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We find the sharp bound for the third Hankel determinant    $$\begin{aligned} H_{3,1}(f):= \left| {\begin{array}{*{20}c} {a_{1} } &amp; {a_{2} } &amp; {a_{3} } \\ {a_{2} } &amp; {a_{3} } &amp; {a_{4} } \\ {a_{3} } &amp; {a_{4} } &amp; {a_{5} } \\ \end{array} } \right| \end{aligned}$$  H 3 , 1 ( f ) : = a 1 a 2 a 3 a 2 a 3 a 4 a 3 a 4 a 5 for analytic functions  f  with    $$a_n:=f^{(n)}(0)/n!,\ n\in \mathbb N,\ a_1:=1,$$  a n : = f ( n ) ( 0 ) / n ! , n ∈ N , a 1 : = 1 , such that    $$\begin{aligned} {{\,\mathrm{Re}\,}}f'(z)&gt;0,\quad z\in \mathbb D:=\{z \in \mathbb C: |z|&lt;1\}. \end{aligned}$$  Re f ′ ( z ) &gt; 0 , z ∈ D : = { z ∈ C : | z | &lt; 1 } .

The sharp bound of the third Hankel determinant for functions of bounded turning