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Univalency of weighted integral transforms of certain functions. (English) Zbl 1098.30016

Let 𝒜 be the class of analytic functions in the unit disc 𝒰 normalized by f(0)=f ' (0)-1=0· For β<1 and γ0, let f𝒫 γ (β)𝒜 if and only if

Ree iη (1-γ)f(z) z+γf ' (z)-β>0,z𝒰,

for some real number η· Also for f𝒜 let define the integral transform

V λ (f)(z)= 0 1 λ(t)f(tz) tdt·

In this paper the authors give sharp conditions when (i) f𝒫 γ (β), γ[1,) V λ (f)𝒫 1 (β ' ); (ii) f𝒫 0 (β) V λ (f)𝒫 1 (β ' )· The second implication is studied only for some special choices of λ(t)·

30C55General theory of univalent and multivalent functions
33C55Spherical harmonics