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.


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