Bound for the fifth coefficient of certain starlike functions

Abstract For two different choices of φ , the sharp bound for the fifth coefficient of a normalized analytic function f satisfying z f ′ ( z ) / f ( z ) ≺ φ ( z ) is obtained by using a bound for a polynomial in the coefficients of functions with positive real part. Our proof uses a characterization of functions with positive real part in terms of certain positive semi-definite Hermitian form and certain well-known coefficient inequalities for functions with positive real part are shown to follow easily from this characterization.