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LetSR  *be the class of starlike functions with real coefficients, i.e., the class of analytic functions f which satisfy the conditionf  ( 0 )  = 0 =  f ′   ( 0 )  − 1    ,Re { z  f ′   ( z )  / f  ( z )  } &gt; 0    , forz ∈ D : = { z ∈ C : | z | &lt; 1 }anda n  : =  f  ( n )    ( 0 )  / n !is real for alln ∈ N    . In the present paper, it is obtained that the sharp inequalities− 4 / 9 ≤  H  3 , 1    ( f )  ≤  3  / 9hold forf ∈   SR  *, whereH  3 , 1    ( f )is the third Hankel determinant of order 3 defined byH  3 , 1    ( f )  =  a 3   (  a 2   a 4  −  a 3 2  )  −  a 4   (  a 4  −  a 2   a 3  )  +  a 5   (  a 3  −  a 2 2  ).

The Sharp Bound of the Hankel Determinant of the Third Kind for Starlike Functions with Real Coefficients