Exact bounds to the Toeplitz determinants of certain order, Zalcman conjecture and Krushkals inequalities for the functions associated with the lemniscate of Bernoulli

Abstract

The main object of this article is to investigate sharp bounds of the Toeplitz determinants of certain order, Zalcman conjecture and Krushkals inequalities for normalized analytic functions in the open unit disk $D$, associated with the familiar subfamily of starlike functions associated with the right half of lemniscate of Bernoulli. The practical tools applied in the derivation of our main results are the coefficient inequalities of the Carathéodory class $P$.