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Zalcman conjectured that |a²_n−a_2n−1|≤(n−1)²,n=2,3,… forf(z)=z+a₂z²+a₃z³+···∈S, the class of normalized holomorphic and univalent functionsf(z) in the unit disk D . We propose a generalized conjecture as follows. Forf(z)=z+a₂z²+a₃z³+···∈S, |a_na_m−a_n+m−1|≤(n−1)(m−1),n,m=2,3,…. We prove that this conjecture holds for starlike functions and univalent functions with real coefficients. We also obtain sharp bounds on |a_na_m−a_n+m−1| for typically real functions.