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Abstract
Certain classes of analytic functions are defined which will generalize new, as well
as well-known, classes of k-uniformly convex and starlike functions. We provide necessary and
sufficent coefficient conditions, distortion bounds, extreme points and radius of starlikeness for
these classes.
Results & Lemmas (4)
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Theorem 1.
Theorem 1. f ∈U(k, β, λ) if and only if ∞ X m=2 (1 + mλ −λ)(m(1 + k) −(k + β))am ≤1 −β (2.1) where 0 ≤β < 1, k ≥2, 0 ≤λ ≤1, and −π < θ ≤π.
Theorem 1. f ∈U(k, β, λ) if and only if ∞ X m=2 (1 + mλ −λ)(m(1 + k) −(k + β))am ≤1 −β (2.1) where 0 ≤β < 1, k ≥2, 0 ≤λ ≤1, and −π < θ ≤π.
Theorem 2.
Theorem 2. If f ∈U(k, β, λ) and |z| ≤r < 1, then we have the sharp bounds r − 1 −β (1 + λ)(2 + k −β)r2 ≤|f(z)| ≤r + 1 −β (1 + λ)(2 + k…
Theorem 2. If f ∈U(k, β, λ) and |z| ≤r < 1, then we have the sharp bounds r − 1 −β (1 + λ)(2 + k −β)r2 ≤|f(z)| ≤r + 1 −β (1 + λ)(2 + k −β)r2 (2.3) and 1 − 2(1 −β) (1 + λ)(2 + k −β)r ≤|f ′(z)| ≤1 + 2(1 −β) (1 + λ)(2 + k −β)r.
Theorem 3.
Theorem 3. Let f1(z) = z and fm(z) = z − 1−β (1+mλ−λ)(m(1+k)−(k+β))zm where λ ≥0, 0 ≤β < 1, k ≥0, and m ≥2. Then f(z) is in U(k, β, λ) if…
Theorem 3. Let f1(z) = z and fm(z) = z − 1−β (1+mλ−λ)(m(1+k)−(k+β))zm where λ ≥0, 0 ≤β < 1, k ≥0, and m ≥2. Then f(z) is in U(k, β, λ) if and only if it can be expressed in the form f(z) = P∞ m=1 γmfm(z) where γm ≥0 and P∞ m=1 γm = 1.
Theorem 5.
Theorem 5. Let the f be in the class U(k, β, λ). Then f is starlike of order δ(0 ≤ δ < 1) in |z| < r2(β, λ, k, δ), where r2(β, λ, k, δ) =…
Theorem 5. Let the f be in the class U(k, β, λ). Then f is starlike of order δ(0 ≤ δ < 1) in |z| < r2(β, λ, k, δ), where r2(β, λ, k, δ) = inf m (1 −δ)(1 + mλ −λ)(m(1 + k) −(k + β)) (m −δ)(1 −β) 1 m−1 , m ≥2. (2.4)