ON A GENERALIZATION OF CLOSE-TO-CONVEXITY OF COMPLEX ORDER
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Abstract
The class $V_k$ of bounded boundary rotation is used to generalize the concept of close-to-convexity of complex order. A function $f:f(z)=z+\sum_{n=2}^\infty a_nz^n$, analytic in the unit disc $E$, belongs to $T_k(b)$, $b \neq 0$ (complex) if and only if there exists a function $g \in V_k$ such that
\[ Re\left\{1+\frac{1}{b}\left(\frac{f'(z)}{g'(z)}-1\right)\right\}>0, \quad z\in E.\]
Some basic properties, rate growth of Hankel determinant and radii problems for the functions in $T_k(b)$ are studied.
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