Sharp Estimate of Fifth Coefficient for Ma Minda Starlike and Convex Functions

Lemma 1. [14, Lemma I] If the functions $1 + \sum_{n=1}^{\infty} b_n z^n$ and $1 + \sum_{n=1}^{\infty} c_n z^n$ are in $\mathcal{P}$ , then the same holds for the function $$1 + \frac{1}{2} \sum_{n=1}^{\infty} b_n c_n z^n.$$

Lemma 2. [14, Lemma II] Let $h(z) = 1 + u_1 z + u_2 z^2 + \cdots$ and $1 + G(z) = 1 + d_1 z + d_2 z^2 + \cdots$ be functions in $\mathcal{P}$ , and set $$\gamma_n = \frac{1}{2^n} \left[ 1 + \frac{1}{2} \sum_{k=1}^n \binom{n}{k} u_k \right], \quad \gamma_0 = 1.$$ If $A_n$ is defined by $$\sum_{n=1}^{\infty} (-1)^{n+1} \gamma_{n-1} G^n(z) = \sum_{n=1}^{\infty} A_n z^n,$$ then $|A_n| \leq 2$ . It is worth recalling the Möbius function $\Psi_{\zeta}$ which maps the unit disk onto the unit disk and given by <span id="page-2-2"></span> $$\Psi_{\zeta}(z) = \frac{z - \zeta}{1 - \overline{\zeta}z}, \quad \zeta \in \mathbb{D}. \tag{5}$$

Lemma 3. [4, Lemma 2.4] If $p \in \mathcal{P}$ , then for some $\zeta_i \in \overline{\mathbb{D}}, i \in \{1, 2, 3\}$ , <span id="page-2-0"></span> $$p_1 = 2\zeta_1, \tag{6}$$ $$p_2 = 2\zeta_1^2 + 2(1 - |\zeta_1|^2)\zeta_2,\tag{7}$$ <span id="page-2-1"></span> $$p_3 = 2\zeta_1^3 + 4(1 - |\zeta_1|^2)\zeta_1\zeta_2 - 2(1 - |\zeta_1|^2)\overline{\zeta_1}\zeta_2^2 + 2(1 - |\zeta_1|^2)(1 - |\zeta_2|^2)\zeta_3.$$ (8) For $\zeta_1, \zeta_2 \in \mathbb{D}$ and $\zeta_3 \in \mathbb{T}$ , there is a unique function $p = L \circ \omega \in \mathcal{P}$ with $p_1, p_2$ and $p_3$ as in (6)-(8), where $$\omega(z) = z\Psi_{-\zeta_1}(z\Psi_{-\zeta_2}(\zeta_3 z)), \quad z \in \mathbb{D}, \tag{9}$$ that is $$p(z) = \frac{1 + (\overline{\zeta_2}\zeta_3 + \overline{\zeta_1}\zeta_2 + \zeta_1)z + (\overline{\zeta_1}\zeta_3 + \zeta_1\overline{\zeta_2}\zeta_3 + \zeta_2)z^2 + \zeta_3z^3}{1 + (\overline{\zeta_2}\zeta_3 + \overline{\zeta_1}\zeta_2 - \zeta_1)z + (\overline{\zeta_1}\zeta_3 - \zeta_1\overline{\zeta_2}\zeta_3 - \zeta_2)z^2 - \zeta_3z^3}, \quad z \in \mathbb{D}.$$ Conversely, if $\zeta_1, \zeta_2 \in \mathbb{D}$ and $\zeta_3 \in \overline{\mathbb{D}}$ are given, then we can construct a (unique) function $p \in \mathcal{P}$ of the form (3) so that $p_i, i \in \{1, 2, 3\}$ , satisfy the identities in (6)-(8). For this, we define $$\omega(z) = \omega_{\zeta_1, \zeta_2, \zeta_3}(z) = z\Psi_{-\zeta_1}(z\Psi_{-\zeta_2}(\zeta_3 z)), \quad z \in \mathbb{D}, \tag{10}$$ where $\Psi_{\zeta}$ is the function given as in (5). Then $\omega \in \mathcal{B}_0$ . Moreover, if we define $p(z) = (1+\omega(z))/(1-\omega(z))$ , $z \in \mathbb{D}$ , then p is represented by (3), where $p_1$ , $p_2$ and $p_3$ satisfy the identities in (6)-(8) (see the proof of [4, Lemma 2.4]).

Theorem 6. Let $\varphi(z)$ be as defined in (2), whose coefficients satisfy the conditions C1 to C4. If $f \in \mathcal{C}(\varphi)$ , then $$|a_5| \le \frac{B_1}{20}.$$ The inequality is sharp. Proof. Since f ∈ C(ϕ), therefore <span id="page-7-0"></span> $$1 + \frac{zf''(z)}{f'(z)} = \varphi((p(z) - 1)/(p(z) + 1)), \tag{24}$$ where p ∈ P is given by [\(3\)](#page-1-0). By comparison of the coefficients of z, z 2 , z 3 in [\(24\)](#page-7-0) with the series expansion of f, ϕ and p, we get <span id="page-7-1"></span> $$a_5 = \frac{B_1}{40}I, (25)$$ where $$I = p_4 + I_1 p_1^4 + I_2 p_1^2 p_2 + I_3 p_1 p_3 + I_4 p_2^2,$$ with I1, I2, I³ and I⁴ given as in [\(13\)](#page-4-3), [\(14\)](#page-4-4) and [\(15\)](#page-4-5). Using the same method as in Theorem [5,](#page-3-3) we obtain $$|I| \leq 2,$$ when B1, B2, B³ and B⁴ satisfy all the conditions C1, C2, C3 and C4. Thus bound of |a⁵| follows from [\(25\)](#page-7-1). Let H(z) = z + a2z ² + a3z ³ ⁺ · · · ∈ S be given by $$1 + \frac{zH''(z)}{H'(z)} = \varphi(z^4),$$ where coefficients of ϕ(z) satisfy the conditions C1 to C4. Clearly, H ∈ C(ϕ) and for the function H, we have a² = a³ = a⁴ = 0 and a⁵ = B1/20. Thus bound is sharp for H.