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Zbl 0818.30013
Fournier, Richard; Ruscheweyh, Stephan
On two extremal problems related to univalent functions.
(English)
[J] Rocky Mt. J. Math. 24, No.2, 529-538 (1994). ISSN 0035-7596

Let $\Lambda: [0, 1]\to \bbfR$ be integrable over $[0, 1]$ and positive in $(0,1)$ and $S$ the class of functions univalent in the unit disk $D$ and normalized as usual. The authors consider for $f\in S$, $$L\sb \Lambda(f)= \inf\Biggl\{\int\sp 1\sb 0 \Lambda(t)\ (\text{Re}(f(tz)/tz- 1/(1+ t)\sp 2) dt\mid z\in D\Biggr\}$$ and $L\sb \Lambda(S)= \inf\{L\sb \Lambda(f)\mid f\in S\}$ resp. $L\sb \Lambda(C)= \inf\{L\sb \Lambda(f)\mid f\in C\}$, where $C\subset S$ denotes the subclass of closed-to-convex functions.\par They ask whether there are functions $\Lambda$ such that $L\sb \Lambda(S)= 0$ and show that for\par $\Lambda(t)/(1- t\sp 2)$ decreasing on $(0,1)$, $L\sb \Lambda(C)= 0$. Furthermore they consider the class $P\sb \beta$ of functions $f$ holomorphic in $D$ normalized in the origin as usual for which $f'(D)- \beta$ lies in a halfplane bounded by a straight line through the origin and functions $$\lambda: [0, 1]\to \bbfR,\quad \int\sp 1\sb 0 \lambda(t) dt= 1,\quad \lambda\ge 0.$$ They determine numbers $\beta= \beta(\lambda)$ such that the conclusion $$f\in P\sb \beta\Rightarrow V\sb \lambda(f) (z)= \int\sp 1\sb 0 \lambda(t) f(tz)/t dt\in S$$ holds and for some special $\lambda$ they find $\beta= \beta(\lambda)$ for which $V\sb \lambda(P\sb \beta)\subset S\sp*$, where $S\sp*$ is the class of starlike functions. For $\lambda(t)= (c+ 1) t\sp c$, $c> -1$, this solves a problem discussed before by many authors.
[K.J.Wirths (Braunschweig)]
MSC 2000:
*30C55 General theory of univalent and multivalent functions
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