Coefficient inequalities for a subclass of Bazilevič functions

Abstract

<jats:title>Abstract</jats:title><jats:p>Let <jats:italic>f</jats:italic> be analytic in <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_dema-2020-0040_eq_001.png" /><jats:tex-math>{\mathbb{D}}=\{z:|z\mathrm{|\hspace{0.17em}\lt \hspace{0.17em}1\}}</jats:tex-math></jats:alternatives></jats:inline-formula> with <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_dema-2020-0040_eq_002.png" /><jats:tex-math>f(z)=z+{\sum }_{n\mathrm{=2}}^{\infty }{a}_{n}{z}^{n}</jats:tex-math></jats:alternatives></jats:inline-formula>, and for <jats:italic>α</jats:italic> ≥ 0 and 0 &lt; <jats:italic>λ</jats:italic> ≤ 1, let <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_dema-2020-0040_eq_003.png" /><jats:tex-math>{ {\mathcal B} }_{1}(\alpha ,\lambda )</jats:tex-math></jats:alternatives></jats:inline-formula> denote the subclass of Bazilevič functions satisfying <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_dema-2020-0040_eq_004.png" /><jats:tex-math>\left|f^{\prime} (z){\left(\frac{z}{f(z)}\right)}^{1-\alpha }-1\right|\lt \lambda </jats:tex-math></jats:alternatives></jats:inline-formula> for 0 &lt; <jats:italic>λ</jats:italic> ≤ 1. We give sharp bounds for various coefficient problems when <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_dema-2020-0040_eq_005.png" /><jats:tex-math>f\in { {\mathcal B} }_{1}(\alpha ,\lambda )</jats:tex-math></jats:alternatives></jats:inline-formula>, thus extending recent work in the case <jats:italic>λ</jats:italic> = 1.</jats:p>

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