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

Let Λ:[0,1] 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 fS,

L Λ (f)=inf 0 1 Λ(t)(Re (f(tz)/tz-1/(1+t) 2 ) d t z D

and L Λ (S)=inf{L Λ (f)fS} resp. L Λ (C)=inf{L Λ (f)fC}, where CS denotes the subclass of closed-to-convex functions.

They ask whether there are functions Λ such that L Λ (S)=0 and show that for

Λ(t)/(1-t 2 ) decreasing on (0,1), L Λ (C)=0. Furthermore they consider the class P β of functions f holomorphic in D normalized in the origin as usual for which f ' (D)-β lies in a halfplane bounded by a straight line through the origin and functions

λ:[0,1], 0 1 λ(t)dt=1,λ0·

They determine numbers β=β(λ) such that the conclusion

fP β V λ (f)(z)= 0 1 λ(t)f(tz)/tdtS

holds and for some special λ they find β=β(λ) for which V λ (P β )S * , where S * is the class of starlike functions. For λ(t)=(c+1)t c , c>-1, this solves a problem discussed before by many authors.

30C55General theory of univalent and multivalent functions