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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.)
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