Radii for Sections of Functions Convex in One Direction

Lemma 2.1. [11] If $$f(z) = z + \sum_{k=2}^{\infty} a_k z^k \in \mathcal{G}(\alpha)$$ , then $|a_n| \leq \frac{\alpha}{n(n-1)}$ , $n \geq 2$ . The inequality is sharp and equality is attained for a function f, where $f'(z) = (1 - z^{n-1})^{\alpha/(n-1)}, n \ge 2$ .

Lemma 2.2. [11, 9] If $f(z) \in \mathcal{G}(\alpha)$ , then the following inequalities and subordination results hold: <span id="page-3-1"></span>(2.6) $$\left| \frac{zf''(z)}{\alpha f'(z)} + \frac{|z|^2}{1 - |z|^2} \right| \le \frac{|z|}{1 - |z|^2},$$ <span id="page-3-2"></span> $$(2.7) f'(z) \prec (1-z)^{\alpha},$$ <span id="page-3-7"></span>(2.8) $$\frac{f(z)}{z} \prec \frac{1 - (1 - z)^{1 + \alpha}}{z(1 + \alpha)},$$ <span id="page-3-5"></span>(2.9) $$\frac{zf'(z)}{f(z)} \prec \frac{(1+\alpha)(1-z)}{1+\alpha-z}.$$

Lemma 2.3. If $\rho \in [0,1)$ and $n \ge 2$ , then the following sequences are decreasing: $$S_1(n,\rho) = \sum_{k=1}^{\infty} \frac{\rho^{k+n}}{(k+n)(k+n-1)}, \quad S_2(n,\rho) = \sum_{k=1}^{\infty} \frac{\rho^{k+n-1}}{k+n-1}.$$

Lemma 2.4. Let $f(z) = z + \sum_{k=2}^{\infty} a_k z^k \in \mathcal{G}(\alpha)$ and $\sigma_n(z; f) = \sum_{k=n+1}^{\infty} a_k z^k$ . Then, for $n \geq 2$ and $|z| = \rho < 1$ , the following inequalities hold: <span id="page-3-6"></span>(2.10) $$|\sigma_n(z;f)| \leq \alpha \left( (1-\rho) \ln(1-\rho) + \rho - \frac{\rho^2}{2} \right),$$ <span id="page-3-3"></span> $$(2.11) |\sigma'_n(z;f)| \leq -\alpha \left(\ln(1-\rho) + \rho\right),$$ <span id="page-3-4"></span> $$(2.12) |\sigma_n''(z;f)| \le \frac{\alpha \rho}{1-\rho}.$$

Lemma 2.5. [1, Theorem 6.2 (Rogosinski's Theorem)] Let $f(z) = \sum_{n=1}^{\infty} a_n z^n$ and $\psi(z) = \sum_{n=1}^{\infty} c_n z^n$ be analytic in $\mathbb{D}$ , and suppose $\psi \prec f$ . Then $$\sum_{n=1}^{n} |c_n|^2 \le \sum_{n=1}^{n} |a_n|^2, \quad n \in \mathbb{N}.$$ <span id="page-4-3"></span>The following result is the special case of [11, Lemma 5].

Lemma 2.6. Let $f \in \mathcal{G}$ and $s_n(z; f)$ be its nth section. Then for all $\rho \in (0, 1)$ and $n \geq 2$ , we have $$\left| \frac{s'_n(z;f)}{f'(z)} - 1 \right| \le |z|^n \left( \frac{1}{n} + \left( \frac{\pi}{\sqrt{6}} + 1 \right) \frac{|z|\sqrt{2\rho - \rho^2}}{\rho^n (1 - \rho)(\rho - |z|)} \right), \quad \text{for } |z| < \rho.$$

Theorem 3.1. Let $f(z) = z + \sum_{k=2}^{\infty} a_k z^k \in \mathcal{G}(\alpha)$ . Then $s_n(z; f)$ is convex of order $\beta$ in the disk $|z| < \rho_{\mathcal{K}}$ for all $n \geq 2$ , where $\rho_{\mathcal{K}}$ is the least positive root of $I_{\alpha,\beta}(\rho) = 0$ in [0,1) for all $\alpha \in (0,1]$ and $\beta \in [0,1)$ , where $I_{\alpha,\beta}(\rho)$ is given by <span id="page-4-0"></span>(3.13) $$I_{\alpha,\beta}(\rho) = (1-\rho)^{\alpha} \left(\rho(1+\alpha-\beta) - (1-\beta)\right) + \alpha\rho \left((2-\beta)\rho - (1-\beta)\right) - \alpha(1-\beta)(1-\rho)\ln(1-\rho).$$

Corollary 3.1. Let $f(z) = z + \sum_{k=2}^{\infty} a_k z^k \in \mathcal{G}$ . Then $s_n(z; f)$ is convex in $|z| < \rho_{\mathcal{K}_1} \approx 0.3578$ for all $n \geq 2$ , where $\rho_{\mathcal{K}_1}$ is the unique root of $I(\rho) = 0$ in [0, 1), where $I(\rho)$ is given by $$I(\rho) = 1 - 2\rho + (1 - \rho)\ln(1 - \rho).$$ Remark 3.1. The Corollary 3.1 improved the result of Maharana et al. [10, Theorem 1], which states that every section $s_n(z; f)$ of $f(z) \in \mathcal{G}$ is convex in the disk |z| < 1/2.

Theorem 3.2. Let $f(z) = z + \sum_{k=2}^{\infty} a_k z^k \in \mathcal{G}(\alpha)$ . Then $s_n(z; f)$ is starlike of order $\beta$ in the disk $|z| < \rho_{\mathcal{S}}$ for all $n \geq 2$ , where $\rho_{\mathcal{S}}$ is the least positive root of $J_{\alpha,\beta}(\rho) = 0$ in [0,1) for all $\alpha \in (0,1]$ and $\beta \in [0,1)$ , where $J_{\alpha,\beta}(\rho)$ is given by (3.19) <span id="page-7-1"></span> $$J_{\alpha,\beta}(\rho) = (1 - (1 - \rho)^{1+\alpha})((1 + \alpha)(1 - \beta - \rho) + \beta\rho) + \alpha(1 + \alpha)(1 + \alpha - \rho) \left( ((3 - \beta)\rho - (2 - \beta)) \ln(1 - \rho) - (2 - \beta)\rho + \left(2 - \frac{\beta}{2}\right)\rho^2 \right).$$

Corollary 3.2. Let $f(z) = z + \sum_{k=2}^{\infty} a_k z^k \in \mathcal{G}$ . Then $s_n(z; f)$ is starlike in $|z| < \rho_{\mathcal{S}_1}$ for all $n \geq 2$ , where $\rho_{\mathcal{S}_1} \approx 0.5698$ is the unique root of $J(\rho) = 0$ in [0, 1), where $J(\rho)$ is given by $$J(\rho) = \rho - \rho^2 + (2 - 3\rho) \ln(1 - \rho).$$

Theorem 3.3. Let $f(z) = z + \sum_{k=2}^{\infty} a_k z^k \in \mathcal{G}(\alpha)$ . Then $s_n(z; f)$ is close-to-convex of order $\beta$ in the disk $|z| < \rho_{\mathcal{C}}$ for all $n \geq 2$ , where $\rho_{\mathcal{C}}$ is the least positive root of $K_{\alpha,\beta}(\rho) = 0$ in [0,1) for all $\alpha \in (0,1]$ and $\beta \in [0,1)$ , where $K_{\alpha,\beta}(\rho)$ is given by <span id="page-11-0"></span>(3.27) $$K_{\alpha,\beta}(\rho) = (1 - \rho)^{\alpha} + \alpha(\ln(1 - \rho) + \rho) - \beta.$$

Corollary 3.3. Let $f(z) = z + \sum_{k=2}^{\infty} a_k z^k \in \mathcal{G}$ . Then $s_n(z; f)$ is close-to-convex in $|z| < \rho_{\mathcal{C}_1}$ for all $n \geq 2$ , where $\rho_{\mathcal{C}_1} \approx 0.6321$ is the unique root of $K(\rho) = 0$ in [0, 1), where $K(\rho)$ is given by $$K(\rho) = 1 + \ln(1 - \rho).$$

Theorem 3.4. Let $f \in \mathcal{G}$ . Then, for each $n \geq 2$ , we have (3.29) $$\left| \frac{s_n(z;f)}{f(z)} - 1 \right| \le |z|^n \left( \frac{1}{n(n+1)} + \frac{2|z|}{\sqrt{3}(1-|z|)} \right), \quad |z| = \rho < 1.$$

Theorem 3.5. Let $f \in \mathcal{G}$ . Then $\Re(s'_n(z;f)) > 0$ in $|z| \le 0.6321$ for $n \ge 17$ .

Theorem 3.6. Let $f \in \mathcal{G}$ . Then $s_n(z; f)$ is starlike in $|z| \leq 0.5698$ for $n \geq 10$ .