# Gamma-Bazilevič Functions

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Department of Mathematics, Faculty of Mathematics and Natural Sciences, Brawijaya University, Jl. Veteran, Malang 65145, East Java, Indonesia

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Department of Mathematics, Swansea University, Bay Campus, Swansea SA1 8EN, UK

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Author to whom correspondence should be addressed.

Received: 6 January 2020 / Revised: 28 January 2020 / Accepted: 29 January 2020 / Published: 2 February 2020

(This article belongs to the Special Issue Complex Analysis and Its Applications 2019)

For $\gamma \ge 0$ and $\alpha \ge 0$ , we introduce the class ${\mathcal{B}}_{1}^{\gamma}\left(\alpha \right)$ of Gamma–Bazilevič functions defined for $z\in \mathbb{D}$ by $Re\left\{{\left[{\displaystyle \frac{z{f}^{\prime}\left(z\right)}{f{\left(z\right)}^{1-\alpha}{z}^{\alpha}}}+{\displaystyle \frac{z{f}^{\prime \prime}\left(z\right)}{{f}^{\prime}\left(z\right)}}+(\alpha -1)\left({\displaystyle \frac{z{f}^{\prime}\left(z\right)}{f\left(z\right)}}-1\right)\right]}^{\gamma}{\left[{\displaystyle \frac{z{f}^{\prime}\left(z\right)}{f{\left(z\right)}^{1-\alpha}{z}^{\alpha}}}\right]}^{1-\gamma}\right\}>0.$ We shown that ${\mathcal{B}}_{1}^{\gamma}\left(\alpha \right)$ is a subset of ${\mathcal{B}}_{1}\left(\alpha \right)$ , the class of ${B}_{1}\left(\alpha \right)$ Bazilevič functions, and is therefore univalent in $\mathbb{D}$ . Various coefficient problems for functions in ${\mathcal{B}}_{1}^{\gamma}\left(\alpha \right)$ are also given.

*Keywords:*univalent; Bazilevič functions; coefficients

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**MDPI and ACS Style**

Fitri, S.; Thomas, D.K. Gamma-Bazilevič Functions. *Mathematics* **2020**, *8*, 175.

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