Abstract
Recently, Takahashi and Nunokawa (Appl. Math. Lett. 16:653-655, 2003) considered the class SS ∗ (α,β) of analytic functions, which satisfy the condition −πβ/2<arg{z f ′ (z)/f(z)}<πα/2 for all z in the unit disc on the complex plane, where 0≤α<1 and 0≤β<1. For α=β the class SS ∗ (α,β) is equal to the well-known class SS ∗ (β) of strongly starlike functions of order β. In this work, we derive a sufficient condition for analytic function to be in the class SS ∗ (α,β). Our theorem is a generalization of the result of Nunokawa et al. (Bull. Inst. Math. Acad. Sin. 31(3):195-199, 2003).MSC:30C45.
Highlights
Let A denote the class of functions with the series expansion ∞f (z) = z + akzk k=in the unit disc U = {z : |z| < }
We denote by S the subclass of A, consisting of univalent functions
We denote by S∗(α) the class of functions starlike of order α
Summary
We denote by S∗(α) the class of functions starlike of order α. Let SS∗(β) denote the class of strongly starlike functions of order β. In [ ] the following inclusion result for the class Gb was obtained. ) says that the domain p(Ur) lies in a sector between two rays arg{w} = –πβ/ and arg{w} = πα/ , and it contacts with the rays at p(z ) and at p(z ) The idea of this proof is that we transform this sector into the unit disc, and we will use Jack’s lemma. + itφ(z) maps the disc Ur onto a domain contained in the unit disc U and tangent to the unit circle at the points F(z ) and at F(z ). ( + itφ(z ))(φ(z ) + it) Taking logarithmic derivative in ( . ), we find that zp (z) α + β zq (z) p(z) q(z)
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