Coefficient problems for certain Close-to-Convex Functions

Lemma 1.1. [22, 23, 29] If ^p ∈ P of the form ^p(z) = 1 + ^c1^z ⁺ ^c2^z ² + c3z ³ ⁺ . . . , with ^c¹ ^≥ ⁰, then $$c_1 = 2\zeta_1$$ $$c_2 = 2\zeta_1^2 + 2(1 - \zeta_1^2)\zeta_2$$ (1.5) and $$c_3 = 2\zeta_1^3 + 4(1 - \zeta_1^2)\zeta_1\zeta_2 - 2(1 - \zeta_1^2)\zeta_1\zeta_2^2 + 2(1 - \zeta_1^2)(1 - |\zeta_2|^2)\zeta_3$$ (1.6) for some ^ζ¹ ^∈ [0, 1] and ^ζ2, ζ³ ^∈ ^D := {^z ^∈ ^C : ^|z|^< ¹} . For ^ζ¹ ^∈ ^D and ^ζ² ^∈ ^T ⁼ {^z ^∈ ^C : ^|z|= 1} , there is a unique function p ∈ P with c¹ and c² as in [\(1.5\)](#page-2-0)-[\(1.6\)](#page-2-3), namely $$p(z) = \frac{1 + (\overline{\zeta_1}\zeta_2 + \zeta_1)z + \zeta_2 z^2}{1 + (\overline{\zeta_1}\zeta_2 - \zeta_1)z - \zeta_2 z^2} \quad (z \in \mathbb{D}).$$ (1.7) For ^ζ1, ζ² ^∈ ^D and ^ζ³ ^∈ ^T, there exists a unique ^p ∈ P with ^c1, c² and ^c³ as in [\(1.5\)](#page-2-0)-[\(1.6\)](#page-2-3), namely, $$\frac{1 + (\overline{\zeta_1}\zeta_2 + \overline{\zeta_2}\zeta_3 + \zeta_1)z + (\overline{\zeta_1}\zeta_3 + \zeta_1\overline{\zeta_2}\zeta_3 + \zeta_2)z^2 + \zeta_3z^3}{1 + (\overline{\zeta_1}\zeta_2 + \overline{\zeta_2}\zeta_3 - \zeta_1)z + (\overline{\zeta_1}\zeta_3 - \zeta_1\overline{\zeta_2}\zeta_3 - \zeta_2)z^2 - \zeta_3z^3} \quad (z \in \mathbb{D}).$$ (1.8) Following result is due to Choi, Kim and Sugawa [6].

Lemma 1.2. [6] For A, B, C ^∈ ^R, let $$Y(A, B, C) := \max \{ |A + Bz + Cz^2| + 1 - |z|^2 : z \in \overline{\mathbb{D}} \}.$$ (1) If AC ≥ 0, then $$Y(A, B, C) = \begin{cases} |A| + |B| + |C|, & |B| \ge 2(1 - |C|), \\ 1 + |A| + \frac{B^2}{4(1 - |C|)}, & |B| < 2(1 - |C|). \end{cases}$$ (2) If AC < 0, then $$Y(A,B,C) = \begin{cases} 1 - |A| + \frac{|B|}{4(1-|C|)}, & -4AC(C^{-2}-1) \le B^2 \land |B| < 2(1-|C|), \\ \\ 1 + |A| + \frac{B^2}{4(1+|C|)}, & B^2 < \min\{4(1+|C|^2), -4AC(C^{-2}-1)\}, \\ \\ R(A,B,C), & otherwise \end{cases}$$ where $$R(A,B,C) = \begin{cases} |A| + |B| - |C|, & |C|(|B| + 4|A|) \le |AB|, \\ -|A| + |B| + |C|, & |AB| \le |C|(|B| - 4|A|), \\ \\ (|C| + |A|)\sqrt{1 - \frac{B^2}{4AC}}, & otherwise. \end{cases}$$

Theorem 2.1. Let $F_f$ be given by (1.3). If $f \in \mathcal{S}_s^*$ , then sharp bound on Hankel determinant for $F_f$ is given by $$|H_{2,1}(F_f)| \le \frac{1}{4}.$$ (2.1) Above inequality is sharp due to the function $$\tilde{f}(z) = \int_0^z (1+t^2)/(1-t^2)^2 dt. \tag{2.2}$$

Theorem 2.2. Let $F_f$ be given by (1.3). If $f \in \mathcal{F}_2$ , then sharp bound on Hankel determinant for $F_f$ is given by $$|H_{2,1}(F_f)| \le \frac{1}{4}.$$ (2.7) Above inequality is sharp due to the function given in Theorem 2.1.

Theorem 2.3. Suppose $X(x) = -48x^4 - 96x^3 - 392x^2 + 24x + 357$ and $F_f$ be given by (1.3). If $f \in \mathcal{F}_1$ , then sharp bound on Hankel determinant for $F_f$ is given by $$|H_{2.1}(F_f)| \le X(\eta)/2304,$$ (2.14) where $\eta \approx 0.0302$ is the unique real root of the equation X'(x) = 0. This result is sharp due to the extremal function $$\tilde{f}(z) = \int_0^z p_1(t)/(1-t)dt,$$ (2.15) where $p_1(z)$ is defined in (2.13).

Theorem 2.4. Suppose $X(x) = -176x^4 - 224x^3 - 264x^2 + 328x + 469$ . Let $F_f$ be given by (1.3). If $f \in \mathcal{F}_3$ , then sharp bound on Hankel determinant for $F_f$ is given by $$|H_{2,1}(F_f)| \le X(\eta)/2304,$$ (2.18) where $\eta \approx 0.3737$ is the unique real root of the equation X'(x) = 0. This result is sharp for the function $$f(z) = \int_0^z p_2(t)/(1-t+t^2)dt,$$ where $p_2(z)$ is given by (2.13).

Theorem 2.5. Suppose $48(17+x)X(x) = (1+x)(x^4+20x^3-114x^2+4x+125)$ . Let $F_f$ be given by (1.3). If $f \in \mathcal{F}_4$ , then sharp bound on Hankel determinant for $F_f$ is given by $$|H_{2,1}(F_f)| \le X(\eta),$$ (2.24) where $\eta \approx 0.381$ is the unique real root of the equation X'(x) = 0. This result is sharp with extremal function $$\tilde{f}(z) = \int_0^z p_3(t)/(1-t)^2 dt,$$ (2.25) where $p_3(z)$ is defined in (2.13).