Generalized Zalcman Conjecture for Starlike and Typically Real Functions

AbstractZalcman conjectured that |a2n−a2n−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, |anam−an+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 |anam−an+m−1| for typically real functions