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.


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