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Abstract
In the present paper, we investigate majorization properties for the class Mα
β(p,γ ) of
uniformly starlike functions and the class Nα
β(p,θ) of spiral-like functions related to an
exponential function, which are defined through the Liu–Owa integral operator Qα
β,p
given by (1.5). Also, some special cases of our main results in a form of corollaries are
shown.
MSC: Primary 30C45; secondary 30C80
Results & Lemmas (5)
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Theorem 2.1
Theorem 2.1 Let the function f ∈Ap and suppose that g ∈Mα β(p,γ ) with |α + β + p – 2| ≥ γ (α + β + p – 1) + e. If Qα β,pf (z) is majorized…
Theorem 2.1 Let the function f ∈Ap and suppose that g ∈Mα β(p,γ ) with |α + β + p – 2| ≥ γ (α + β + p – 1) + e. If Qα β,pf (z) is majorized by Qα β,pg(z) in U, that is, Qα β,pf (z) ≪Qα β,pg(z) (z ∈U), then, for |z| ≤r1, we have Qα–1 β,p f (z) ≤ Qα–1 β,p g(z)
Theorem 3.1
Theorem 3.1 Let the function f ∈Ap and assume that g ∈Nα β (p,θ) with |α + β – 1| ≥ |tanθ||α + β| + e. If Qα β,pf (z) is majorized by Qα…
Theorem 3.1 Let the function f ∈Ap and assume that g ∈Nα β (p,θ) with |α + β – 1| ≥ |tanθ||α + β| + e. If Qα β,pf (z) is majorized by Qα β,pg(z) in U, that is, Qα β,pf (z) ≪Qα β,pg(z) (z ∈U), then, for |z| ≤r2, we have Qα–1 β,p f (z) ≤ Qα–1 β,p g(z)
Corollary 4.1
Corollary 4.1 Let the function f ∈A and assume that g ∈Mα β(γ ) with |α + β – 1| ≥γ (α + β) + e. If Qα βf (z) is majorized by Qα βg(z) in…
Corollary 4.1 Let the function f ∈A and assume that g ∈Mα β(γ ) with |α + β – 1| ≥γ (α + β) + e. If Qα βf (z) is majorized by Qα βg(z) in U, then, for |z| ≤r3, we have Qα–1 β f (z) ≤ Qα–1 β g(z) , where r3 := r1(1,α,β,γ ) is the smallest positive root of the equation r2er –
Corollary 4.2
Corollary 4.2 Let the function f ∈Ap and assume that g ∈Mα β(p) with |α + β + p – 2| ≥e. If Qα β,pf (z) is majorized by Qα β,pg(z) in U,…
Corollary 4.2 Let the function f ∈Ap and assume that g ∈Mα β(p) with |α + β + p – 2| ≥e. If Qα β,pf (z) is majorized by Qα β,pg(z) in U, then, for |z| ≤r4, we have Qα–1 β,p f (z) ≤ Qα–1 β,p g(z) , where r4 := r1(p,α,β,0) is the smallest positive root of the equation r2er – |α + β + p – 2|r2 – er – 2r + |α + β + p – 2| = 0 (p ∈N;α ≥0;β > –1). Taking θ = 0 in Theorem 3.1, we state the following corollary.
Corollary 4.3
Corollary 4.3 Let the function f ∈Ap and suppose that g ∈Nα β (p) with |α + β – 1| ≥e. If Qα β,pf (z) is majorized by Qα β,pg(z) in U,…
Corollary 4.3 Let the function f ∈Ap and suppose that g ∈Nα β (p) with |α + β – 1| ≥e. If Qα β,pf (z) is majorized by Qα β,pg(z) in U, then, for |z| ≤r5, we have Qα–1 β,p f (z) ≤ Qα–1 β,p g(z) , where r5 := r2(α,β,0) is the smallest positive root of the equation r2er – |α + β – 1|r2 – er – 2r + |α + β – 1| = 0 (α ≥0;β > –1). (4.1)
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