Abstract
We introduce the subclass UT (Φ, Ψ; α, β) of analytic functions with negative coeffi-
cients. Coefficient inequalities, distortion theorems, closure theorems, radii of close-to-convexity,
starlikeness, and convexity for functions belonging to the class UT (Φ, Ψ; α, β) are obtained. We
also determine integral operators for functions in this class and some properties involving mod-
ified Hadamard products of several functions belonging to the class U ∗
T (Φ, Ψ, α, β).
Results & Lemmas (17)
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Theorem 1.
Theorem 1. Given α(1 ≤α < 1), k ≥0. If ∞ X n=2 [(1 + k)λn −(α + k)µn] |an| ≤1 −α (2.1) then f(z) ∈U(Φ, Ψ; α, k).
Theorem 1. Given α(1 ≤α < 1) , k ≥0. If ∞ X n=2 [(1 + k)λn −(α + k)µn] |an| ≤1 −α (2.1) then f(z) ∈U(Φ, Ψ; α, k).
Theorem 2.
Theorem 2. f(z) ∈UT (Φ, Ψ, α, k) for α(−1 ≤α < 1) and k ≥0, iff ∞ X n=2 [(1 + k)λn −(α + k)µn]an ≤1 −α. (2.2) The result (2.2) is sharp.
Theorem 2. f(z) ∈UT (Φ, Ψ, α, k) for α(−1 ≤α < 1) and k ≥0 , iff ∞ X n=2 [(1 + k)λn −(α + k)µn]an ≤1 −α. (2.2) The result (2.2) is sharp.
Theorem 3.
Theorem 3. Let the function f(z) ∈UT (Φ, Ψ; α, k). If σn(α, k) ∞ n=2 is a non- decreasing sequence, then, for |z| = r < 1 r − 1 −α σ2(α,…
Theorem 3. Let the function f(z) ∈UT (Φ, Ψ; α, k). If {σn(α, k)}∞ n=2 is a non- decreasing sequence, then, for |z| = r < 1 r − 1 −α σ2(α, k)r2 ≤|f(z)| ≤r + 1 −α σ2(α, k)r2 (3.1) and if {σn(α, k)/n}∞ n=2is a non-decreasing sequence, then, for |z| = r < 1 1 −2(1 −α) σ2(α, β) r ≤|f ′(z)| ≤1 + 2(1 −α) σ2(α, β) r.
Corollary 1.
Corollary 1. The disk |z| < 1 is mapped onto a domain that contains the disk |w| < σ2(α,k)−(1−α) σ2(α,k) by any f(z) ∈UT (Φ, Ψ; α, k). The…
Corollary 1. The disk |z| < 1 is mapped onto a domain that contains the disk |w| < σ2(α,k)−(1−α) σ2(α,k) by any f(z) ∈UT (Φ, Ψ; α, k). The theorem is sharp with extermal function f(z) given by (3.3).
Theorem 4.
Theorem 4. Let the function fi(z), i = 1, 2,..., m, defined by fi(z) = z − ∞ X n=2 an,izn (an,i ≥0) (4.1)
Theorem 4. Let the function fi(z), i = 1, 2, . . . , m, defined by fi(z) = z − ∞ X n=2 an,izn (an,i ≥0) (4.1)
Theorem 5.
Theorem 5. Let the functions fi(z) be defined by (4.1) be in the class UT (Φ, Ψ, α, k) for every i = 1, 2,..., m. Then the functions h(z) =…
Theorem 5. Let the functions fi(z) be defined by (4.1) be in the class UT (Φ, Ψ, α, k) for every i = 1, 2, . . . , m. Then the functions h(z) = m X i=1 cifi(z) (ci ≥0) (4.6) is also in the same class UT (Φ, Ψ, α, k) where m P i=1 ci = 1.
Theorem 6.
Theorem 6. Let f1(z) = z and fn(z) = z − 1 −α σn(α, k)zn (n ≥2). (4.7) Then f(z) ∈UT (Φ, Ψ; α, k) if and only if it can be expressed in the…
Theorem 6. Let f1(z) = z and fn(z) = z − 1 −α σn(α, k)zn (n ≥2). (4.7) Then f(z) ∈UT (Φ, Ψ; α, k) if and only if it can be expressed in the form: f(z) = ∞ X n=1 δnfn(z) (4.8) where δn ≥0
Theorem 6.
Theorem 6. 5. Radii of Close-to-convexity, Starlikeness and Convexity
Theorem 6. 5. Radii of Close-to-convexity, Starlikeness and Convexity
Theorem 7.
Theorem 7. Let the function f(z) ∈UT (Φ, Ψ; α, k). Then f(z) is close-to-convex of order ρ(0 ≤ρ < 1) in |z| < r1, where r1 = inf n σn(α,…
Theorem 7. Let the function f(z) ∈UT (Φ, Ψ; α, k). Then f(z) is close-to-convex of order ρ(0 ≤ρ < 1) in |z| < r1, where r1 = inf n σn(α, k)(1 −ρ) n(1 −α) 1/(n−1) (n ≥2). (5.1) The result is sharp, the extermal function f(z) being given by (2.3).
Theorem 7
Theorem 7 follows easily from (5.3).
Theorem 7 follows easily from (5.3).
Theorem 8.
Theorem 8. Let the function f(z) ∈UT (Φ, Ψ; α, k). Then f(z) is starlike of order ρ(0 ≤ρ < 1) in |z| < r2, where r2 = inf n σn(α, k)(1 −ρ)…
Theorem 8. Let the function f(z) ∈UT (Φ, Ψ; α, k). Then f(z) is starlike of order ρ(0 ≤ρ < 1) in |z| < r2, where r2 = inf n σn(α, k)(1 −ρ) (n −ρ)(1 −α) 1/(n−1) (n ≥2). (5.4) The result is sharp, with the extermal function f(z) given by (2.3).
Theorem 8
Theorem 8 follows easily from (5.6).
Theorem 8 follows easily from (5.6).
Corollary 2.
Corollary 2. Let the function f(z) ∈UT (Φ, Ψ; α, k). Then f(z) is convex of order ρ(0 ≤ρ < 1) in |z| < r3, where r3 = inf n σn(α, k)(1…
Corollary 2. Let the function f(z) ∈UT (Φ, Ψ; α, k). Then f(z) is convex of order ρ(0 ≤ρ < 1) in |z| < r3, where r3 = inf n σn(α, k)(1 −ρ) n(n −ρ)(1 −α) 1/(n−1) (n ≥2). (5.7)
Theorem 9.
Theorem 9. Let the functions f(z) defined by (1.3) be in the class UT (Φ, Ψ; α, k), and let c be real number such that c > −1. Then the…
Theorem 9. Let the functions f(z) defined by (1.3) be in the class UT (Φ, Ψ; α, k), and let c be real number such that c > −1. Then the function F(z) ∈UT (Φ, Ψ; α, k), where F(z) = c + 1 zc z Z 0 tc−1f(t)dt (6.1)
Theorem 10.
Theorem 10. Let the function F(z) = z − ∞ P n=2 anzn (an ≥0) be in the class UT (Φ, Ψ; α, k) and let c be a real number such that c > −1.…
Theorem 10. Let the function F(z) = z − ∞ P n=2 anzn (an ≥0) be in the class UT (Φ, Ψ; α, k) and let c be a real number such that c > −1. Then the function given by (6.1) is univalent in |z| < r4, where r4 = inf n σn(α, k)(c + 1) n(c + n) |β| 1/(n−1) (n ≥2). (6.2)
Theorem 11.
Theorem 11. Let each of the functions fj(z) (j = 1, 2) defined by (4.1) be in the class UT (Φ, Ψ; α, k). Let ∆(n) = (σn(α, k))2 −(1 −α)2[(1…
Theorem 11. Let each of the functions fj(z) (j = 1, 2) defined by (4.1) be in the class UT (Φ, Ψ; α, k). Let ∆(n) = (σn(α, k))2 −(1 −α)2[(1 + k)λn −kµn] (σn(α, k))2 −(1 −α)2µn , (7.2) If ∆(n) is an increasing function of n(n ≥2), then (f1 ∗f2)(z) ∈UT (Φ, Ψ, γ, k), for γ = (σ2(α, k))2 −(1 −α)2[(1 + k)λ2 −kµ2] (σ2(α, k))2 −(1 −α)2µ2 . (7.3) The result is sharp.
Theorem 12.
Theorem 12. Let each of the functions fj(z) (j = 1, 2) defined by (4.1) be in the class UT (Φ, Ψ; α, k). Let Ω(n) = 1 2(σn(α, k))2 −[(1 +…
Theorem 12. Let each of the functions fj(z) (j = 1, 2) defined by (4.1) be in the class UT (Φ, Ψ; α, k). Let Ω(n) = 1 2(σn(α, k))2 −[(1 + k)λn −kµn](1 −α)2 1 2(σn(α, k))2 −µn(1 −α)2 , (7.6)
Function classes studied:
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