Some properties of certain close-to-convex harmonic mappings

Lemma 2.1. ([23, p. 41]) For a function $p \in \mathcal{P}$ of the form (1.2), the sharp inequality $|p_n| \leq 2$ holds for each $n \geq 1$ . Equality holds for the function $$p(z) = \frac{1+z}{1-z}.$$

Lemma 2.2. ([24, Theorem 1]) Let $p \in \mathcal{P}$ be of the form (1.2) and $\mu \in \mathbb{C}$ . Then $$|p_n - \mu p_k p_{n-k}| \le 2 \max\{1, |2\mu - 1|\} \quad (1 \le k \le n - 1).$$ If $|2\mu - 1| \ge 1$ , then the inequality is sharp for the function $$p(z) = \frac{1+z}{1-z}$$ or its rotations. If $|2\mu - 1| < 1$ , then the inequality is sharp for $$p(z) = \frac{1+z^n}{1-z^n}$$ or its rotations. <span id="page-3-1"></span>Lemma 2.3. ([\[56\]](#page-16-1)) Let f = h + g ∈ M(α, ζ, n). Then the coefficients a^k (k ∈ N \ {1}) of h satisfy <span id="page-3-2"></span><span id="page-3-0"></span> $$|a_k| \le \frac{1}{k!} \prod_{j=2}^k (j - 2\alpha) \ (k \in \mathbb{N} \setminus \{1\}). \tag{2.1}$$ Moreover, the coefficients b^k (k = n + 1, n + 2, · · · ; n ∈ N) of g satisfy $$|b_{n+1}| \le \frac{|\zeta|}{n+1}$$ and $|b_{k+n}| \le \frac{|\zeta|}{(k+n)(k-1)!} \prod_{j=2}^{k} (j-2\alpha) \ (k \in \mathbb{N} \setminus \{1\}; \ n \in \mathbb{N}).$ (2.2) The bounds are sharp for the extremal function given by $$f(z) = \int_0^z \frac{dt}{(1 - \delta t)^{2 - 2\alpha}} + \overline{\int_0^z \frac{\zeta t^n}{(1 - \delta t)^{2 - 2\alpha}} dt} \quad (|\delta| = 1; z \in \mathbb{D}).$$ (2.3) <span id="page-3-6"></span><span id="page-3-5"></span>Lemma 2.4. ([\[56\]](#page-16-1)) Let f ∈ M(α, ζ, n) with 0 ≤ α < 1 and 0 ≤ ζ < ¹ 2n−1 (n ∈ N). Then $$\Phi(r; \alpha, \zeta, n) \le |f(z)| \le \Psi(r; \alpha, \zeta, n) \quad (r = |z| < 1), \tag{2.4}$$ where $$\Phi(r;\alpha,\zeta,n) = \begin{cases} \log(1+r) - \frac{\zeta r^{n+1} {}_{2}F_{1}(1,\,n+1;\,n+2;\,-r)}{n+1} & (\alpha=1/2), \\ \frac{(1+r)^{2\alpha-1} - 1}{2\alpha-1} - \frac{\zeta r^{n+1} {}_{2}F_{1}(n+1,\,2-2\alpha;\,n+2;\,-r)}{n+1} & (\alpha\neq1/2), \end{cases}$$ and $$\Psi(r;\alpha,\zeta,n) = \begin{cases} -\log(1-r) + \frac{\zeta r^{n+1} {}_{2}F_{1}(1, n+1; n+2; r)}{n+1} & (\alpha = 1/2), \\ \frac{1 - (1-r)^{2\alpha-1}}{2\alpha - 1} + \frac{\zeta r^{n+1} {}_{2}F_{1}(n+1, 2-2\alpha; n+2; r)}{n+1} & (\alpha \neq 1/2). \end{cases}$$ All these bounds are sharp, the extremal function is fα,ζ,n = h^α + gα,ζ,n or its rotations, where $$f_{\alpha,\zeta,n}(z) = \begin{cases} -\log(1-z) + \frac{\overline{\zeta z^{n+1} {}_{2}F_{1}(1, n+1; n+2; z)}}{n+1} & (\alpha = 1/2), \\ \frac{1 - (1-z)^{2\alpha-1}}{2\alpha - 1} + \frac{\overline{\zeta z^{n+1} {}_{2}F_{1}(n+1, 2-2\alpha; n+2; z)}}{n+1} & (\alpha \neq 1/2). \end{cases}$$ $$(2.5)$$ The following two results are due to Ma and Minda [\[42\]](#page-15-3). <span id="page-3-4"></span>Lemma 2.5. Let f ∈ C(φ). Then zf00(z)/f⁰ (z) ≺ zK00(z)/K⁰ (z) and f 0 (z) ≺ K⁰ (z), where K is given by [\(1.4\)](#page-1-1). <span id="page-3-3"></span>Lemma 2.6. Assume that f ∈ C(φ) and |z| = r < 1. Then $$K'(-r) \le |f'(z)| \le K'(r),$$ (2.6) where K is given by [\(1.4\)](#page-1-1). Equality holds for some z 6= 0 if and only if f is a rotation of K.

Lemma 2.7. ([13]) Let $f(z) = \sum_{n=0}^{\infty} a_n z^n$ and $g(z) = \sum_{n=0}^{\infty} b_n z^n$ be two analytic functions in $\mathbb{D}$ and $g \prec f$ . Then $$\sum_{n=0}^{\infty} |b_n| r^n \le \sum_{n=0}^{\infty} |a_n| r^n \tag{2.7}$$ for $|z| = r \le 1/3$ . Remark 2.1. Lemma 2.7 continues to hold for quasi-subordination (cf. [5]). Moreover, the bound 1/3 is optimal as shown by [48, Lemma 1].

Theorem 3.1. Let $f \in \mathcal{M}(\alpha, \zeta, n)$ . Then $$|T_2(n)[h]| \le \left(\frac{1}{n!} \prod_{j=2}^n (j-2\alpha)\right)^2 + \left(\frac{1}{(n+1)!} \prod_{j=2}^{n+1} (j-2\alpha)\right)^2 \quad (n \in \mathbb{N} \setminus \{1\}), \quad (3.1)$$ <span id="page-4-2"></span>and <span id="page-4-3"></span><span id="page-4-1"></span> $$|T_2(n)[g]| \le \frac{1}{[(2n-1)(n+1)]^2}.$$ (3.2) The inequalities in (3.1) and (3.2) are sharp for the extremal function given by (2.3).

Theorem 3.2. Let $f \in \mathcal{M}(\alpha, \zeta, 1)$ . Then $$|T_3(1)[h]| \le \begin{cases} \frac{1}{9} \left( 8\alpha^4 - 34\alpha^3 + 71\alpha^2 - 72\alpha + 36 \right) & \left( -\frac{1}{2} \le \alpha \le \frac{1}{2} \right), \\ \frac{1}{9} \left( -2\alpha^3 + 25\alpha^2 - 44\alpha + 30 \right) & \left( \frac{1}{2} \le \alpha < 1 \right), \end{cases}$$ (3.7) <span id="page-5-1"></span>and <span id="page-5-0"></span> $$|T_3(1)[g]| \le \frac{1}{3}(1-\alpha).$$ (3.8) The inequality in (3.7) is sharp for the function h given by (3.4), and the inequality in (3.8) is sharp for the function g defined by $$g(z) = \int_0^z \frac{\zeta t}{(1 - \delta t)^{2 - 2\alpha}} dt \quad (|\delta| = 1; |\zeta| \le 1; z \in \mathbb{D}).$$ (3.9)

Theorem 3.3. Let $f \in \mathcal{M}(\alpha, \zeta, 2)$ . Then $$|T_3(2)[h]| \le \begin{cases} \frac{1}{108} \left(1 - \alpha\right)^3 (2\alpha^2 - 7\alpha + 12)(10\alpha^2 - 27\alpha + 36\right) & \left(-\frac{1}{2} \le \alpha \le \frac{1}{7}\right), \\ \frac{5}{108} \left(1 - \alpha\right)^3 (2\alpha^2 - 7\alpha + 12)(2\alpha^2 - 4\alpha + 7) & \left(\frac{1}{7} \le \alpha < 1\right), \end{cases}$$ (3.16) <span id="page-6-2"></span>and $$|T_3(2)[g]| = |2b_3^2b_4| \le \frac{1}{243}(1-\alpha).$$ (3.17) The inequality in (3.16) is sharp for the function h given by (3.4), and the inequality in (3.17) is sharp for the function g defined by $$g(z) = \int_0^z \frac{\zeta t^2}{(1 - \delta t)^{2 - 2\alpha}} dt \quad \left( |\delta| = 1; \, |\zeta| \le \frac{1}{3}; \, z \in \mathbb{D} \right). \tag{3.18}$$ •

Proposition 4.1. Let $f \in \mathcal{HC}_n(\phi)$ . Then <span id="page-7-4"></span> $$L(\zeta, n, r) \le |f(z)| \le R(\zeta, n, r), \tag{4.1}$$ <span id="page-7-3"></span>where $$L(\zeta, n, r) = -K(-r) - |\zeta| \int_0^r t^n K'(-t) dt, \tag{4.2}$$ and $$R(\zeta, n, r) = K(r) + |\zeta| \int_{0}^{r} t^{n} K'(t) dt.$$ (4.3) The bounds are sharp for the extremal function $f_{\zeta} = h_{\zeta} + \overline{g_{\zeta}}$ with $h_{\zeta} = K$ , where K satisfies (1.4) or its rotations and $g_{\zeta}$ satisfies $g'_{\zeta} = \zeta z^n h'_{\zeta}$ .

Proposition 4.2. Let $f \in \mathcal{HC}_n(\phi)$ and $S_r$ be the area of the image $f(\mathbb{D}_r)$ ( $\mathbb{D}_r := \{z \in \mathbb{C} : |z| < r \leq 1\}$ ). Then <span id="page-8-6"></span> $$2\pi \int_0^r t \left(1 - |\zeta|^2 t^{2n}\right) (K'(-t))^2 dt \le S_r \le 2\pi \int_0^r t \left(1 - |\zeta|^2 t^{2n}\right) (K'(t))^2 dt. \tag{4.10}$$

Theorem 4.2. Let $f \in \mathcal{HC}_n(\phi)$ and $S_r$ be the area of the image $f(\mathbb{D}_r)$ . Then the inequality $$M_f(r) + \frac{S_r}{2\pi} \le d(f(0), \partial f(\mathbb{D}))$$ holds for $|z| = r \leq \min\{1/3, \widetilde{r}_f\}$ , where $\widetilde{r}_f$ is the smallest positive root in (0,1) of <span id="page-12-0"></span> $$L(\zeta, n, 1) = M_K(r) + |\zeta| \int_0^r t^n M_{K'}(t) dt + \int_0^r t \left(1 - |\zeta|^2 t^{2n}\right) (K'(t))^2 dt,$$ and $L(\zeta, n, 1)$ is given by (4.2) with r = 1.