A Conjecture on H 3(1) for Certain Starlike Functions

Abstract

ABSTRACT We prove a conjecture concerning the third Hankel determinant, proposed by Kumar and Kamaljeet in [A cardioid domain and starlike functions, Anal. Math. Phys. 11 (2021), Art. 54], which states that |H 3(1)| ≤ 1/9 is sharp for the class S ℘ * = { z f ′ ( z ) / f ( z ) ≺ φ ( z ) : = 1 + z e z } . In addition, we also establish bounds for sixth and seventh coefficient, and |H 4(1)| for functions in S ℘ * . The general bounds for two and three folds symmteric functions related with the Ma-Minda classes S * ( φ ) of starlike functions are also obtained.