Higher-order Schwarzian derivatives of starlike functions with respect to symmetrical points and a subclass of functions with bounded turning
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In theory of univalent functions, the estimates on higher-order Schwarzian derivatives of univalent analytic functions $f$ in the unit disk $\mathbf{D}$ have received wide attention and lots of sharp results on these have been given, such as the starlike and convex function classes. In this paper, we investigate the bounds for two kinds of higher-order Schwarzian derivatives of $f$ at zero with $f(0)=f'(0)-1=0$ when $f(z)$ satisfies $\Re\big(\frac{zf'(z)}{f(z)-f(-z)}\big)>0$ or $\Re f'(z)>0,$ where $z\in\mathbf{D}.$
Abstract
In theory of univalent functions, the estimates on higher-order Schwarzian derivatives of univalent analytic functions $f$ in the unit disk $\mathbf{D}$ have received wide attention and lots of sharp results on these have been given, such as the starlike and convex function classes. In this paper, we investigate the bounds for two kinds of higher-order Schwarzian derivatives of $f$ at zero with $f(0)=f'(0)-1=0$ when $f(z)$ satisfies $\Re\big(\frac{zf'(z)}{f(z)-f(-z)}\big)>0$ or $\Re f'(z)>0,$ where $z\in\mathbf{D}.$