1. Some properties associated to a certain class of starlike functions.
- Author
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Masih, Vali Soltani, Ebadian, Ali, and Yalçin, Sibel
- Subjects
STAR-like functions ,GEOMETRIC function theory ,ANALYTIC functions ,GEOMETRIC approach ,UNIVALENT functions ,CHARACTERISTIC functions - Abstract
Let 𝓐 denote the family of analytic functions f with f(0) = f′(0) – 1 = 0, in the open unit disk Δ. We consider a class S c s ∗ (α) := f ∈ A : z f ′ (z) f (z) − 1 ≺ z 1 + α − 1 z − α z 2 , z ∈ Δ , $$\begin{array}{} \displaystyle \mathcal{S}^{\ast}_{cs}(\alpha):=\left\{f\in\mathcal{A} : \left(\frac{zf'(z)}{f(z)}-1\right)\prec \frac{z}{1+\left(\alpha-1\right) z-\alpha z^2}, \,\, z\in \Delta\right\}, \end{array}$$ where 0 ≤ α ≤ 1/2, and ≺ is the subordination relation. The methods and techniques of geometric function theory are used to get characteristics of the functions in this class. Further, the sharp inequality for the logarithmic coefficients γ
n of f ∈ S c s ∗ $\begin{array}{} \mathcal{S}^{\ast}_{cs} \end{array}$ (α): ∑ n = 1 ∞ γ n 2 ≤ 1 4 1 + α 2 π 2 6 − 2 L i 2 − α + L i 2 α 2 , $$\begin{array}{} \displaystyle \sum_{n=1}^{\infty}\left|\gamma_n\right|^2 \leq \frac{1}{4\left(1+\alpha\right)^2}\left(\frac{\pi^2}{6}-2 \mathrm{Li}_2\left(-\alpha\right)+ \mathrm{Li}_2\left(\alpha^2\right)\right), \end{array}$$ where Li2 denotes the dilogarithm function are investigated. [ABSTRACT FROM AUTHOR]- Published
- 2019
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