The Sharp Bound for the Hankel Determinant of the Third Kind for Convex Functions

We prove the sharp inequality |H3,1(f)|≤ 4/135 for convex functions, that is, for analytic functions f with aₙ:=f⁽ⁿ⁾(0)/n!,~n∈ N , such that eqnarrayRe\1+zf(z)f(z)\>0 for~z∈ D:=∈ C:|z|<1\,eqnarray where H3,1(f) is the third Hankel determinant eqnarrayH3,1(f):=|array@ccc@a₁ & a₂ & a₃\\ a₂ & a₃ & a₄\\ a₃ & a₄ & a₅array|.eqnarray