On Coefficient Functionals for Functions with Coefficients Bounded by 1

Paweł Zaprawa, Anna Futa and Magdalena Jastrzębska
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Paweł Zaprawa: Faculty of Mechanical Engineering, Lublin University of Technology, ul. Nadbystrzycka 36, 20-618 Lublin, Poland
Anna Futa: Institute of Mathematics, Maria Curie-Skodowska University, pl. Marii Curie-Skodowskiej 1, 20-031 Lublin, Poland
Magdalena Jastrzębska: Department of Applied Mathematics, Lublin University of Technology, ul. Nadbystrzycka 38, 20-618 Lublin, Poland

Mathematics, 2020, vol. 8, issue 4, 1-14

Abstract: 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 Q R ( 1 2 ) of analytic functions with real coefficients which satisfy the condition Re f ( z ) z > 1 2 for z in the unit disk Δ . It is known that all coefficients of f ∈ Q R ( 1 2 ) 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 Q R ( 1 2 ) 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.

Keywords: coefficient problems; analytic functions; Schwarz functions; starlike functions; functions convex in one direction (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2020
References: View references in EconPapers View complete reference list from CitEc
Citations: View citations in EconPapers (1)

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