Bulletin of the Korean Mathematical Society (대한수학회보)

Korean Mathematical Society (대한수학회)
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SUFFICIENT CONDITIONS AND RADII PROBLEMS FOR A STARLIKE CLASS INVOLVING A DIFFERENTIAL INEQUALITY

  • Received : 2019.12.12
  • Accepted : 2020.04.23
  • Published : 2020.11.30
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Abstract

Let 𝒜n be the class of analytic functions f(z) of the form f(z) = z + ∑∞k=n+1 αkzk, n ∈ ℕ defined on the open unit disk 𝔻, and let $${\Omega}_n:=\{f{\in}{\mathcal{A}}_n:\|zf^{\prime}(z)-f(z)\|<{\frac{1}{2}},\;z{\in}{\mathbb{D}}\}$$. In this paper, we make use of differential subordination technique to obtain sufficient conditions for the class Ωn. Writing Ω := Ω1, we obtain inclusion properties of Ω with respect to functions which map 𝔻 onto certain parabolic regions and as a consequence, establish a relation connecting the parabolic starlike class 𝒮P and the uniformly starlike UST. Various radius problems for the class Ω are considered and the sharpness of the radii estimates is obtained analytically besides graphical illustrations.

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References

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