Abstract In this paper we discuss coefficient problems for functions in the class 𝒞 0 ⁢ ( k ) {{\mathcal{C}}_{0}(k)} . This family is a subset of 𝒞 {{\mathcal{C}}} , the class of close-to-convex functions, consisting of functions which are convex in the positive direction of the real axis. Our main aim is to find some bounds of the difference of successive coefficients depending on the fixed second coefficient. Under this assumption we also estimate | a n + 1 | - | a n | {|a_{n+1}|-|a_{n}|} and | a n | {|a_{n}|} . Moreover, it is proved that Re ⁡ { a n } ≥ 0 {\operatorname{Re}\{a_{n}\}\geq 0} for all f ∈ 𝒞 0 ⁢ ( k ) {f\in{\mathcal{C}}_{0}(k)} .

中文翻译：

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逼近凸函数的连续系数

摘要 在本文中，我们讨论了类 𝒞 0 ⁢ ( k ) {{\mathcal{C}}_{0}(k)} 中函数的系数问题。这个族是 𝒞 {{\mathcal{C}}} 的一个子集，它是近凸函数类，由在实轴正方向上凸的函数组成。我们的主要目标是根据固定的第二系数找到连续系数差异的一些界限。在这个假设下，我们还估计 | + 1 | - | 一个 | {|a_{n+1}|-|a_{n}|} 和 | 一个 | {|a_{n}|} 。此外，证明了 Re ⁡ { an } ≥ 0 {\operatorname{Re}\{a_{n}\}\geq 0} 对于所有 f ∈ 𝒞 0 ⁢ ( k ) {f\in{\mathcal{C }}_{0}(k)}。