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Some Results on the Janowski's Starlike Functions of Complex Order

Y. Polatoglu, M. Bolcal · 2000 📄 find paper →

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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)

Each result is auto-extracted from the paper via OCR - the math is shown verbatim and may carry occasional transcription artifacts. Click to expand the full statement.

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) = 
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