Schwarzian Norm Estimates for Some Classes of Analytic Functions
Tóm tắt
Let
$${\mathcal {A}}$$
denote the class of analytic functions f in the unit disk
$${\mathbb {D}}=\{z\in {\mathbb {C}}:|z|<1\}$$
normalized by
$$f(0)=0$$
,
$$f'(0)=1$$
. In the present article, we obtain the sharp estimates of the Schwarzian norm for functions in the classes
$${\mathcal {G}}(\beta )=\{f\in {\mathcal {A}}:\mathrm{Re\,}[1+zf''(z)/f'(z)]<1+\beta /2\}$$
, where
$$\beta >0$$
and
$${\mathcal {F}}(\alpha )=\{f\in {\mathcal {A}}:\mathrm{Re\,}[1+zf''(z)/f'(z)]>\alpha \}$$
, where
$$-1/2\le \alpha \le 0$$
. We also establish two-point distortion theorem for functions in the classes
$${\mathcal {G}}(\beta )$$
and
$${\mathcal {F}}(\alpha )$$
.
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