## On two extremal problems related to univalent functions.(English)Zbl 0818.30013

Let $$\Lambda: [0, 1]\to \mathbb{R}$$ 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 $$f\in S$$, $L_ \Lambda(f)= \inf\Biggl\{\int^ 1_ 0 \Lambda(t)\;(\text{Re}(f(tz)/tz- 1/(1+ t)^ 2) dt\mid z\in D\Biggr\}$ and $$L_ \Lambda(S)= \inf\{L_ \Lambda(f)\mid f\in S\}$$ resp. $$L_ \Lambda(C)= \inf\{L_ \Lambda(f)\mid f\in C\}$$, where $$C\subset S$$ denotes the subclass of closed-to-convex functions.
They ask whether there are functions $$\Lambda$$ such that $$L_ \Lambda(S)= 0$$ and show that for
$$\Lambda(t)/(1- t^ 2)$$ decreasing on $$(0,1)$$, $$L_ \Lambda(C)= 0$$. Furthermore they consider the class $$P_ \beta$$ 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 $\lambda: [0, 1]\to \mathbb{R},\quad \int^ 1_ 0 \lambda(t) dt= 1,\quad \lambda\geq 0.$ They determine numbers $$\beta= \beta(\lambda)$$ such that the conclusion $f\in P_ \beta\Rightarrow V_ \lambda(f) (z)= \int^ 1_ 0 \lambda(t) f(tz)/t dt\in S$ holds and for some special $$\lambda$$ they find $$\beta= \beta(\lambda)$$ for which $$V_ \lambda(P_ \beta)\subset S^*$$, where $$S^*$$ is the class of starlike functions. For $$\lambda(t)= (c+ 1) t^ c$$, $$c> -1$$, this solves a problem discussed before by many authors.

### MSC:

 30C55 General theory of univalent and multivalent functions of one complex variable
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### References:

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