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 a₍: =f^ (n) (0) /n!, ~n N, such that eqnarrayRe\1+zf^{ (z) f^{ (z) }\}>0 for~z D: =\z 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