本文讨论了带辐角$β$（$-π/2<β<π/2$）的逼近凸函数的系数估计.利用Herglotz表示公式，得到了系数的精确界.特别地，我们解决了A. W. Goodman和E. B.萨夫在$\cite{GS}$。最后通过一些复杂的计算得到第三个系数的显式估计。
英文摘要：
This paper deals with coefficient estimates for close-to-convex functions with argument $β$ ($-π/2<β<π/2$). By using Herglotz representation formula, sharp bounds of coefficients are obtained. In particluar, we solve the problem posed by A. W. Goodman and E. B. Saff in $\cite{GS}$. Finally some complicted computations yield the explicit estimate of the third coefficient.
On logarithmic coefficients of some close-to-convex functions
The logarithmic coefficients $γ_n$ of an analytic and univalent function $f$ in the unit disk $\mathbb{D}=\{z\in\mathbb{C}:|z|<1\}$ with the normalization $f(0)=0=f'(0)-1$ is defined by $\log \frac{f(z)}{z}= 2\sum_{n=1}^{\infty} γ_n z^n$ . Recently, D.K. Thomas [On the logarithmic coefficients of close to convex functions , { \it Proc. Amer. Math. Soc.} { \bf 144} (2016), 1681--1687] proved that $|γ_3|\le \frac{7}{12}$ for functions in a subclass of close-to- convex functions (with argument $0$ ) and claimed that the estimate is sharp by providing a form of a extremal function. In the present …
A motivation comes from { \em M. Ismail and et al.: A generalization of starlike functions, Complex Variables Theory Appl., 14 (1990), 77--84} to study a generalization of close-to- convex functions by means of a $q$ -analog of a difference operator acting on analytic functions in the unit disk $\mathbb{D}=\{z\in \mathbb{C}:\,|z|<1\}$ . We use the terminology { \em $q$ -close-to- convex functions } for the $q$ -analog of close-to- convex functions . The $q$ -theory has wide applications in special functions and quantum physics which makes the study interesting and pertinent in this field. In t…
