# On Coefficient Functionals for Functions with Coefficients Bounded by 1

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Faculty of Mechanical Engineering, Lublin University of Technology, ul. Nadbystrzycka 36, 20-618 Lublin, Poland

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Institute of Mathematics, Maria Curie-Skodowska University, pl. Marii Curie-Skodowskiej 1, 20-031 Lublin, Poland

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Department of Applied Mathematics, Lublin University of Technology, ul. Nadbystrzycka 38, 20-618 Lublin, Poland

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Author to whom correspondence should be addressed.

Received: 31 January 2020 / Revised: 17 March 2020 / Accepted: 25 March 2020 / Published: 1 April 2020

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

In this paper, we discuss two well-known coefficient functionals ${a}_{2}{a}_{4}-{{a}_{3}}^{2}$ and ${a}_{4}-{a}_{2}{a}_{3}$ . The first one is called the Hankel determinant of order 2. The second one is a special case of Zalcman functional. We consider them for functions in the class ${\mathcal{Q}}_{\mathbb{R}}\left(\frac{1}{2}\right)$ of analytic functions with real coefficients which satisfy the condition $\mathrm{Re}f\left(z\right)z>{\textstyle \frac{1}{2}}$ for z in the unit disk $\mathsf{\Delta}$ . It is known that all coefficients of $f\in {\mathcal{Q}}_{\mathbb{R}}\left(\frac{1}{2}\right)$ are bounded by 1. We find the upper bound of ${a}_{2}{a}_{4}-{{a}_{3}}^{2}$ and the bound of $|{a}_{4}-{a}_{2}{a}_{3}|$ . We also consider a few subclasses of ${\mathcal{Q}}_{\mathbb{R}}\left(\frac{1}{2}\right)$ and we estimate the above mentioned functionals. In our research two different methods are applied. The first method connects the coefficients of a function in a given class with coefficients of a corresponding Schwarz function or a function with positive real part. The second method is based on the theorem of formulated by Szapiel. According to this theorem, we can point out the extremal functions in this problem, that is, functions for which equalities in the estimates hold. The obtained estimates significantly extend the results previously established for the discussed classes. They allow to compare the behavior of the coefficient functionals considered in the case of real coefficients and arbitrary coefficients.
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*Keywords:*coefficient problems; analytic functions; Schwarz functions; starlike functions; functions convex in one direction

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**MDPI and ACS Style**

Zaprawa, P.; Futa, A.; Jastrzębska, M. On Coefficient Functionals for Functions with Coefficients Bounded by 1. *Mathematics* **2020**, *8*, 491.

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