Abstract
For a normalized analytic functionfdefined on the unit disc𝔻, letϕ(f,f′,f′′;z)be a function of positive real part in𝔻,ψ(f,f′,f′′;z)need not have that property in𝔻, andχ=ϕ+ψ. For certain choices ofϕandψ, a sharp radius constantρis determined,0<ρ<1, so thatχ(ρz)/ρmaps𝔻onto a specified region in the right half-plane.
Highlights
Let A be the class of functions f analytic in D = {z ∈ C : |z| < 1} and normalized by f(0) = 0 = f(0) − 1.Let S be its subclass consisting of univalent functions
For φ(z) := φ(f, f, f; z) = zf(z)/f(z) and ψ(z) := ψ(f, f, f; z) = z2f(z)/f(z), with f ∈ ST(1/2), several radius results for the sum φ+ψ to be in certain regions in the complex plane are obtained in the following result
Let f ∈ ST(1/2); let χ : D → C be defined by χ (z) z2f (z) f (z) zf (z), f (z) χi (z) = χ, i = 1, 2, . . . , 6
Summary
(e) For the function h given by (21), it follows from (22) and (23) that arg h (z) It follows from Lemma 2 that χi satisfies the required condition.
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