Abstract
The purpose of this paper is to give exact value of the radius of starlikeness and distortion theorem, Koebe domain for the class of Janowski's starlike functions of complex order. We note that the class of Janowski's starlike functions of complex order contain many interesting subclasses of univalent functions.
Results & Lemmas (3)
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LEMMA 2.1.
LEMMA 2.1. Let f(z) ∈S∗(A, B, b),then z f ′(z) f(z) − 1 − h B2 −b(AB −B2) i r2 1 −B2r2 ≤|b|(A −B)r 1 −B2r2.
LEMMA 2.1. Let f(z) ∈S∗(A, B, b),then z f ′(z) f(z) − 1 − h B2 −b(AB −B2) i r2 1 −B2r2 ≤|b|(A −B)r 1 −B2r2 .
THEOREM 2.1. · radius
THEOREM 2.1. The radius of starlikeness of the class S∗(A, B, b) is rs = 2 |b|(A −B) + q |b|2(A −B)2 + 4[B2 + (AB −B2)Re b]. This radius is…
THEOREM 2.1. The radius of starlikeness of the class S∗(A, B, b) is rs = 2 |b|(A −B) + q |b|2(A −B)2 + 4[B2 + (AB −B2)Re b] . This radius is sharp. Because the extremal function is f∗(z) = z(1+Bz) b(A−B) B B ̸= 0 ebAz
THEOREM 3.1.
THEOREM 3.1. If f(z) ∈S∗(A, B, b),then (3.1) F(r; −A, −B, |b|) ≤|f(z)| ≤F(r; A, B, |b|) where F(r; A, B, |b|) = r(1+Br) |b|(A−B) B if B…
THEOREM 3.1. If f(z) ∈S∗(A, B, b),then (3.1) F(r; −A, −B, |b|) ≤|f(z)| ≤F(r; A, B, |b|) where F(r; A, B, |b|) = r(1+Br) |b|(A−B) B if B ̸= 0 ; re|b|Ar if B = 0 This bound are sharp. Because the extremal function is f∗(z) =