On convex and starlike functions in a sector. (English) Zbl 0864.30008

If \(f\) is analytic in the unit disk \(D\), normalized by \(f(0)=0\), \(f'(0)=1\), and \[ \text{Re}\{zf'(z)/f(z)\}>0 \] for all \(z\in D\) then \(f\) is univalent and maps \(D\) onto a domain which is starshaped with respect to the origin. Similarly, the condition \(\text{Re}\{1+zf''(z)/f'(z)\}>0\) implies that \(f\) is univalent and maps \(D\) onto a convex domain. Let \(h(z)=(1+z)/(1-z)\). Then this paper considers the special classes \(C(\alpha)\) and \(S^*(\beta)\), defined to be the classes of normalized functions \(f\), analytic in \(D\) and satisfying \(\{1+zf''(z)/f'(z)\}\prec h(z)^\alpha\) and \(\{zf'(z)/f(z)\}\prec h(z)^\beta\), respectively, in \(D\). The authors exhibit real values functions \(\alpha(\beta)\) and \(\beta(\gamma)\) and prove that \(C(\alpha(\beta))\subset S^*(\beta)\) and that if \(f\in S^*(\beta(\gamma))\) then \(f(z)/z\prec h(z)^\gamma\). They show that these results are sharp. The numbers \(\alpha\) and \(\beta\) must be greater than zero and less than or equal to one. The function \(\beta(\gamma)={2\over\pi}\arctan(\gamma)\) is actually defined for all nonnegative \(\gamma\), but the result is vacuous if \(\gamma>1\). These facts are stated in the theorems of this carefully written paper.


30C45 Special classes of univalent and multivalent functions of one complex variable (starlike, convex, bounded rotation, etc.)