The sharp bound of the third order Hankel determinant for inverse of Ozaki close-to-convex functions

Lemma 1.1. If $p \in \mathcal{P}$ , is of the form (1.5) with $c_1 \geq 0$ , such that $c_1 \in [0,2]$ then $$2c_2 = c_1^2 + \nu \mu,$$ $$4c_3 = c_1^3 + 2c_1\nu\mu - c_1\nu\mu^2 + 2\nu\left(1 - |\mu|^2\right)\rho,$$ $$8c_4 = c_1^4 + 3c_1^2\nu\mu + (4 - 3c_1^2)\nu\mu^2 + c_1^2\nu\mu^3 + 4\nu(1 - |\mu|^2)(1 - |\rho|^2)\psi + 4\nu(1 - |\mu|^2)(c_1\rho - c\mu\rho - \bar{\mu}\rho^2),$$ where $\nu := 4 - c_1^2$ , for some $\mu$ , $\rho$ and $\psi$ such that $|\mu| \le 1$ , $|\rho| \le 1$ and $|\psi| \le 1$ .

Lemma 1.2. Let $\psi_1, \psi_2, \psi_3, \psi_4 : [0,2] \to \mathbb{R}$ be a function definded by $$\psi_1(c) := -160c^2 + 16c^3 + 20c^4 - 4c^5 + \frac{5c^6}{4}$$ $$\psi_2(c) := 32c + 48c^2 + 32c^3 + 14c^4 - 10c^5 - \frac{13c^6}{2}$$ $$\psi_3(c) := -256 + 64c + 276c^2 - 48c^3 - 82c^4 + 8c^5 + \frac{29c^6}{4}$$ $$\psi_4(c) := 320 - 32c - 272c^2 - 32c^3 + 76c^4 + 10c^5 - 7c^6$$ $$\psi_5(c) := -64 - 64c + 48c^2 + 32c^3 - 12c^4 - 4c^5 + c^6$$ Then following are true (a) $$\psi_1(c) \leq 0$$ . for $c \in [0, 2]$ (b) $$\psi_1(c) + \psi_2(c) \le 0$$ . for $c \in (\frac{87137}{250000}, 2]$ (c) $$\psi_1(c) + \psi_2(c) = 0$$ . for $c \in (\frac{87137}{250000}, \frac{4511}{4000}]$ (d) $\psi_1(c) + \psi_2(c) + \psi_3(c) = 0$ . for $c \in (\frac{87137}{250000}, \frac{4511}{4000}]$ (d) $\psi_1(c) + \psi_2(c) + \psi_3(c) + 0.6\psi_4(c) \leq 0$ . for $c \in [\frac{4511}{4000}, 2]$ (d) $$\psi_1(c) + \psi_2(c) + \psi_3(c) + 0.6\psi_4(c) \le 0$$ . for $c \in \left[\frac{4511}{4000}, 2\right]$ (e) $$\psi_5(c) \leq 0$$ . for $c \in [0, 2]$ Proof(a). Since $4-c^2>0$ for $c\in[0,2]$ $$\psi_1(c) := -160c^2 + 16c^3 + 20c^4 - 4c^5 + \frac{5c^6}{4}$$ $$= -48c^2 - \frac{3c^6}{4} - 2c^2(4 - c^2)(14 - 2c + c^2) \le 0$$ $$Proof(b)$$ . Let $c = \frac{87137}{250000}t, 1 < t \le \frac{500000}{87137}$ . Then $$\psi_1(c(t)) + \psi_2(c(t)) := 11.1535t - 13.6064t^2 + 2.03249t^3 + 0.501798t^4 - 0.072018t^5 - 0.00941315t^6$$ $$\leq 0.0094t(1-t)(17.189-7.9049t+t^2)(69.0293+16.5646t+t^2) \leq 0$$ $$Proof(c)$$ . Let $c = \frac{87137}{250000}t, 1 < t \le \frac{563875}{174274}$ . Then $$\psi_1(c(t)) + \psi_2(c(t)) + \psi_3(c(t)) := -256. + 33.4606t + 19.9237t^2 - 0.708421t^4$$ $$-0.0308649t^5 + 0.00358596t^6$$ $$\leq 0.00358596(-18.403 + t)(8.71052 + t)$$ $$(15.6059 - 7.89366t + t^2)(28.5372 + 8.97902t + t^2) < 0$$ П Proof(d). Let $c = \frac{4511}{4000}t$ , $1 \le t \le \frac{8000}{4511}$ . Then $$\begin{split} \psi_1 + \psi_2 + \psi_3 + 06\psi_4 = &64 + 72.176t - 137.357t^2 - 45.8974t^3 \\ &+ 45.2907t^4 + 7.29666t^5 - 10.286t^6 \\ \leq &10.286(1-t)(0.511168+t) \\ &(4.07379 - 3.46659t + t^2)(3.06843 + 3.21981t + t^2) \leq 0 \end{split}$$ Proof(e). Since $4-c^2>0$ for $c\in[0,2]$ $$\psi_1 = (4 - c^2)^2 (-4 - 4c + c^2) \le 0$$

Lemma 1.3. Let $\Psi: [0, \frac{87137}{250000}] \times (0, 0.25) \to \mathbb{R}$ be a function definded by $$\Psi(c,x) = 320 + \psi_1(c) + \psi_2(c)x + \psi_3(c)x^2 + \psi_4(c)x^3 + \psi_5(c)x^4$$ where $\psi_1, \psi_2, \psi_3, \psi_4, \psi_5$ define as lemma 1.2 for $c \in [0, \frac{87137}{250000}]$ , Then $\Psi(c, x) \leq 320$ for $0 \leq c \leq \frac{87137}{250000}$ and 0 < x < 0.25

Lemma 1.4. Let $\Psi : \left[0, \frac{87137}{250000}\right] \times [0.25, 1] \to \mathbb{R}$ be a function definded by $\Phi(c, x) = \phi_1(x) + \phi_2(x)c + \phi_3(x)c^2 + \phi_4(x)c^3 + \phi_5(x)c^4 + \phi_6(x)c^5 + \phi_7(x)c^6$ where for $x \in [0.25, 1]$ , $$\phi_1(x) := -256x^2 + 320x^3 - 64x^4$$ $$\phi_2(x) := 32x + 64x^2 - 32x^3 - 64x^4$$ $$\phi_3(x) := -160 + 48x + 276x^2 - 272x^3 + 48x^4$$ $$\phi_4(x) := 16 + 32x - 48x^2 - 32x^3 + 32x^4$$ $$\phi_5(x) := 20 + 14x - 82x^2 + 76x^3 - 12x^4$$ $$\phi_6(x) := -4 - 10x + 8x^2 + 10x^3 - 4x^4$$ $$\phi_7(x) := \frac{5}{4} - \frac{13x}{2} + \frac{29x^2}{4} - 7x^3 + x^4$$ Then $\Phi(c, x) < 0$ for $0 \le c \le \frac{87137}{250000}$ and $0.25 \le x < 1$

Lemma 1.5. Let $\Psi: \left(\frac{87137}{250000}, \frac{4511}{4000}\right] \times (0, 0.6) \to \mathbb{R}$ be a function definded as in lemma 1.2. Then $\Psi(c, x) \leq 320$ for $\frac{87137}{250000} < c \leq \frac{4511}{4000}$ and 0 < x < 0.6 Proof. Since $\phi_4(c) > 0$ , $\phi_1(c)$ , $\phi_1(c) + \phi_2(c) < 0$ , $\phi_1(c) + \phi_2(c) + \phi_3(c) < 0$ , $\phi_1(c) + \phi_2(c) + \phi_3(c) < 0$ and $\phi_5(c) < 0$ in for $\frac{87137}{250000} < c \le \frac{4511}{4000}$ $$\Psi(c,x) \le 320 + (\phi_1(c) + \phi_2(c) + \phi_3(c) + 0.6\phi_4(c))x^2 + \phi_5(c)x^4 < 320.$$

Lemma 1.6. Let $\Gamma: \left(\frac{87137}{250000}, 1\right] \times [0.6, 1] \to \mathbb{R}$ be a function definded by $\Gamma(c,x) = \gamma_1(x) + \gamma_2(x)c + \gamma_3(x)c^2 + \gamma_4(x)c^3 + \gamma_5(x)c^4 + \gamma_6(x)c^5 + \phi_7(x)c^6$ where for $x \in [0.6, 1]$ , $$\gamma_1(x) := -256x^2 + 320x^3 - 64x^4$$ $$\gamma_2(x) := 96x^2 - 32x^3 - 64x^4$$ $$\gamma_3(x) := 164x^2 - 272x^3 + 48x^4$$ $$\gamma_4(x) := -32x^3 + 32x^4$$ $$\gamma_5(x) := -48x^2 + 76x^3 - 12x^4$$ $$\gamma_6(x) := -6x^2 + 10x^3 - 4x^4$$ $$\gamma_7(x) := 2x^2 - 7x^3 + x^4$$ Then $\Phi(c,x) < 0$ for $\frac{87137}{250000} < c \le 1$ and $0.6 \le x \le 1$

Lemma 1.7. Let $\Psi: \left(1, \frac{4511}{4000}\right] \times [0.6, 1] \to \mathbb{R}$ be a function definded as in lemma 1.2. Then $\Psi(c, x) \leq 320$ for $1 < c \leq \frac{4511}{4000}$ and $0.6 \leq x \leq 1$ Proof. Since $\phi_1(c)$ , $\phi_1(c) + \phi_2(c) < 0$ , $\phi_1(c) + \phi_2(c) + \phi_3(c) < 0$ for $1 < c \le 1.12775$ $\Psi(c, x) \le 320 + (\phi_1(c) + \phi_2(c) + \phi_3(c))x^2 + \phi_4(c)x^3 + \phi_5(c)x^4$ $= 320 + (-256 + 96c + 164c^2 - 48c^4 - 6c^5 + 2c^6)x^2$ $+ (320 - 32c - 272c^2 - 32c^3 + 76c^4 + 10c^5 - 7c^6)x^3$ $+ (-64 - 64c + 48c^2 + 32c^3 - 12c^4 - 4c^5 + c^6)x^4$ $\le 320 - 23x^2 + 63x^3 - 53x^4 < 318.459.$

Lemma 1.8. Let $\Psi: (\frac{4511}{4000}, 2] \times [0, 1] \to \mathbb{R}$ be a function definded as in lemma 1.2. Then $\Psi(c, x) \leq 320$ for $\frac{4511}{4000} < c \leq 2$ and $0 \leq x \leq 1$