Second Hankel Determinant of Logarithmic Coefficients of Convex and Starlike Functions

Abstract

We begin the study of Hankel matrices whose entries are logarithmic coefficients of univalent functions and give sharp bounds for the second Hankel determinant of logarithmic coefficients of convex and starlike functions.

Second Hankel Determinant of Logarithmic Coefficients of Convex and Starlike Functions of Order Alpha

“…To see that the last inequality in (3.9) holds observe that ψ is decreasing. Indeed, we have For α = 0 we get the estimate for the class S c of convex functions [12]. The inequality is sharp.…”

“…Based on the ideas mentioned above, in [12] was begun the research study of the Hankel determinant H q,n (F f /2) which entries are logarithmic coefficients of f , i.e.,…”

Coefficient problems for certain Close-to-Convex Functions

“…One can observe that H 2,1 (f ) = a 3 − a 2 2 , is the Fekete-Szegö functional, which is further generalised to a 3 − µa 2 2 , where µ is a complex number. In 1916, Bieberbach [11] estimated H 2,1 (f ) for the class S. Several authors have studied second Hankel determinant (see [16,21,25]), denoted by H 2,2 (f ) for certain subclasses of A. Many authors have studied the problem of calculating max f ∈F |H 2,2 (f )| for various subfamilies F ⊂ A [13,15,21].…”

On the Second Hankel determinant of Logarithmic Coefficients for certain univalent functions

“…Identifying the widespread applications of logarithmic coefficients, recently, Kowalczyk and Lecko [12] together proposed the study of the Hankel determinant whose entries are logarithmic coefficients of f ∈ S, which is given by…”

“…Kowalczyk and Lecko [12] obtained the sharp bound of second Hankel determinant of F f /2, i.e., H 2,1 (F f /2) for starlike and convex functions. The problem of computing the sharp bounds of H 2,1 (F f /2) has been considered in [4] for various subclasses of S.…”