Abstract
In this paper the authors generalize the notion Janowski-starlike functions of complex order on the unit disk $$\Delta $$ to the unit ball $$B^n$$ in higher dimensions, written as $$\mathcal {S}^*_{B^n}[A,B, \lambda ]$$. The growth and distortion theorems for Janowski-starlike mappings of complex order $$\lambda $$ are characterized in Sect. 2. Finally, we prove that if f belongs to a subclass of Janowski-starlike function of complex order on the unit disk $$\Delta $$ in $$\mathbb {C}$$, then the Roper–Suffridge extension operator and the modified Roper–Suffridge extension operator are all Janowski-starlike mappings of complex order $$\lambda $$ on the unit ball $$B^n$$ in Sect. 3. Related extension results for $$\mathcal {S}^*_{B^n}[A,B, \lambda ]$$ are also given.
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