The sharp bound of the third Hankel determinant for Convex functions of order -1/2

We prove the sharp inequality |H3,1( f)| ⩽1/16 for the third Hankel determinant H3,1( f) for convex functions of order −1/2 i.e., functions f analytic in z ∈D := {z ∈C : |z| < 1} with an := f (n)(0)/n!, n ∈N, a1 := 1, such that Re  1+ z f ′′(z) f ′(z)  > −1 2, z ∈D, thus proving a recent conjecture. Mathematics subject classification (2020): 30C45, 30C50.