On close-to-convex functions

Lemma 2.1. [11, Vol. I, Page 67] Let $$f \in \mathcal{S}$$ and $z = re^{i\theta} \in \mathbb{D}$ . If $m'(r) < |f'(z)| < M'(r)$ . where m'(r) and M'(r) are real-valued functions of r in [0,1), then $$\int_0^r m'(t)dt \le |f(z)| \le \int_0^r M'(t)dt.$$ The following result was proved by Choi et al. [6]. But here we have written a part of it for our convenient.

Lemma 2.2. [6] For $$A, B \in \mathbb{C}$$ and $K, L, M \in \mathbb{R}$ , let $$\Omega(A, B, K, L, M) = \max_{\substack{|u_1 \leq 1 \\ |u_1| \leq 1}} (|A|(1 - |u_1|^2) + |B|(1 - |v_1|^2) + |Ku_1^2 + Lv_1^2 + 2Mu_1v_1|),$$ Further consider the following three conditions involving A, B, K, L, M: (A1) $$|A| \ge \max\{|K|\sqrt{1 - \frac{M^2}{KL}}, |M| - |K|\};$$ (B1) $|B| \ge \max\{|L|\sqrt{1 - \frac{M^2}{KL}}, |M| - |L|\};$ (B2) $|L| + |M| \le |B| < |L|\sqrt{1 - \frac{M^2}{KL}}.$ If $$KL \ge 0$$ , and $D = (|K| - |A|)(|L| - |B|) - M^2$ then $$\Omega(A,B,K,L,M) = \begin{cases} |A| + |B| & \text{if } |A| + |B| \ge |K| + |L| \text{ and } D \ge 0, \\ |A| + |L| - \frac{M^2}{|K| - |L|} & \text{if } |A| > |M| + |K| \text{ and } D < 0, \\ |B| + |K| - \frac{M^2}{|L| - |B|} & \text{if } |B| > |M| + |L| \text{ and } D < 0, \\ |K| + 2|M| + |L| & \text{otherwise.} \end{cases}$$ If KL < 0, then $\Omega(A, B, K, L, M) = |A| + |B| + \max\{0, R\}$ , where $$R = \begin{cases} 0, & \textit{when A1 \& B1 holds}, \\ |L| - |B| + \frac{M^2}{|A| + |K|}, & \textit{when A1 holds but B1 \& B2 does not hold.} \end{cases}$$

Theorem 3.1. If $f \in \widetilde{\mathcal{K}}_u$ and $z = re^{i\theta}$ , $0 \le r < 1$ , then <span id="page-3-4"></span> $$(3.1) re^{-r} \le |f(z)| \le re^r.$$ <span id="page-3-3"></span>(3.2) $$e^{-r}(1-r) \le |f'(z)| \le e^r(1+r),$$ Moreover, all the inequalities are sharp.

Theorem 4.1. The radius of convexity of $\widetilde{\mathcal{K}}_u$ is $(3-\sqrt{5})/2$ .

Theorem 5.1. Let $f \in \widetilde{\mathcal{K}}_u$ be given by (1.1). Then for every $\lambda \in \mathbb{R}$ 1. Let $$f \in \widetilde{\mathcal{K}}_u$$ be given by (1.1). Then for every $\lambda \in \mathbb{R}$ $$\Phi_{\lambda}(f) = |a_3 - \lambda a_2^2| \le \begin{cases} \frac{1}{2} - \lambda & \text{if } \lambda \le -\frac{1}{3}, \\ \frac{14 - 3\lambda}{6(4 + 3\lambda)} & \text{if } -\frac{1}{3} \le \lambda \le \frac{1}{6}, \\ \frac{1}{2} & \text{if } \frac{1}{6} \le \lambda \le 1, \\ \lambda - \frac{1}{2} & \text{if } \lambda \ge 1. \end{cases}$$ If the inequalities are sharp. Moreover, all the inequalities are sharp.

Theorem 6.1. Let $f \in \widetilde{\mathcal{K}}_u$ be of the form (1.1). Then $$||P_f|| \le \frac{9}{4},$$ and the estimate is sharp.