Second Hankel determinant for close-to-convex functions
The aim of the article is to establish a sharp bound for the second-order Hankel determinant (i.e., \(|a_1a_4-a_3^2|\)) for functions that are analytic in the unit disc \(\Delta\) of the complex plane and have a power series development of the form \(f(z)=z+\sum_{k=2}^\infty a_kz^k\) and are, additionally, close-to-convex. The main result states that for every close-to-convex function of the above-mentioned form there is a sharp inequality involving the first three coefficients: \[ |a_2a_4-a_3^2|\leq A+B\varrho+C\varrho^2 \] where \(p,b\in[0,2], \varrho\in[0,1]\), \(H(b)=\frac{1}{3}(2+\frac{7}{18}b^2+\frac{25}{36}b^3)\) for \(b\in[0,6/7]\), \(H(b)=\frac{1}{9}(11b-2b^3)\) for \(b\in[6/7,2]\), and \(A,B,C\) are defined as follows: \[ A=\frac{1}{288}(4-p^2)(2-p)(16-p-9b), \] \[ B=\frac{1}{244}(4-p^2)(16-b^2+2bp+p^2), \] \[ C=\frac{1}{72}(4-b^2)(2+b^2)+\frac{1}{8}pH(b)+\frac{1}{288}p^2(4+10b^2+5bp+p^2)+\frac{1}{16}(b+p)(4-p^2). \] Utilizing some elementary techniques for finding bounds of certain functions, the authors show that \(A+B\varrho+C\varrho^2\leq A+B+C\) and then they compute the maximum of this expression, that leads to the result \(|a_2a_4-a_3^2|\leq 1.242\dots\). In the last section of the paper there are some further remarks on the second Hankel determinant for univalent functions.
scientific article; zbMATH DE number 6806459
scientific article