On the coefficients estimate of K-quasiconformal harmonic mappings

Theorem 1.1. Let $\mathcal{F}_0$ be any one of the three classes $\mathcal{S}_H^{*0}(K)$ , $\mathcal{K}_H^0(K)$ and $\mathcal{T}_H^0(K)$ . Then Conjecture B holds with $\mathcal{F}_0$ in place of $\mathcal{S}_H^0(K)$ . The constants A(n,k) and B(n,k) may be simplified to the following forms $$A(n,k) = \frac{1}{n(1-k)^3} \left( n^2 + (-2n^2 - 2n + 1)k + (n+1)^2 k^2 - k^{n+1} - k^{n+2} \right)$$ and $$B(n,k) = \frac{k}{n(1-k)^3} \Big( (n-1)^2 + (-2n^2 + 2n + 1)k + n^2k^2 - k^n - k^{n+1} \Big),$$ respectively. In order to show that A(n, k) and B(n, k) are increasing functions of k, we need the following lemma.

Lemma 1.1. For $x \in [0,1)$ and n > 2, let $$\phi_n(x) = \frac{M_n(x)}{(1-x)^3}$$ and $\psi_n(x) = \frac{x}{(1-x)^3} M_{n-1}(x)$ for all $x \in [0,1)$ , where $$(1.8) M_n(x) = n^2 + (-2n^2 - 2n + 1)x + (n+1)^2 x^2 - x^{n+1} - x^{n+2}.$$ Then $\phi_n$ and $\psi_n$ are strictly increasing on [0,1).

Corollary 1.1. Let $\mathcal{F}$ be any one of the three classes $\mathcal{S}_H^*(K)$ , $\mathcal{K}_H(K)$ and $\mathcal{T}_H(K)$ . Then the coefficients of each function $f \in \mathcal{F}$ (K > 1) satisfy the sharp inequalities $$|a_n| \le E_n(k) := A(n, k_0) + \frac{1 - \sqrt{1 - k_0^2}}{k_0} B(n, k_0),$$ $$|b_n| \le F_n(k) := B(n, k_0) + \frac{1 - \sqrt{1 - k_0^2}}{k_0} A(n, k_0),$$ for $n=2,3,\ldots$ , where $k_0=\frac{2k}{1+k^2}$ , A(n,k) and B(n,k) are defined by (1.5) and (1.6), respectively. Equalities occur for the function $Q_k$ defined by (1.10) $$Q_k(z) = P_k(z) + \frac{\sqrt{K_0} - 1}{\sqrt{K_0} + 1} \overline{P_k(z)},$$ where $P_k(z) \in \mathcal{F}_0^0$ is defined by (1.4), $\mathcal{F}_0^0$ is any one of the three classes $\mathcal{S}_H^{*0}(K_0)$ , $\mathcal{K}_H^0(K_0)$ and $\mathcal{T}_H^0(K_0)$ and $K_0 = K^2$ . The limiting case of Corollary 1.1 as $k \to 1^-$ gives the following well-known result.

Corollary 1.2. Let $\mathcal{F}$ be any one of the three classes $\mathcal{S}_H^*$ , $\mathcal{K}_H$ and $\mathcal{T}_H$ . Then the coefficients of each function $f = h + \overline{g} \in \mathcal{F}$ satisfy the sharp inequalities $$|a_n| < \frac{2n^2 + 1}{3}$$ and $|b_n| < \frac{2n^2 + 1}{3}$ for $n = 2, 3, \dots$

Theorem 1.2. The coefficients of each function $f \in \mathcal{C}^0_H(K)$ satisfy the sharp inequalities $$|a_n| \le a(n,k), |b_n| \le b(n,k) \text{ and } |a_n| - |b_n| \le 1 \text{ for } n = 2,3,\ldots,$$ where $$a(n,k) = \frac{n - k(n+1) + k^{n+1}}{n(1-k)^2}$$ and $b(n,k) = \frac{k}{n(1-k)^2}(n-1-nk+k^n)$ . Specially, if n = 2, we have the sharp inequalities $$|a_2| \le \frac{3K+1}{2K+2}$$ and $|b_2| \le \frac{K-1}{2K+2}$ . Equalities occur for the function $P \in \mathcal{C}^0_H(K)$ defined by $$(1.11) P(z) = h_k(z) + \overline{g_k(z)} = z + \sum_{n=2}^{\infty} a(n,k)z^n - \sum_{n=2}^{\infty} b(n,k)\overline{z}^n,$$ where $$h_k(z) = \frac{z}{(1-k)(1-z)} - \frac{k}{(1-k)^2} \log\left(\frac{1-kz}{1-z}\right),$$ and $$g_k(z) = \frac{-kz}{(1-k)(1-z)} - \frac{k}{(1-k)^2} \log\left(\frac{1-z}{1-kz}\right).$$ Theorem 1.2 leads also to sharp coefficient bounds for mappings of the class $\mathcal{C}_H(K)$ . In order to state this result, we need the following lemma.

Lemma 1.2. For $x \in [0,1)$ and $n \ge 2$ , let $$G_n(x) = \frac{L_n(x)}{(1-x)^2}$$ and $H_n(x) = \frac{x}{(1-x)^2} L_{n-1}(x)$ for all $x \in [0, 1)$ , where (1.12) $$L_n(x) = n - (n+1)x + x^{n+1}$$ Then $G_n$ and $H_n$ are strictly increasing on [0,1). Proof. Clearly, $L_n(x) = n(1-x) - x(1-x^n) = (1-x)[n-x(1+x+\cdots x^{n-1})] > 0$ on [0, 1). Moreover, $L'_n(x) = -(n+1)(1-x^n)$ , $L''_n(x) = n(n+1)x^{n-1}$ and $L'''_n(x) = (n-1)n(n+1)x^{n-2}$ so that $L'_n(x) < 0$ on [0,1), $L''_n(x) > 0$ on (0,1), and $L'''_n(x) > 0$ on (0,1). Moreover, $L'_n(1) = 0$ and $L''_n(1) > 0$ . Now, a computation gives $$G'_n(x) = \frac{F_n(x)}{(1-x)^3}$$ for all $x \in [0,1)$ , where $F_n(x) = (1 - x)L'_n(x) + 2L_n(x)$ . Again, $$F'_n(x) = (1-x)L''_n(x) + L'_n(x)$$ and $F''_n(x) = (1-x)L'''_n(x)$ , showing that $F''_n(x) \ge 0$ for all $x \in [0,1)$ and $n \ge 2$ . Therefore, $F'_n(x) \le F'_n(1) = 0$ for all $x \in [0,1)$ , which in turn implies that $F_n(x) \ge F_n(1) = 0$ for all $x \in [0,1)$ , Thus, $G_n(x)$ is strictly increasing on (0,1). For the proof the second part, we find that $H_2(x) = x$ and so, there is nothing to prove. We may thus, assume that $n \geq 3$ and we have $$H'_n(x) = \frac{J_n(x)}{(1-x)^3}$$ for all $x \in [0,1)$ , where $J_n(x) = (1-x)[xL'_{n-1}(x) + L_{n-1}(x)] + 2xL_{n-1}(x)$ . We need to show that $J_n(x) \ge 0$ on [0,1). Indeed, $J_n(0) = n-1 > 0$ and we may rewrite $J_n(x)$ as $$J_n(x) = x[(1-x)L'_{n-1}(x) + 2L_{n-1}(x)] + (1-x)L_{n-1}(x).$$ By the previous argument, it is clear that $J_n(x) > 0$ on (0,1) and for $n \ge 3$ . Thus, $H_n(x)$ is strictly increasing on (0,1). Remark 1.3. First we observe that $nG_n(k) = a(n,k)$ and $nH_n(k) = b(n,k)$ . By Lemma 1.2, a(n,k) and b(n,k) are increasing functions of $k \in [0,1)$ . If $k \to 1^-$ , then $$\lim_{k \to 1^{-}} a(n, k) = \frac{n+1}{2} \text{ and } \lim_{k \to 1^{-}} b(n, k) = \frac{n-1}{2}.$$ Hence, in this case, Theorem 1.2 coincides with the first part of Theorem C.

Corollary 1.3. The coefficients of each function $f \in C_H(K)$ (K > 1) satisfy the sharp inequalities $$|a_n| \le C_n(k) = a(n, k_0) + \left(\frac{1 - \sqrt{1 - k_0^2}}{k_0}\right) b(n, k_0)$$ and $$|b_n| \le D_n(k) = b(n, k_0) + \left(\frac{1 - \sqrt{1 - k_0^2}}{k_0}\right) a(n, k_0)$$ for n = 2, 3, ..., where $k_0 = \frac{2k}{1+k^2}$ . Specially, if n = 2, we have the sharp inequalities $$|a_2| \le C_2(k) = \frac{1+k+2k^2}{1+k^2} = \frac{2K^2-K+1}{K^2+1}$$ and $$|b_2| \le D_2(k) = \frac{k(2+k+k^2)}{1+k^2} = \frac{(K-1)(2K^2+K+1)}{(K+1)(K^2+1)}.$$ Equalities occur for the function Q defined by (1.13) $$Q(z) = P(z) + \frac{\sqrt{K_0} - 1}{\sqrt{K_0} + 1} \overline{P(z)},$$ where $P(z) \in \mathcal{C}_H^0(K_0)$ is defined by (1.11) and $K_0 = K^2$ . Remark 1.4. As before, we remark that $C_n(k)$ and $D_n(k)$ are strictly increasing functions of $k \in [0, 1)$ . If $k \to 1^-$ , then, as in the case of the proof of Corollary 1.2, we have $$\lim_{k \to 1^{-}} C_n(k) = n$$ and $\lim_{k \to 1^{-}} D_n(k) = n$ . Thus, in the limiting case, Corollary 1.3 coincides with the second part of Theorem C. 1.6. Typically real K-quasiconformal harmonic mappings. By (1.4), since $P_k(0) = 0$ , $h'(0) + \overline{g'(0)} = 1 > 0$ and $\frac{g'(z)}{h'(z)} = kz$ , we know that $P_k \in \mathcal{T}_H^0(K)$ (cf. [5] or [14, Theorem A]). We consider the coefficients estimate of $f \in \mathcal{T}_H(K)$ and $f \in \mathcal{T}_H^0(K)$ and show that Conjecture B holds for $\mathcal{T}_H^0(K)$ . More precisely, we have

Theorem 1.3. If $f = h + \overline{g} \in \mathcal{T}_H^0(K)$ , then $a_1 = 1$ , and we have the sharp inequalities stated in (1.7) hold. We present the proofs of Theorems 1.1, 1.2, and 1.3 in Section 2.