Abstract :
Suppose f is a spirallike function of type β (or starlike function of order α) on the unit disk D in C.
Let Ωn,p1,p2,...,pn = {z = (z1, z2, . . . , zn) ∈ Cn: n
j=1 |zj |pj < 1}, where 1 p1 2 (or 0 < p1 2),
pj 1, j = 2, . . . , n, are real numbers. In this paper, we prove that
Φn,β2,γ2,...,βn,γn(f )(z) = f (z1), f (z1)
z1 β2
f (z1) γ2 z2, . . . , f (z1)
z1 βn
f (z1) γn zn
preserves spirallikeness of type β (or starlikeness of order α) on Ωn,p1,p2,...,pn, where βj ∈ [0, 1], γj ∈
[0, 1
pj ], and βj + γj 1, pj is the same as above, we choose the branches such that
f (z1)
z1 βj z1=0 = 1, f (z1) γj z1=0 = 1, j= 2, . . . , n.
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