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Starlikeness of integral transforms and duality. (English) Zbl 1244.30008

Let D be the complex unit disc and 𝒜 the class of functions f(z)=z+a 2 z 2 +. For f𝒜 also satisfing the condition

Re e iφ (1-α+2γ)f(z) z+(α-2γ)f ' (z)+γzf '' (z)-β>0

for suitable φ, α, β and γ, the authors give sufficient conditions so that the function defined by

V λ (f)(z)= 0 z λ(t)f(tz) tdt

(with λ chosen so that the above formula generalizes some results of other authors, but also provide new results) is starlike. Particular cases of λ are taken into account. Some consequences are also given. One of them gives a sharp estimate for the real constant β<1 that ensures starlikeness of a function f𝒜 that satisfies the condition Re (f ' (z)+αzf '' (z)+γz 2 f ''' (z))>β.

30C45Special classes of univalent and multivalent functions
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