Coefficient estimates for close-to-convex functions with argument $β$

Theorem 1. Suppose $f(z) = z + \sum_{n=2}^{\infty} a_n z^n \in \mathcal{CL}(\beta)$ for $\alpha \beta \in (-\pi/2, \pi/2)$ , then the sharp inequality $$|a_n| \le \frac{2\cos\beta}{n} \max_{|u|=1} \left| \frac{n}{1 + e^{-2i\beta}} + \sum_{k=1}^{n-1} k u^{n-k} \right|.$$ (3) holds for $n = 2, 3, \cdots$ . Extremal functions are given by $$f'(z) = \frac{1}{(1 - yz)^2} \frac{1 + e^{-2i\beta}yu_nz}{1 - yu_nz}$$ for $y \in \partial \mathbb{D}$ , where $u_n \in \partial \mathbb{D}$ is a point at which the above maximum is attained. We mention here that it seems that there are no extremal functions other than the form given above in Theorem 1. Theorem A follows from Theorem 1 immediately by the elementary inequality $$\left| \frac{n}{1 + e^{-2i\beta}} + \sum_{k=1}^{n-1} k u^{n-k} \right| \le \frac{n}{2\cos\beta} + \frac{n(n-1)}{2}$$ for any $u \in \partial \mathbb{D}$ . The expression in (3) is implicit. When n=3, we can give a more concrete estimate and also show the extremal functions are unique;

Theorem 2. Suppose $f(z) = z + \sum_{n=2}^{\infty} a_n z^n \in \mathcal{CL}(\beta)$ , then the sharp inequality $$|a_3| \le \frac{2\cos\beta}{3}\sqrt{5 + \frac{9}{4\cos^2\beta} + \frac{13}{1 - t_0}}\tag{4}$$ holds, where $t_0$ is the unique root of the equation $$t^{3} - \left(\frac{4}{3}\cos^{2}\beta + 6\right)t^{2} + \left(\frac{40}{9}\cos^{2}\beta + 9\right)t + 4\cos^{2}\beta - 4 = 0$$ (5) in $0 \le t < 1$ . Equality holds in (4) if and only if $$f'(z) = \frac{1}{(1 - yz)^2} \frac{1 + e^{-2i\beta}yu_3z}{1 - yu_3z}$$ for some $y \in \partial \mathbb{D}$ , where $$u_3 = \begin{cases} 1 - \frac{t_0}{2} - i\sqrt{t_0 - \frac{t_0^2}{4}} \frac{\beta}{|\beta|}, & \text{when } \beta \neq 0; \\ 1, & \text{when } \beta = 0. \end{cases}$$ Remark 1 Comparing Theorem A and Theorem 2, it is not difficult to see that $$1 + 2\cos\beta = \frac{2\cos\beta}{3}\sqrt{5 + \frac{9}{4\cos^2\beta} + \frac{13}{1 - t_0}}$$ if and only if $$t_0 = \frac{9 - 9\cos\beta}{9 + 4\cos\beta}.$$ Since this $t_0$ is a root of (5) in [0,1) only when $\beta = 0$ , Theorem A is sharp only when $\beta = 0$ for n = 3. Finally we give an example to show how Theorem 2 works. Example. Let $\beta = \pi/4$ . Applying Mathematica, we may get the root of equation (5) which belongs to [0,1) is $0.201\cdots$ , therefore in this case $$|a_3| \lesssim 2.394$$ which is less than $1 + \sqrt{2} \approx 2.414$ by Theorem A.

Lemma 1. (see [3] p. 52) If $f \in S^*$ , then there exists a Borel probability measure $\nu$ on $\partial \mathbb{D}$ such that $$f(z) = \int_{\partial \mathbb{D}} \frac{z}{(1 - yz)^2} d\nu(y).$$

Definition 1. A function $f \in A_1$ is said to be close-to-convex (denoted by $f \in CL$ ) if there exist a starlike function g and a real number $\beta \in (-\pi/2, \pi/2)$ such that $$\frac{zf'}{g} \in \mathcal{P}_{\beta}.$$ This definition involving a real number $\beta$ is slightly different from the original one due to Kaplan [5]. An equivalent definition of $\mathcal{CL}$ by using Kaplan class and some related sets of univalent functions can be found in [6]. If we specify the real number $\beta$ in the above definition, the corresponding function is called a close-to-convex function with argument $\beta$ and we denote the class of all such functions by $\mathcal{CL}(\beta)$ (see [1, II, Definition 11.4]). Note that the union of class $\mathcal{CL}(\beta)$ over $\beta \in (-\pi/2, \pi/2)$ is precisely $\mathcal{CL}$ while the intersection is the class of convex functions. These results were given in [2] without proof. Since the former one is obvious, we will only give an outline of the proof of the latter one. Choose a sequence $\{\beta_n\} \subset (-\pi/2, \pi/2)$ such that $\beta_n \to \pi/2$ as $n \to \infty$ . The assertion follows from the facts that the class of starlike functions is compact in the sense of locally uniform convergence and any function sequence $\{p_n\}$ where $p_n \in \mathcal{P}_{\beta_n}$ converges to the constant function 1 locally uniform as $\beta_n \to \pi/2$ . In the literature, when studying the close-to-convex functions, some authors focus only on the case $\beta=0$ . A. W. Goodman and E. B. Saff [2] were the first to point out explicitly that $\mathcal{CL}(\beta)$ and $\mathcal{CL}$ are different when $\beta\neq 0$ and more deeply the class $\mathcal{CL}(\beta)$ has no inclusion relation with respect to $\beta$ . Therefore it is useful to consider the individual class $\mathcal{CL}(\beta)$ . The present paper follows their way in this direction and improves their result concerning the class $\mathcal{CL}(\beta)$ ; Theorem A (Goodman-Saff [2]) Suppose $$f(z) = z + \sum_{n=2}^{\infty} a_n z^n \in \mathcal{CL}(\beta)$$ for a $\beta \in (-\pi/2, \pi/2)$ . Then $$|a_n| < 1 + (n-1)\cos\beta.$$ for $n=2,3,\cdots$ . If either n=2 or $\beta=0$ , the inequality is sharp. In the above mentioned paper, they also stated that the problem of finding the maximum for $|a_n|$ in the class $\mathcal{CL}(\beta)$ was difficult for $n \geq 3$ . With regard to their problem, in the present paper we shall establish the following theorems: