On starlike and convex functions of complex order with fixed second coefficient (Q326934)

Summary: Let \(F_{p}(b,M)\) denote the class of functions \(f(z)=z+\sum_{k=2}^{\infty}a_{k}z^{k}\) which are analytic in the open unit disc \(U=\{z:|z|<1\}\) and satisfy the inequality \[ \left| \frac{b-1+\frac{zf'(z)}{f(z)}}{b}-M\right| <M \] for \(b\neq 0\), complex, \(M>\dfrac{1}{2}\), \(|a_{2}|=2p\), \(0\leq p\leq \big( \frac{1+m}{2}\big) |b|\), \(m=1-\frac{1}{M}\) and for all \(z\in U\). Further \(f(z)\) is in the class \(G_{p}(b,M)\) if \(zf'(z)\) is in the class \(F_{p}(b,M)\). In the present paper, we obtain lower bounds for the classes introduced above and apply them to determine \(\gamma\)-spiral radiu for functions of the class \(F_{p}(b,M)\) and \(\gamma\)-convex radius for functions of the class \(G_{p}(b,M)\).