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Integral transforms of a class of analytic functions. (English) Zbl 1146.30009

Let Δ be the complex unit disc and 𝒜 be the class of all analytic functions in Δ, normalized with the conditions f(0)=f ' (0)-1=0. A function in 𝒜 is said to be in the class 𝒫 λ (β) if Re [e iϕ (f ' (z)+γzf '' (z)-β]>0 in Δ (ϕ, γ0 and β<1). For a nonnegative real-valued integrable function λ(t) satisfying the normalizing condition 0 1 λ(t)dt=1 and f𝒜 let

F(z)=V λ (f)(z)= 0 1 λ(t)f(tz) tdt

and

Λ γ (t)= t 1 λ(s) s 1/γ ds,γ>0
Π γ (t)= t 1 Λ γ (s)s 1/γ-2 dsforgamma>0andΠ γ (t)= t 1 λ(s) sdsforγ=0

If

β 1-β=- 0 1 λ(t)g γ (t)dt

for some λ0 and β<1 and if, in addition Π γ (t)/(1-t 2 ) is decreasing on (0,1), the authors prove their principal result, that states that V λ (𝒫 λ (β))S * , where S * is the subclass of 𝒜 consisting of starlike functions in Δ. Some other results on the class 𝒫 γ (β) and applications are also given.

MSC:
30C45Special classes of univalent and multivalent functions