Classes of functions whose derivatives have positive real part (Q1823335)

Let f(z) be analytic in the unit disk \(\Delta =\{z:\) \(| z| <1\}\), and let \(f(0)=0\), \(f'(0)=1\) and \(f'(z)\neq 0\) in \(\Delta\). Let \(\lambda\) be a fixed real number and let H(\(\lambda)\) denote the subclass of functions f satisfying \[ Re\{(1-\lambda)f'(z)+\lambda (1+zf''(z)/f'(z))\}>0 \] for all z in \(\Delta\). It is known that if \(f\in H(\lambda)\) with \(\lambda\leq 0\) then Re f\({}'(z)>0\) for z in \(\Delta\), and hence f is univalent in \(\Delta\). The case \(\lambda =1\) gives the convex functions which are also univalent in \(\Delta\). The univalence of f in the case \(\lambda >0\), \(\lambda\neq 1\) appears to be an open question. In this paper the authors consider the subclass M(\(\lambda)\) of H(\(\lambda)\) consisting of functions of the form \(f(z)=z- \sum^{\infty}_{n=2}a_ nz^ n\), with \(a_ n\geq 0\) for all \(n=2,3,... \). The class M(\(\lambda)\) is shown for all real \(\lambda\) to contain only starlike functions whose derivatives have positive real parts. Relationships between M(\(\lambda)\) and other classes of univalent functions are investigated, and coefficient and distortion bounds for functions in M(\(\lambda)\) (\(\lambda\geq 0)\) are established.