Let f be analytic in the unit disk $\mathbb {D}=\{z\in \mathbb {C}:|z|<1 \}$ and let ${\mathcal S}$ be the subclass of normalised univalent functions with $f(0)=0$ and $f'(0)=1$ , given by $f(z)=z+\sum _{n=2}^{\infty }a_n z^n$ . Let F be the inverse function of f, given by $F(\omega )=\omega +\sum _{n=2}^{\infty }A_n \omega ^n$ for $|\omega |\le r_0(f)$ . Denote by $ \mathcal {S}_p^{* }(\alpha )$ the subset of $ \mathcal {S}$ consisting of the spirallike functions of order $\alpha $ in $\mathbb {D}$ , that is, functions satisfying $$\begin{align*}{\mathrm{Re}} \ \bigg\{e^{-i\gamma}\dfrac{zf'(z)}{f(z)}\bigg\}>\alpha\cos \gamma, \end{align*}$$ for $z\in \mathbb {D}$ , $0\le \alpha <1$ and $\gamma \in (-\pi /2,\pi /2)$ . We give sharp upper and lower bounds for both $ |a_3|-|a_2| $ and $ |A_3|-|A_2| $ when $f\in \mathcal {S}_p^{* }(\alpha )$ , thus solving an open problem and presenting some new inequalities for coefficient differences.

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关于螺旋函数的注释

让F在单位盘中进行解析$\mathbb {D}=\{z\in \mathbb {C}:|z|<1 \}$然后让${\数学S}$是归一化单价函数的子类$f(0)=0$和$f'(0)=1$, 由$f(z)=z+\sum _{n=2}^{\infty }a_n z^n$. 让F是的反函数F, 由$F(\omega )=\omega +\sum _{n=2}^{\infty }A_n \omega ^n$为了$|\omega |\le r_0(f)$. 表示为$ \mathcal {S}_p^{* }(\alpha )$的子集$ \数学{S}$由有序的螺旋函数组成$\阿尔法$在$\mathbb {D}$，即满足的函数$$\begin{align*}{\mathrm{Re}} \ \bigg\{e^{-i\gamma}\dfrac{zf'(z)}{f(z)}\bigg\}>\alpha \cos \gamma, \end{对齐*}$$为了$z\in \mathbb {D}$,$0\le \alpha <1$和$\gamma \in (-\pi /2,\pi /2)$. 我们为两者给出了明确的上限和下限$ |a_3|-|a_2| $和$ |A_3|-|A_2| $什么时候$f\in \mathcal {S}_p^{* }(\alpha )$，从而解决了一个开放问题并提出了一些新的系数差异不等式。