## 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.

### MSC:

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

### Keywords:

convex functions; starshaped