## Integral means for univalent functions with negative coefficients.(English)Zbl 0889.30010

Let $$T$$ denote the class of functions $$f$$ analytic and univalent in the unit disk $$E= \{z:|z|< 1\}$$, of the form $$f(z)= z-\sum^\infty_{n= 2}a_nz^n$$, $$a_n\geq 0$$. It is shown that $$\int^{2\pi}_0 |f(re^{i\theta})|^\lambda d\theta$$ attains its maximum within the family $$T$$ for $$f(z)= z-{z^2\over 2}$$, for each $$\lambda>0$$. The extreme points of $$T$$ are $$f_1(z)= z$$, $$f_n(z)= z-{z^n\over n}$$, $$n= 2,3,\dots$$ . For integral means of derivatives of univalent functions, it is shown that $$r$$ as $$r= |z|$$ changes, so does the extremal integral means function. It is proved that for $$\lambda>0$$, $$f\in T$$, $\int^{2\pi}_0|f^k(re^{i\theta})|^\lambda d\theta\leq \int^{2\pi}_0|f^{(k)}_n(re^{i\theta})|^\lambda d\theta\text{ when } {n-k\over n-1}\leq r\leq {n- k+1\over n},$ when $$f_n$$ $$(n\geq k=2,3,4,\dots)$$ are the extreme points of $$T$$.
Regarding $$T^*(\alpha)$$, $$C(\alpha)$$, the subfamilies of $$T$$ consisting of functions starlike of order $$\alpha$$, convex of order $$\alpha$$ respectively, results obtaining extremal integral mean solutions are stated.

### MSC:

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