F_{1}(\zeta)}}{1 - |b_{1}|^{2}}$$ $$= \frac{H_{1}(\zeta) - \overline{b_{1}}G_{1}(\zeta)}{1 - |b_{1}|^{2}} + \frac{\overline{G_{1}(\zeta) - b_{1}H_{1}(\zeta)}}{1 - |b_{1}|^{2}}$$ $$= H_{2}(\zeta) + \overline{G_{2}(\zeta)}$$ and $F_2 \in \mathcal{S}_H^0$ , where $b_1 = G_1'(0) = g'(z)/\overline{h'(z)}$ . Again, a straightforward computation shows that $$|H_2''(0)| = \frac{|H_1''(0) - \overline{b_1}G_1''(0)|}{1 - |b_1|^2}$$ $$= t \left| (1 - |z|^2) \frac{h''(z)\overline{h'(z)} - g''(z)\overline{g'(z)}}{|h'(z)|^2 - |g'(z)|^2} - 2\overline{z} \right|$$ $$= t|(1 - |z|^2)P_f(z) - 2\overline{z}| \le 2\alpha_0,$$ which implies the second inequality in (3.1). Next we consider stable harmonic univalent (resp. convex) mappings. A sense-preserving harmonic mapping $f = h + \overline{g}$ in $\mathbb{D}$ is called SHU (resp. SHC) if $h + \lambda \overline{g}$ is univalent (resp. convex) in $\mathbb{D}$ for every $|\lambda| = 1$ . The following result has some similarities with the classical estimate of the SD for SHU and SHC mappings in [12, Theorem 2], but the method of proof is different and so can also be adapted to prove [12, Theorem 2].

Theorem 3.2. Let $f = h + \overline{g}$ be a sense-preserving harmonic mapping in $\mathbb{D}$ . If f is SHU (resp. SHC), then we have $$||P_{h+\varepsilon g}|| \le 6 \ (resp.\ 4) \quad and \quad ||P_{h+\varepsilon \overline{g}}|| \le 6 \ (resp.\ 4)$$ for each $\varepsilon \in \overline{\mathbb{D}}$ . All estimates are sharp. Proof. If $f = h + \overline{g}$ is SHU (resp. SHC) in $\mathbb{D}$ , then both $h + \varepsilon g$ and $h + \varepsilon \overline{g}$ are univalent (resp. convex) in $\mathbb{D}$ for each $\varepsilon \in \overline{\mathbb{D}}$ (see [20]). It follows from Bieberbach's criterion (resp. [41, Theorem 1]) that $||P_{h+\varepsilon g}|| \leq 6$ (resp. 4) for each $\varepsilon \in \overline{\mathbb{D}}$ . Fix $\varepsilon \in \overline{\mathbb{D}}$ and let $f_{\varepsilon} = h + \varepsilon \overline{g}$ . For all $z_0 \in \mathbb{D}$ , it follows from [21, Lemma 1] that $P_{f_{\varepsilon}}(z_0) = P_{h-\varepsilon \overline{\omega(z_0)}g}(z_0)$ and thus, $$(1 - |z_0|^2)|P_{f_{\varepsilon}}(z_0)| = (1 - |z_0|^2)|P_{h-\varepsilon\overline{\omega(z_0)}g}(z_0)|,$$ where $\omega = g'/h'$ . This implies that $||P_{f_{\varepsilon}}|| \leq \sup_{\lambda \in \overline{\mathbb{D}}} ||P_{h+\lambda g}||$ and the assertion follows. To show that all estimates are sharp, it is enough to consider the analytic functions $$k(z) = \frac{z}{(1-z)^2}$$ and $l(z) = \frac{1+z}{1-z}$ that belong to the families of SHU and SHC mappings with $||P_k|| = 6$ and $||P_l|| = 4$ , respectively. Combining Corollary 2.1 and Theorem 3.1 (resp. Theorem 3.2), we can obtain a few similar results as that of Corollary 2.2 for univalent harmonic mappings (resp. SHU and SHC mappings). However, we do not include these statements here. Below we consider the converse of Theorem 3.1.

Theorem 3.3. Let $f = h + \overline{g}$ be a sense-preserving harmonic mapping in $\mathbb{D}$ . If $||P_f|| \le M$ , then f is univalent in the hyperbolic disk $D_h(z,t)$ for each $z \in \mathbb{D}$ . Consequently, f is ULU in $\mathbb{D}$ and its uniformly hyperbolic radius $\rho(f)$ is no less than t. Here $t = 2 \tanh^{-1} (1/(8(M+1)))$ .

Lemma 4.1. ([17, p. 44] and [40, Theorem 2]) Let f be a locally univalent analytic function in $\mathbb{D}$ . Then the following are equivalent. - (1) $f \in ULU$ ; - (2) $f \in ULC$ ; - (3) $||P_f|| < \infty$ ; - (4) $||S_f|| < \infty$ ; - (5) There exists a constant m > 0, and a univalent analytic function F in $\mathbb{D}$ such that $f' = (F')^m$ . To describe our results, we introduce the following abbreviations analogous to the paper [20]. Below, let $f = h + \overline{g}$ be a sense-preserving harmonic mapping in $\mathbb{D}$ and $\varepsilon_1$ , $\varepsilon_2 \in \overline{\mathbb{D}}$ with $\varepsilon_1 \neq \varepsilon_2$ . If $h + \varepsilon_1 \overline{g}$ is ULU (resp. ULC) in $\mathbb{D}$ if and only if $h + \varepsilon_2 \overline{g}$ is ULU (resp. ULC) in $\mathbb{D}$ , then we say that f is SHULU (resp. SHULC). Similarly, if $h + \varepsilon_1 g$ is ULU (resp. ULC) in $\mathbb{D}$ , then we say that h + g is SAULU (resp. SAULC). If $||P_{h+\varepsilon_1 \overline{g}}||$ (resp. $||S_{h+\varepsilon_1 \overline{g}}||$ ) is bounded if and only if $||P_{h+\varepsilon_2 \overline{g}}||$ (resp. $||S_{h+\varepsilon_2 \overline{g}}||$ ) is bounded, then we say that f has SBHPSN (resp. SBHSN). Similarly, if $||P_{h+\varepsilon_1 g}||$ (resp. $||S_{h+\varepsilon_1 g}||$ ) is bounded if and only if $||P_{h+\varepsilon_2 g}||$ (resp. $||S_{h+\varepsilon_2 g}||$ ) is bounded, then we say that h + g has SBAPSN (resp. SBASN).

Theorem 4.1. (Equivalent conditions) Let $f = h + \overline{g}$ be a sense-preserving harmonic mapping in $\mathbb{D}$ . Then the following conditions are equivalent. - (1) h + q is SAULU; - (2) h + g is SAULC; - (3) h + g has SBAPSN; - (4) h + q has SBASN; - (5) For any two points $\varepsilon_1$ , $\varepsilon_2 \in \overline{\mathbb{D}}$ with $\varepsilon_1 \neq \varepsilon_2$ , there exists a constant $m_1 > 0$ , and a univalent analytic function $F_1$ such that $(h + \varepsilon_1 g)' = (F_1')^{m_1}$ if and only if there exists a constant $m_2 > 0$ , and a univalent analytic function $F_2$ such that $$(h + \varepsilon_2 g)' = (F_2')^{m_2};$$ - (6) f is SHULU; - (7) f is SHULC; - (8) f has SBHPSN: - (9) f has SBHSN.

Proposition 5.1. For the functions $H_{a,b}$ and $H_a$ defined by (5.2), we have the following properties: - (1) $||P_{H_{a,b}}|| = 2 \max\{|a|, |b|\}$ . Thus, if $\max\{|a|, |b|\} \le 1/2$ , then the functions $H_{a,b}$ are univalent in $\mathbb{D}$ . If $\max\{|a|, |b|\} > 3$ , then the functions $H_{a,b}$ are not univalent in $\mathbb{D}$ . - (2) If $\min\{|a|,|b|\} + |a-b| \le 1$ , then the functions $H_{a,b}$ are close-to-convex and univalent in $\mathbb{D}$ . - (3) If $a \leq 0 \leq b \leq a+3$ , then the functions $H_{a,b}$ are convex in one direction and univalent in $\mathbb{D}$ . Furthermore, if $a \leq 0 \leq b \leq a+2$ , then the functions $H_{a,b}$ are convex in $\mathbb{D}$ . - (4) The function $H_a$ is univalent in $\mathbb{D}$ if and only if $|a| \leq 1$ .

Proposition 5.2. For all $\theta \in \mathbb{R}$ , the family of harmonic mappings $F_{a,b,\theta}$ defined by (5.1) has the following properties: - (1) $||P_{F_{a,a,\theta}}|| = 2|a| + 1 = ||P_{H_a}|| + 1 \text{ for all } a \in \mathbb{R}.$ - (2) If $a \in (-\infty, -1] \cup [0, \infty)$ , then $||P_{F_{a,a+1,\theta}}|| = |2a+1| = ||P_{H_{a,a+1}}|| 1$ . However, $||P_{F_{a,a+1,\theta}}|| = 1 \text{ for each } a \in (-1,0).$ - (3) If $a \leq 0 \leq b \leq a+3$ , then the functions $F_{a,b,\theta}$ are close-to-convex and univalent in $\mathbb{D}$ . - (4) The functions $F_{a,a+1,\theta}$ are univalent in $\mathbb{D}$ if and only if $a \in [-1,0]$ .

Theorem 5.1. The following conditions are equivalent. - (1) $\lambda \geq 1$ ; - (2) There exists a $\omega \in \mathcal{A}_0(\lambda)$ with $|\omega'(0)| = 1$ ; - (3) The set $\{\mu \cdot I : |\mu| = 1\}$ is contained in $\mathcal{A}_0(\lambda)$ ; - (4) Every automorphism $\sigma$ of the unit disk is an admissible dilatation in $\mathcal{B}_H(\lambda)$ .

Proposition 6.1. A harmonic mapping $f \in \mathcal{H}$ belongs to $\mathcal{B}_H(\lambda)$ if and only if for each pair of points z, $z_0$ in $\mathbb{D}$ , the inequality $$|A(z) - A(z_0)| \le \lambda d_h(z, z_0)$$ holds, where $A(z) = \log J_f(z)$ .

Theorem 6.1. (Distortion theorem) Let $f = h + \overline{g} \in \mathcal{B}_H(\lambda)$ for some $\lambda \geq 0$ with $b_1 = g'(0)$ , and let $H_{a,b}$ and $H_a$ be defined by (5.2). Then for each $z \in \mathbb{D}$ , we have - $(1) (1 |b_1|^2) H'_{\lambda}(-|z|) \le J_f(z) \le (1 |b_1|^2) H'_{\lambda}(|z|);$ $(2) \sqrt{1 |b_1|^2} H'_{\frac{\lambda}{2}}(-|z|) \le |h'(z)| \le (1 + |b_1 z|) H'_{\frac{\lambda-1}{2}, \frac{\lambda+1}{2}}(|z|);$ - (3) $|g'(z)| \le (|z|^2 + |b_1|) H'_{\frac{\lambda-1}{2}, \frac{\lambda+1}{2}}(|z|);$ - $(4) -\sqrt{1-|b_1|^2}H_{\frac{\lambda}{2}}(-|z|) \le |h(z)| \le (1-|b_1|)H_{\frac{\lambda-1}{2},\frac{\lambda+1}{2}}(|z|) + |b_1|H_{\frac{\lambda+1}{2}}(|z|);$ - (5) $|g(z)| \le H_{\frac{\lambda+1}{2}}(|z|) (1-|b_1|)H_{\frac{\lambda-1}{2},\frac{\lambda+1}{2}}(|z|);$ - (6) $|f(z)| \leq (1 + |b_1|)H_{\frac{\lambda+1}{2}}(|z|)$ . Furthermore, if f is univalent in $\mathbb{D}$ , then $$-(1-|b_1|)H_{\frac{\lambda+1}{2}}(-|z|) \le |f(z)| \le (1+|b_1|)H_{\frac{\lambda+1}{2}}(|z|).$$ The estimates in (1) are sharp for all $\lambda \geq 0$ . The right sides of (2)-(6) are sharp for all $\lambda \geq 1$ and the left side of (6) is sharp for $\lambda = 1$ . Moreover, if $f \in \mathcal{B}_H^0(\lambda)$ , then the left sides of (2) and (4) are sharp for all $\lambda \geq 0$ .

Corollary 6.1. (Growth and covering theorem) Let $f = h + \overline{g} \in \mathcal{B}_H(\lambda)$ with $b_1 = g'(0)$ , and let $H_{a,b}$ and $H_a$ be defined by (5.2). If $\lambda > 1$ , then f, h and g satisfy the same growth condition $$f(z) (h(z), g(z)) = O(1 - |z|)^{\frac{1-\lambda}{2}} as |z| \to 1.$$ If $\lambda < 1$ , then f (resp. h, q) is bounded by $$(1+|b_1|)H_{\frac{\lambda+1}{2}}(1) \ (resp. \ (1-|b_1|)H_{\frac{\lambda-1}{2},\frac{\lambda+1}{2}}(1)+|b_1|H_{\frac{\lambda+1}{2}}(1), \ H_{\frac{\lambda+1}{2}}(1)-(1-|b_1|)H_{\frac{\lambda-1}{2},\frac{\lambda+1}{2}}(1)).$$ For all $\lambda > 0$ , the image $h(\mathbb{D})$ contains the disk $\{|z| < -\sqrt{1-|b_1|^2}H_{\frac{\lambda}{2}}(-1)\}$ . If $f \in \mathcal{B}_H(\lambda) \cap \mathcal{S}_H$ , then the image $f(\mathbb{D})$ contains the disk $\{|z| < -(1-|b_1|)H_{\frac{\lambda+1}{2}}(-1)\}$ . If $f \in \mathcal{B}_H(\lambda) \cap \mathcal{S}_H^0$ for some $\lambda \in [0, 1]$ , then $$-H_{\frac{\lambda+1}{2}}(-1) \ge -H_1(-1) = 2\log 2 - 1 = 0.38629\cdots$$ This result is an improvement over the non-sharp known result that $f(\mathbb{D}) \supseteq \{w : |w| < 1/16\}$ if $f \in \mathcal{S}_H^0$ . In Corollary 6.1, the case $\lambda = 1$ is critical. By Theorem 6.1, we have that, for $f \in \mathcal{B}_H(1)$ , $$|f(z)| \le (1+|b_1|)H_1(|z|) = (1+|b_1|)(-2\log(1-|z|)-|z|), \quad z \in \mathbb{D},$$ which shows that functions in $\mathcal{B}_H(1)$ need not be bounded. The following result gives a sufficient condition for the boundedness of mappings in $\mathcal{B}_H(1)$ .

Proposition 6.2. Let $f = h + \overline{g}$ be a sense-preserving harmonic mapping in $\mathbb{D}$ . If f satisfies the condition $$\beta(f) := \overline{\lim_{|z| \to 1^{-}}} ((1 - |z|^{2})|P_{f}(z)| - 1) \log \frac{1}{1 - |z|^{2}} < -2,$$ then f, h and g are bounded in $\mathbb{D}$ . Proof. Without loss of generality, we can assume that g(0) = 0. It follows from (1.2) that $\beta(A \circ f) = \beta(f)$ for any affine harmonic mapping A defined in (1.2). Let $A \circ f = H + \overline{G}$ . It is easy to check that both h and g are bounded in $\mathbb{D}$ if and only if both H and G are bounded in $\mathbb{D}$ . Note that $A \circ f$ is also sense-preserving in $\mathbb{D}$ . Thus, it is enough to consider the case $f = h + \overline{g} \in \mathcal{H}_0$ and prove that both h and g are bounded in $\mathbb{D}$ . By assumption, there exist $\beta < -2$ and $r_0 \in (1 - 1/(2e), 1)$ such that (6.1) $$|P_f(z)| \le \frac{1}{1 - |z|^2} + \frac{\beta}{(1 - |z|^2)\log(1/1 - |z|^2)}$$ for $z \in D_{r_0} = \{z : r_0 < |z| < 1\}$ . Fix $z \in D_{r_0}$ and let $\Gamma$ be a line segment from z to $z_0 := r_0 e^{i \arg z}$ in the proof of Proposition 6.1. Then we have (6.2) $$|\log J_f(z)| \le 2 \int_{r_0}^{|z|} |P_f(\zeta)| |d\zeta| + C_1,$$ where $C_1 = \max_{\theta \in [0,2\pi]} |J_f(r_0 e^{i\theta})| < \infty$ . By (6.1) and (6.2), we see that $$|\log J_f(z)| \le \log \frac{1+|z|}{1-|z|} + \int_{r_0}^{|z|} \frac{2\beta dt}{(1-t^2)\log(1/(1-t^2))} + C_1$$ $$\le \log \frac{1+|z|}{1-|z|} + \int_{r_0}^{|z|} \frac{\beta dt}{(1-t)\log(1/(2(1-t)))} + C_1$$ $$= \log \frac{1+|z|}{1-|z|} + \beta \log \log \frac{1}{2(1-|z|)} + C_2,$$ where $C_2 = C_1 - \beta \log \log \frac{1}{2(1-r_0)}$ . Exponentiating the last inequality shows that $$|J_f(z)| = |h'(z)|^2 (1 - |\omega_f(z)|^2) \le e^{C_2} \frac{1 + |z|}{1 - |z|} \left( \log \frac{1}{2(1 - |z|)} \right)^{\beta}.$$ Using that $f \in \mathcal{B}_H^0(1)$ , we have $|\omega_f(z)| \leq |z|$ in $\mathbb{D}$ and thus, we find that $$|g'(z)| < |h'(z)| = \left(\frac{J_f(z)}{1 - |\omega_f(z)|^2}\right)^{1/2} \le \frac{e^{C_2/2}}{1 - |z|} \left(\log \frac{1}{2(1 - |z|)}\right)^{\beta/2}.$$ Since $\beta/2 < -1$ , the function $[\log(1/(2(1-t)))]^{\beta/2}/(1-t)$ is integrable on the interval $[r_0, 1)$ . It follows that both h and g are bounded in $\mathbb{D}$ so that f is also bounded in $\mathbb{D}$ . $\square$ Remark. Let $f = h + \overline{g}$ be a sense-preserving harmonic mapping in $\mathbb{D}$ . If h and g are unbounded in $\mathbb{D}$ , then the boundedness of f is uncertain. For instance, let's recall the function $F_{0,1,\theta}$ defined by (5.1): $$F_{0,1,\theta}(z) = H_{0,1}(z) + \overline{G_{0,1,\theta}(z)} = -\log(1-z) + \overline{e^{i\theta}(-z - \log(1-z))},$$ we see that $H_{0,1}$ and $G_{0,1,\theta}$ are unbounded in $\mathbb{D}$ . However, $F_{0,1,0}(z) = -\overline{z} - 2\log|1-z|$ and $F_{0,1,\pi}(z) = \overline{z} - 2\arg(1-z)$ are unbounded and bounded in $\mathbb{D}$ , respectively. Furthermore, it follows from Proposition 5.2 that $||P_{F_{0,1,\theta}}|| = 1$ for any $\theta \in \mathbb{R}$ . By Theorem 6.1 and [16, Theorem 5.1], we conclude the Hölder continuity of mappings in $\mathcal{B}_H(\lambda)$ .

Theorem 6.2. Let $f = h + \overline{g} \in \mathcal{B}_H(\lambda)$ for some $\lambda \in [0,1)$ . Then $h + \varepsilon g$ is Hölder continuous of exponent $\frac{1-\lambda}{2}$ in $\mathbb{D}$ for each $\varepsilon \in \overline{\mathbb{D}}$ . Moreover, f is Hölder continuous of exponent $\frac{1-\lambda}{2}$ in $\mathbb{D}$ .

Theorem 7.1. If $f \in \mathcal{B}_H(\lambda)$ , then we have $$(7.1) |a_2| \le \frac{1}{2} \min \left\{ (1 - |b_1|^2)\lambda + 2|b_1b_2|, \min_{\varepsilon \in \overline{\mathbb{D}}} \{ |1 + \varepsilon b_1|(\lambda + 1) + 2|\varepsilon b_2| \} \right\}.$$ If $f \in \mathcal{B}_{H}^{0}(\lambda)$ , then $|a_{2}| \leq \lambda/2$ and the estimate is sharp for all $\lambda > 0$ .

Theorem 7.2. Let $f = h + \overline{g} \in \mathcal{B}_H(\lambda)$ . Then, for any a > 0 and a real number p, we have (7.2) $$I_p(r, h' + \varepsilon g') = O\left((1 - r)^{-\alpha(|p|(\lambda + 1)/2) - a}\right),$$ for each $\varepsilon \in \overline{\mathbb{D}}$ and thus, in particular, $$|a_n| + |b_n| = O\left(n^{\alpha((\lambda+1)/2)-1+a}\right).$$ For p > 0, we get that (7.3) $$I_p(r,f) = O\left((1-r)^{p-\alpha(|p|(\lambda+1)/2)-a}\right).$$ Here $$\alpha(\lambda) = \frac{\sqrt{1+4\lambda^2}-1}{2}$$ .

Theorem 7.3. Let $f = h + \overline{g} \in \mathcal{B}_H(\lambda)$ for some $\lambda$ with $1.982 < \lambda \leq 5$ . If there exists a constant $\varepsilon \in \overline{\mathbb{D}}$ such that $h + \varepsilon g$ is univalent in $\mathbb{D}$ , then $|a_n + \varepsilon b_n| = O(n^{(\lambda - 3)/2})$ as $n \to \infty$ . In particular, if h is univalent in $\mathbb{D}$ , then $|a_n| = O(n^{(\lambda - 3)/2})$ as $n \to \infty$ . Moreover, if $h + \varepsilon g$ is univalent in $\mathbb{D}$ for every $|\varepsilon| = 1$ , then $|a_n| + |b_n| = O(n^{(\lambda - 3)/2})$ as $n \to \infty$ . The three estimates are sharp. Proof. If $f \in \mathcal{B}_H(\lambda)$ for some $\lambda$ with $1.982 < \lambda \le 5$ , then as before we have $f_{\varepsilon} \in \mathcal{B}_A(\frac{\lambda+1}{2})$ (1.491 $< \frac{\lambda+1}{2} \le 3$ ) for each $\varepsilon \in \overline{\mathbb{D}}$ . The results follow from [24, Theorem 3.2]. To show the sharpness, we construct a family of functions $$T_{\lambda,\theta}(z) = t_{\lambda}(z) + \overline{e^{i\theta}zt_{\lambda}(z)} = \sum_{n=1}^{\infty} a_n z^n + \sum_{n=2}^{\infty} \overline{b}_n \overline{z}^n, \quad z \in \mathbb{D},$$ where $\lambda \in (1.982, 5], \theta \in \mathbb{R}$ and $$t_{\lambda}(z) = \frac{1 - (1 - z)^{(1 - \lambda)/2}}{(1 - \lambda)/2}.$$ First, we show that $T_{\lambda,\theta} \in \mathcal{B}_H(\lambda)$ for all $\lambda > 1$ and $\theta \in \mathbb{R}$ . It suffices to prove that $||P_{T_{\lambda,\theta}}|| = \lambda$ due to $T_{\lambda,\theta} \in \mathcal{H}$ . By computation, we find that $$P_{T_{\lambda,\theta}}(z) = \frac{1+\lambda}{2} \cdot \frac{1}{1-z} - \frac{\overline{z}}{1-|z|^2}.$$ Also, we note that $||P_{t_{\lambda}}|| = 1 + \lambda$ . If we get $||P_{T_{\lambda,\theta}}|| \leq \lambda$ , then it follows from (2.1) that $||P_{T_{\lambda,\theta}}|| = \lambda$ . Indeed we may let $z = x + iy \in \mathbb{D}$ . By computation, we obtain $$4\lambda^{2}|1-z|^{2} - \left|(1+\lambda)(1-|z|^{2}) - 2\overline{z}(1-z)\right|^{2}$$ $$= (\lambda - 1)[(1-x)^{3}(1-x+\lambda(3+x)) + 2(3+3\lambda-2x+(1-\lambda)x^{2})y^{2} - (\lambda - 1)y^{4}]$$ $$\geq (\lambda - 1)[2(3+3\lambda-2x+(1-\lambda)x^{2})y^{2} - 2(\lambda - 1)y^{2}]$$ $$\geq 2(\lambda - 1)(3+3\lambda-2+1-\lambda-\lambda+1)y^{2} \geq 0,$$ which clearly implies that $$(1 - |z|^2)|P_{T_{\lambda,\theta}}(z)| \le \lambda$$ and thus, $||P_{T_{\lambda,\theta}}|| \leq \lambda$ . Next, we show that the functions $T_{\lambda,\theta}$ are univalent in $\mathbb{D}$ for each $1 < \lambda \leq 5$ and all $\theta \in \mathbb{R}$ . A simple computation shows that $$\operatorname{Re}\left(1+z\frac{t_{\lambda}''(z)}{t_{\lambda}'(z)}\right) = \operatorname{Re}\left(1+\frac{1+\lambda}{2}\cdot\frac{z}{1-z}\right) > 1-\frac{1+\lambda}{4} \ge -\frac{1}{2}, \quad z \in \mathbb{D},$$ for $1 < \lambda \le 5$ . According to a well-known result, the function $t_{\lambda}$ is univalent and convex in one direction (and hence, close-to-convex) in $\mathbb{D}$ . Note that the dilatation of $T_{\lambda,\theta}$ is $e^{i\theta}z$ . It follows from [9, Theorem 1] that the functions $T_{\lambda,\theta}$ are univalent in $\mathbb{D}$ for each $1 < \lambda \le 5$ and all $\theta \in \mathbb{R}$ . Therefore, $T_{\lambda,\theta}$ is SHU and thus, $t_{\lambda} + \varepsilon z t_{\lambda}$ is univalent in $\mathbb{D}$ for each $\varepsilon \in \overline{\mathbb{D}}$ . Finally, by Stirling's formula, we have $$|a_n| = \frac{2\Gamma((\lambda + 2n - 1)/2)}{(\lambda - 1)n!\Gamma((\lambda - 1)/2)} \sim \frac{2}{\lambda - 1}n^{(\lambda - 3)/2} \text{ as } n \to \infty.$$ Note that $b_n = e^{i\theta}a_{n-1}$ for each n > 1. Hence, $|a_n| + |b_n| = |a_n| + |a_{n-1}| = O(n^{(\lambda - 3)/2})$ as $n \to \infty$ . Given a harmonic mapping $f \in \mathcal{H}$ , let $\gamma(f)$ denote the infimum of exponents $\gamma$ such that $|a_n| + |b_n| = O(n^{\gamma - 1})$ as $n \to \infty$ , that is, $$\gamma(f) = \overline{\lim_{n \to \infty}} \frac{\log n(|a_n| + |b_n|)}{\log n}.$$ For the subset X of $\mathcal{H}$ , we let $\gamma(X) = \sup_{f \in X} \gamma(f)$ . There are some investigations about $\gamma(f)$ (resp. $\gamma(X)$ ) if f (resp. X) is restricted to be analytic or special families of analytic functions. The reader can refer to [10, 24, 28] and [31, Chapter 10] for some details on this problem. For each $\lambda \in (0, \infty)$ and $\varepsilon \in \overline{\mathbb{D}}$ , we introduce $$\mathcal{A}_H(\lambda,\varepsilon) =: \left\{ f_{\varepsilon} : f_{\varepsilon}(z) = \frac{h(z) + \varepsilon g(z)}{1 + \varepsilon g'(0)} \text{ and } f = h + \overline{g} \in \mathcal{B}_H(\lambda) \right\}$$ and obtain the following theorem.

Theorem 7.4. For each $\lambda \in (0, \infty)$ and $\varepsilon \in \overline{\mathbb{D}}$ , we have $$\max\{(\lambda - 1)/2, 0\} \le \gamma(\mathcal{B}_H(\lambda)) \quad (\gamma(\mathcal{A}_H(\lambda, \varepsilon))) \le \alpha((\lambda + 1)/2),$$ where $\alpha(\lambda) = \frac{\sqrt{1+4\lambda^2}-1}{2}$ . In particular, $\gamma(\mathcal{B}_H(\lambda)) = O((\lambda+1)^2)$ and $\gamma(\mathcal{A}_H(\lambda,\varepsilon)) = O((\lambda+1)^2)$ as $\lambda \to 0$ . We continue the discussion by mentioning a connection with integral means for univalent analytic functions. For a univalent harmonic mapping $f = h + \overline{g} \in \mathcal{S}_H$ , a complex number $\varepsilon \in \overline{\mathbb{D}}$ and a real number p, we let $$\beta_{f_{\varepsilon}}(p) = \overline{\lim_{r \to 1^{-}}} \frac{\log I_{p}(r, f'_{\varepsilon})}{\log \frac{1}{1 - r}}.$$ Clearly, for a univalent analytic function $f \in \mathcal{A} \cap \mathcal{S}_H$ , $$\beta_f(p) = \overline{\lim_{r \to 1^-}} \frac{\log I_p(r, f')}{\log \frac{1}{1-r}}.$$ Brennan conjectured that $\beta_f(-2) \leq 1$ for univalent analytic functions f (see [31, Charpter 8]). As a corollary to Theorem 7.2, we have

Theorem 7.5. For $f \in \mathcal{B}_H(\lambda)$ and a real number p, $$\beta_{f_{\varepsilon}}(p) \le \alpha(|p|(\lambda+1)/2) = \frac{\sqrt{1+p^2(1+\lambda)^2}-1}{2}$$ holds for each $\varepsilon \in \overline{\mathbb{D}}$ . In particular, the Brennan conjecture is true for every univalent harmonic mapping f with $||P_f|| \leq \sqrt{2} - 1$ . 8. The space $\mathcal{B}_H(\lambda)$ and the Hardy space For a harmonic mapping $f = h + \overline{g}$ in $\mathbb{D}$ , the Bloch seminorm is given by (see [14]) $$||f||_{\mathscr{B}_H} = \sup_{z \in \mathbb{D}} (1 - |z|^2) (|h'(z)| + |g'(z)|),$$ and f is called a (harmonic) Bloch mapping when $||f||_{\mathscr{B}_H} < \infty$ . Let BMOA (resp. BMOH) denote the class of analytic functions (resp. harmonic mappings) that have bounded mean oscillation on the unit disk $\mathbb{D}$ (see [1]). Kim [23] showed some relationships among $\mathcal{B}_A(\lambda)$ , $H^p$ and BMOA (see also [25]). Combined with the study on Bloch, BMO and univalent harmonic mappings (see [1]), a generalization of Kim's result is given in [32]. Basic properties about analytic Bloch functions may be obtained from [5, 31]. Our results are based on the following observation. It follows from Theorem 6.1 (6) that the inequality $$|f(z)| \le (1+|b_1|) \int_0^{|z|} \left(\frac{1+t}{1-t}\right)^{\frac{\lambda+1}{2}} dt, \quad z \in \mathbb{D},$$ holds for every $f \in \mathcal{B}_H(\lambda)$ , which implies that - f is bounded when $\lambda < 1$ , - $f(z) = O(-\log(1-|z|)) (|z| \to 1)$ when $\lambda = 1$ , and - $f(z) = O((1-|z|)^{1-\frac{\lambda+1}{2}})$ ( $|z| \to 1$ ) when $\lambda > 1$ . On the other hand, the proofs of our results are similar to that of results of [32, Section 4]. Let $$\mathcal{T}_H(\lambda) = \{ f = h + \overline{g} \in \mathcal{H} : ||T_f|| \le 2\lambda \}$$ with $$||T_f|| := \sup_{z \in \mathbb{D}, \ \theta \in [0, 2\pi]} (1 - |z|^2) \left| \frac{h''(z) + e^{i\theta} g''(z)}{h'(z) + e^{i\theta} g'(z)} \right|.$$ For the one parameter family $\mathcal{T}_H(\lambda)$ , the authors showed its relationship with Hardy spaces in [32, Section 4]. Note that $||T_f|| = \sup_{\theta \in [0,2\pi]} ||P_{h+e^{i\theta}g}||$ . If $f \in \mathcal{B}_H(\lambda)$ , then it is easy to see that $f \in \mathcal{T}_H(\frac{\lambda+1}{2})$ from (2.1). Therefore, applying the above observation and replacing $h + e^{i\theta}g$ ( $\theta \in [0,2\pi]$ ) (resp. $\lambda$ ) to $h + \varepsilon g$ ( $\varepsilon \in \overline{\mathbb{D}}$ ) (resp. $(\lambda + 1)/2$ ) in corresponding proofs of [32, Section 4], then we can easily obtain the following results. So we omit their proofs.

Theorem 8.1. (1) If $\lambda < 1$ , then $\mathcal{B}_H(\lambda) \cap \mathcal{S}_H \subset h^{\infty}$ . - (2) If $\lambda = 1$ , then $\mathcal{B}_H(\lambda) \cap \mathcal{S}_H \subset BMOH$ . - (3) If $\lambda > 1$ , then $\mathcal{B}_H(\lambda) \cap \mathcal{S}_{H_K} \subset h^p$ for every $0 , where <math>K \ge 1$ and $\mathcal{S}_{H_K} = \{ f = h + \overline{g} \in \mathcal{S}_H : f \text{ is } K\text{-quasiconformal} \}.$

Theorem 8.2. Let $\lambda \geq 1$ . Then $\mathcal{B}_H(\lambda) \subset h^p$ with $0 , where <math>p_0(\lambda) = \infty$ if $\lambda = 1$ . Remark. Theorem 8.2 can be directly obtained by choosing $p - \alpha(p(\lambda + 1)/2) > 0$ in Theorem 7.2.

Corollary 8.1. A uniformly locally univalent harmonic mapping f in $\mathbb{D}$ is contained in the Hardy space $h^p$ for some p = p(f) > 0.

Theorem 9.1. (Subordination principle I) Let $f = h + \overline{g}$ be a harmonic mapping in $\mathbb{D}$ and $F = H + \overline{G} \in \mathcal{B}_H$ . If $h' + \overline{g'} \preceq H' + \overline{G'}$ , then we have $||P_f|| \leq ||P_F||$ . In this case, f is ULU in $\mathbb{D}$ . Proof. By assumption, there exists a function $\phi \in \mathcal{A}_D$ such that $h' = H' \circ \phi$ and $g' = G' \circ \phi$ . Therefore, f is sense-preserving in $\mathbb{D}$ since F is sense-preserving in $\mathbb{D}$ . Moreover, we have $P_h = (P_H \circ \phi)\phi'$ and $$\omega_f(z) = \frac{g'(z)}{h'(z)} = \frac{G'(w)}{H'(w)} = \omega_F(w), \quad w = \phi(z).$$ Consequently, $$\frac{\overline{\omega_f(z)}\omega_f'(z)}{1-|\omega_f(z)|^2} = \frac{\overline{\omega_F(w)}\omega_F'(w)}{1-|\omega_F(w)|^2}\phi'(z).$$ It follows from (1.1) that $P_f = (P_F \circ \phi)\phi'$ . By Schwarz-Pick's lemma, $$|\phi'(z)| \le \frac{1 - |\phi(z)|^2}{1 - |z|^2}$$ and using this, we find that $$(1 - |z|^2)|P_f(z)| = (1 - |z|^2)|\phi'(z)P_F(\phi(z))| \le (1 - |\phi(z)|^2)|P_F(\phi(z))| \le ||P_F||.$$ The desired conclusion follows. Often, the property of a sense-preserving harmonic mapping is mainly decided by its analytic part. As another example of it, we have

Theorem 9.2. (Subordination principle II) Let $f = h + \overline{g}$ be a sense-preserving harmonic mapping in $\mathbb{D}$ and $F = H + \overline{G} \in \mathcal{B}_H$ such that $h' \leq H'$ . Then we have $||P_f|| \leq ||P_F|| + 2$ . Thus, f is ULU in $\mathbb{D}$ .

Definition 7.1. For $0 , the Hardy space <math>H^p$ is the set of all functions f analytic in $\mathbb{D}$ for which $||f||_p := \sup\{M_p(r, f) : 0 < r < 1\} < +\infty$ , where $M_p(r, f)$ is defined as above. Let $h^p$ denote the analogous space of harmonic mappings f in $\mathbb{D}$ with $||f||_p$ defined similarly (see [18]). In [4], Aleman and Martín constructed convex harmonic mappings that do not belong to $h^{1/2}$ which settles the question raised by Duren [18]. It is worth pointing out that the space $h^p$ is well-behaved for $p \ge 1$ whereas $H^p$ is comparatively well-behaved for all p > 0, such is not the case for $h^p$ , 0 . Since $f \in \mathcal{B}_H(\lambda)$ implies that $f_{\varepsilon} \in \mathcal{B}_A((\lambda + 1)/2)$ for each $\varepsilon \in \overline{\mathbb{D}}$ , it follows from the result of [24, p. 190] that $$|a_n + \varepsilon b_n| = O(n^{(\lambda+1)/2-1})$$ uniformly for $\varepsilon \in \overline{\mathbb{D}}$ as $n \to \infty$ and thus, we obtain that $|a_n| + |b_n| = O(n^{(\lambda-1)/2})$ as $n \to \infty$ . Especially, $|a_n| = O(n^{(\lambda-1)/2})$ and $|b_n| = O(n^{(\lambda-1)/2})$ as $n \to \infty$ . Moreover, if $\lambda < 1$ and f is univalent in $\mathbb{D}$ , then, by Corollary 6.1, f is bounded and thus, Area $$(f(\mathbb{D})) = \pi \Big( \sum_{n=1}^{\infty} n(|a_n|^2 - |b_n|^2) \Big) < \infty,$$ which implies that $\sqrt{|a_n|^2 - |b_n|^2} = o(n^{-1/2})$ as $n \to \infty$ . Combining the results from [24, Section 3] and the implication (1.3), we can get a series of results. We omit detailed proofs, but it might be appropriate to include some necessary explanations. In fact, we only need to modify the conditions by replacing the parameter $\lambda$ in the theorems of [24, Section 3] by $(\lambda + 1)/2$ at appropriate places.