Abstract
For $α\geq 0$, $δ>0$, $β<1$ and $γ\geq 0$, the class $\mathcal{W}_β^δ(α,γ)$ consist of analytic and normalized functions $f$ along with the condition \begin{align*} {\rm Re\,} e^{iφ}(\dfrac{}{}(1\!-\!α\!+\!2γ)\!({f}/{z})^δ+(α\!-\!3γ\!+\!γ[\dfrac{}{}(1-{1}/δ)({zf'}/{f})+ {1}/δ(1+{zf''}/{f'})]).\\ .\dfrac{}{}({f}/{z})^δ\!({zf'}/{f})-β)>0, \end{align*} where $φ\in\mathbb{R}$ and $|z|<1$, is taken into consideration. The class $\mathcal{S}^\ast_s(ζ)$ be the subclass of the univalent functions, defin
Results & Lemmas (17)
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Theorem 2.1.
Theorem 2.1. Let µ ≥0, ν ≥0 are defined in (2.1), δ ≥1 and 1 −1 δ ≤ζ ≤ 1 −1 2δ . Let β <1 satisfy β (1 −β) = − Z 1 0 λ(t)gδ
Theorem 2.1. Let µ ≥0, ν ≥0 are defined in (2.1), δ ≥1 and 1 −1 δ ≤ζ ≤ 1 −1 2δ . Let β <1 satisfy β (1 −β) = − Z 1 0 λ(t)gδ
Theorem 2.2.
Theorem 2.2. Let γ ≥0(µ ≥0, ν ≥0), δ ≥1 and 1−1 δ ≤ζ ≤ 1−1 2δ . Assume that the functions Λδ ν(t) and Πδ µ,ν(t), defined in (2.7) and…
Theorem 2.2. Let γ ≥0(µ ≥0, ν ≥0), δ ≥1 and 1−1 δ ≤ζ ≤ 1−1 2δ . Assume that the functions Λδ ν(t) and Πδ µ,ν(t), defined in (2.7) and (2.8), respectively are positive on t ∈(0, 1) and integrable on t ∈[0, 1]. If β < 1 satisfy (2.9) and tδ/µ−1Πδ µ,ν(t)
Theorem 3.1.
Theorem 3.1. Let β < 1 satisfy (2.9) and let λ(t) be real-valued, non-negative and integrable function for t ∈(0, 1). Further assume that…
Theorem 3.1. Let β < 1 satisfy (2.9) and let λ(t) be real-valued, non-negative and integrable function for t ∈(0, 1). Further assume that the functions Λδ ν(t) and Πδ µ,ν(t) defined in (2.7) and (2.8), respectively are positive on (0, 1) and integrable on [0, 1]. Then for f(z) ∈Wδ β(α, γ), the function Fδ = V δ λ (f)(z) belongs to the class S∗ s (ζ) with 1−1 δ ≤ ζ ≤ 1−1
Theorem 3.2.
Theorem 3.2. Let β < 1 satisfy (2.9) and let λ(t) be real-valued, non-negative and integrable function for t ∈(0, 1). Further assume that…
Theorem 3.2. Let β < 1 satisfy (2.9) and let λ(t) be real-valued, non-negative and integrable function for t ∈(0, 1). Further assume that the functions Λδ ν(t) and Πδ µ,ν(t) defined in (2.7) and (2.8), respectively are positive on (0, 1) and integrable on [0, 1]. Then for f(z) ∈Wδ β(α, γ), the function Fδ = V δ λ (f)(z) belongs to the class S∗ 1 −1 δ or z1−δ(Fδ(z))δ ∈S∗, δ ≥1, whenever tλ′(t) λ(t) ≤
Theorem 4.2
Theorem 4.2], the bound is 1 + 1 µ, where µ ≥1, which is clearly less than or equal to 2. 4. Applications In this section, using the…
Theorem 4.2], the bound is 1 + 1 µ, where µ ≥1, which is clearly less than or equal to 2. 4. Applications In this section, using the conditions derived in Section 3, applications for various choices of λ(t) are considered such that the conditions under which the generalized integral op- erator (1.2), for respective choice maps Wδ β(α, γ) to S∗ s(ζ) are examined. To start with consider λ(t) = (1 + c)tc, c > −1, (4.1) the integral operator (1.2) defined by the above weight function λ(t) is known as
Theorem 4.1.
Theorem 4.1. Let γ ≥0 (µ ≥0, ν ≥0), ξ ∈[0, 1/2], δ ≥1 and c > −1. Further let β < 1 satisfy (2.9), where λ(t) is given in (4.1). Then for…
Theorem 4.1. Let γ ≥0 (µ ≥0, ν ≥0), ξ ∈[0, 1/2], δ ≥1 and c > −1. Further let β < 1 satisfy (2.9), where λ(t) is given in (4.1). Then for f(z) ∈Wδ β(α, γ), the function z1−δ Bδ c(f)(z) δ belongs to the class S∗(ξ), whenever c ≤
Theorem 3.1
Theorem 3.1 for ξ ∈(0, 1/2], and Theorem 3.2 for ξ = 0, the result is immediate. □
Theorem 3.1 for ξ ∈(0, 1/2], and Theorem 3.2 for ξ = 0, the result is immediate. □
Corollary 4.1.
Corollary 4.1. Let γ ≥0(µ ≥0, ν ≥0), ξ ∈[0, 1/2] and δ ≥1. Let β < 1 satisfy β 1 −β = − Z 1 0 gδ µ,ν(t)dt, where gδ µ,ν(t) is given in…
Corollary 4.1. Let γ ≥0(µ ≥0, ν ≥0), ξ ∈[0, 1/2] and δ ≥1. Let β < 1 satisfy β 1 −β = − Z 1 0 gδ µ,ν(t)dt, where gδ µ,ν(t) is given in (2.5) for γ > 0, and (2.4) for γ = 0. Moreover, F(z) ∈A satisfies Re " z F(z) z
Theorem 2.2
Theorem 2.2 for the given cases, it is enough to show that the function a(t) defined in (3.6) decreases for t ∈(0, 1). Now taking the…
Theorem 2.2 for the given cases, it is enough to show that the function a(t) defined in (3.6) decreases for t ∈(0, 1). Now taking the logarithmic derivative of a(t) will give a′(t) a(t) = 2(t + ξ + ξt) (1 −t2)b(t) b(t) + (1 −t2)b′(t) 2(t + ξ + ξt) . It is easy to see that the terms (t+ξ +ξt), (1−t2), a(t) and b(t) are positive for all values of t ∈(0, 1) and ξ ∈[0, 1/2]. Thus a′(t) ≤0 is equivalent of obtaining r(t) ≤0, where r(t) := b(t) + (1 −t2)b′(t) 2(t + ξ + ξt). Clearly r(1) = 0, theref
Theorem 4.2.
Theorem 4.2. Let γ ≥0 (µ ≥0, ν ≥0), ξ ∈[0, 1/2], δ ≥1 and a, b, c > 0. Let β < 1 satisfy (2.9), where λ(t) is given by (4.6). Then for f(z)…
Theorem 4.2. Let γ ≥0 (µ ≥0, ν ≥0), ξ ∈[0, 1/2], δ ≥1 and a, b, c > 0. Let β < 1 satisfy (2.9), where λ(t) is given by (4.6). Then for f(z) ∈Wδ β(α, γ), the function (z1−δ Hδ a,b,c(f)(z) δ) belongs to the class ∈S∗(ξ), whenever (i). (c−a−b) ≥1 and 0 < b ≤1 for γ = 0 and δ ≤α, (ii). (c−a−b) ≥0 and 0 < b ≤ 6−δ µ −δ ν
Theorem 2.2
Theorem 2.2, the result follows directly. □
Theorem 2.2, the result follows directly. □
Theorem 4.3.
Theorem 4.3. Let γ ≥0 (µ ≥, ν ≥0), k > −1, p ≥1, ξ ∈[0, 1/2] and δ ≥1. Let β < 1 satisfy (2.9), where λ(t) is given in (4.8). Then for f(z)…
Theorem 4.3. Let γ ≥0 (µ ≥, ν ≥0), k > −1, p ≥1, ξ ∈[0, 1/2] and δ ≥1. Let β < 1 satisfy (2.9), where λ(t) is given in (4.8). Then for f(z) ∈Wδ β(α, γ), the function z1−δ F δ k,p(f)(z) δ belongs to the class ∈S∗(ξ), whenever (i). p ≥2 and −1 < k ≤0 for γ = 0 and δ ≤α, (ii). p ≥1 and −1 < k ≤5 −δ µ −δ ν
Theorem 4.4.
Theorem 4.4. Let γ ≥0 (µ ≥0, ν ≥0), δ ≥1, ξ ∈[0, 1/2] and a, b, c > 0. Let β < 1 satisfy β 1−β = − Γ(c) Γ(a)Γ(b)Γ(c−a−b+1) Z 1 0…
Theorem 4.4. Let γ ≥0 (µ ≥0, ν ≥0), δ ≥1, ξ ∈[0, 1/2] and a, b, c > 0. Let β < 1 satisfy β 1−β = − Γ(c) Γ(a)Γ(b)Γ(c−a−b+1) Z 1 0 tb−1(1−t)c−a−b 2F1 c−a, 1−a c−a−b+1 ; 1−t gδ µ,ν(t)dt,
Corollary 4.2.
Corollary 4.2. Let γ ≥0 (µ ≥0, ν ≥0), δ ≥1, ξ ∈[0, 1/2] and b, c > 0. Let β < 1 satisfy β (1 −β) = − Γ(c) Γ(b)Γ(c −b) Z 1 0 tb−1(1…
Corollary 4.2. Let γ ≥0 (µ ≥0, ν ≥0), δ ≥1, ξ ∈[0, 1/2] and b, c > 0. Let β < 1 satisfy β (1 −β) = − Γ(c) Γ(b)Γ(c −b) Z 1 0 tb−1(1 −t)c−b−1gδ µ,ν(t)dt, where gδ µ,ν(t) is given in (2.5) for γ > 0, and (2.4) for γ = 0. Then for f(z) ∈Wδ β(α, γ), the function z1−δ Lδ b,c(f)(z)
Corollary 4.3.
Corollary 4.3. Consider γ ≥0 (µ ≥0, ν ≥0), b > 0, c > 0 and δ ≥1. Let β0 < β < 1, where β0 = 1 − 1 2 1 − 5F4 1, b, (2 −ξ), δ
Corollary 4.3. Consider γ ≥0 (µ ≥0, ν ≥0) , b > 0, c > 0 and δ ≥1. Let β0 < β < 1, where β0 = 1 − 1 2 1 − 5F4 1, b, (2 −ξ), δ
Theorem 4.5.
Theorem 4.5. Let a > −1, b > −1, γ ≥0 (µ ≥0, ν ≥0), ξ ∈[0, 1/2] and δ ≥1. Let β < 1 satisfy (2.9), where λ(t) is given in (4.10). Then for…
Theorem 4.5. Let a > −1, b > −1, γ ≥0 (µ ≥0, ν ≥0), ξ ∈[0, 1/2] and δ ≥1. Let β < 1 satisfy (2.9), where λ(t) is given in (4.10). Then for f(z) ∈Wδ β(α, γ), the function z1−δ Gδ a,b(f)(z) δ belongs to the class S∗(ξ), whenever −1 < b = a ≤ 5 −δ µ −δ ν , γ > 0 (µ > 0, ν > 0), δ ≤min{µ, ν}, 0,
Theorem 4.6.
Theorem 4.6. Let γ ≥0 (µ ≥0, ν ≥0), k ≥0, ξ ∈[0, 1/2] and δ ≥1. Let β < 1 satisfy (2.9), where λ(t) is given in (4.12). Then for f(z) ∈Wδ…
Theorem 4.6. Let γ ≥0 (µ ≥0, ν ≥0), k ≥0, ξ ∈[0, 1/2] and δ ≥1. Let β < 1 satisfy (2.9), where λ(t) is given in (4.12). Then for f(z) ∈Wδ β(α, γ), the function z1−δ T δ k (f)(z) δ belongs to the class ∈S∗(ξ), whenever (i). k ≥0, for γ > 0 (δ ≤µ, δ ≤ν), (ii). k ∈[2/3, 1] ∪[3, ∞), for γ = 0 (µ = 0, ν = α(1 ≤δ ≤α)).
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