Denote by $${\mathcal {A}}$$ , the class of functions f, analytic in $${\mathbb {D}} =\{z:|z|<1\}$$ and given by $$f(z)=z+\sum _{n=2}^{\infty }a_{n}z^{n}$$ for $$z\in {\mathbb {D}}$$ , and by $${\mathcal {S}}$$ the subset of $${\mathcal {A}}$$ whose elements are univalent in $${\mathbb {D}}$$ . The class $${\mathcal {B}}_{1}(\alpha )\subset {\mathcal {S}}$$ , of Bazilevic functions is defined by $$Re\dfrac{zf^{\prime }(z)}{f(z)}\left( \dfrac{f(z)}{z}\right) ^{\alpha }>0$$ , for $$\alpha \ge 0$$ and $$z\in {\mathbb {D}}$$ . We give sharp bounds for $$|\gamma _{n}|$$ , where $$\log \dfrac{f(z)}{z}=2\sum _{n=1}^{\infty }\gamma _{n}z^{n}$$ , when $$n=1,2,3$$ , and $$ \alpha \ge 0$$ , and obtain the sharp bound for $$|\gamma _4|$$ when $$0\le \alpha \le \alpha ^{*}(\alpha ^{*}\approx 1.5464),$$ together with another bound for $$|\gamma _{4}|$$ when $$\alpha \ge 0.$$ Sharp bounds for some initial coefficients of the inverse function when $$f\in {\mathcal {B}} _{1}(\alpha )$$ are also found, which augment known results.

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关于 $$\mathbf {{\mathcal {B}}_1}(\varvec{\alpha })$$ Bazilevic 函数的系数

用 $${\mathcal {A}}$$ 表示，函数 f 的类，在 $${\mathbb {D}} =\{z:|z|<1\}$$ 中解析并由 $$ 给出f(z)=z+\sum _{n=2}^{\infty }a_{n}z^{n}$$ 为 $$z\in {\mathbb {D}}$$ ，由 $$ {\mathcal {S}}$$ $${\mathcal {A}}$$ 的子集，其元素在 $${\mathbb {D}}$$ 中是单价的。Bazilevic 函数的类 $${\mathcal {B}}_{1}(\alpha )\subset {\mathcal {S}}$$ 由 $$Re\dfrac{zf^{\prime }( z)}{f(z)}\left( \dfrac{f(z)}{z}\right) ^{\alpha }>0$$ ，对于 $$\alpha \ge 0$$ 和 $$z \in {\mathbb {D}}$$ 。我们为 $$|\gamma _{n}|$$ 给出了明确的界限，其中 $$\log \dfrac{f(z)}{z}=2\sum _{n=1}^{\infty }\ gamma _{n}z^{n}$$ ，当 $$n=1,2,3$$ 和 $$ \alpha \ge 0$$ ，并获得 $$|\gamma _4| 的锐界$$ 当 $$0\le \alpha \le \alpha ^{*}(\alpha ^{*}\approx 1.5464),