On the Fekete–Szegö type functionals for functions which are convex in the direction of the imaginary axis

Abstract

In this paper we consider two functionals of the Fekete–Szegö type: Φ f (μ) = a2a4 − μa3 and Θ f (μ) = a4 −μa2a3 for analytic functions f (z) = z +a2z +a3z + . . ., z ∈∆, (∆= {z ∈C : |z| < 1}) and for real numbers μ. For f which is univalent and convex in the direction of the imaginary axis, we find sharp bounds of the functionals Φ f (μ) and Θ f (μ). It is possible to transfer the results onto the class KR(i ) of functions convex in the direction of the imaginary axis with real coefficients as well as onto the class T of typically real functions. As corollaries, we obtain bounds of the second Hankel determinant in KR(i ) and T . 2020 Mathematics Subject Classification. 30C50. Funding. The project/research was financed in the framework of the project Lublin University of Technology Regional Excellence Initiative, funded by the Polish Ministry of Science and Higher Education (contract no.030/RID/2018/19). Manuscript received 14th August 2019, revised 24th June 2020, accepted 3rd November 2020.