On the Fekete-Szegő problem for alpha-quasi-convex functions (Q2711240)

Let \(Q_\alpha\) \((\alpha\geq 0)\) denote the class of normalized analytic alpha-quasi-convex functions \(f\), defined in the unit disc, \(D=\{z: |z|<1\}\), by the condition NEWLINE\[NEWLINE\text{Re}\left[ (1-\alpha){f'(z)\over g'(z)}+ \alpha{\bigl( zf'(z)\bigr)' \over g'(z)}\right] >0,NEWLINE\]NEWLINE where \(f(z)=z+ \sum^\infty_{n=2} a_nz^n\) and where \(g(z)=z+ \sum^\infty_{n=2} b_nz^n\) is a convex univalent function in \(D\). Sharp upper bounds are obtained for \(|a_3-\mu a^2_2|\), when \(\mu\geq 0\).