Abstract
We use the concept of q-differentiation to define a class Eq(β, δ) of analytic and univalent
functions. The investigations thereafter includes coefficient estimates, inclusion property and some
conditions for membership of some analytic functions to be in the class Eq(β, δ). Our results generalize
some known and new ones.
Results & Lemmas (17)
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Lemma 2.1
Lemma 2.1 ( [14]). Let g(z) = ∞ P m=1 amzm ≺G(z) = ∞ P m=1 bmzm, z ∈UD where G(z) is univalent in UD and G(UD) is a convex domain, then…
Lemma 2.1 ( [14]). Let g(z) = ∞ P m=1 amzm ≺G(z) = ∞ P m=1 bmzm, z ∈UD where G(z) is univalent in UD and G(UD) is a convex domain, then |am| ≤|b1|, m ∈N. Equality holds for the function g(z) = G(τzm), |τ| = 1. The lemmas that follow are the q-analogous versions of the original ones as referenced.
Lemma 2.2
Lemma 2.2 ( [6]). Let p(z) be analytic in UD such that p(0) = 1. If Re zDq(p(z)) p(z) + 1 > 3δ −1 2δ, z ∈UD, then for α = (δ −1)/δ (δ…
Lemma 2.2 ( [6]). Let p(z) be analytic in UD such that p(0) = 1. If Re zDq(p(z)) p(z) + 1 > 3δ −1 2δ , z ∈UD, then for α = (δ −1)/δ (δ ∈[1/2, 1)), Re p(z) > 2α. The constant 2α is the best possible.
Lemma 2.3
Lemma 2.3 ( [5]). Let u = u1+u2i and v = v1+v2i such that γ(u, v): C2 −→C is a complex-valued function such that
Lemma 2.3 ( [5]). Let u = u1+u2i and v = v1+v2i such that γ(u, v) : C2 −→C is a complex-valued function such that
Theorem 3.1.
Theorem 3.1. Let β ∈(−π, π] and δ ∈[0, 1), if condition (6) holds, then Eq(β, δ) ⊂BT q(δ). BT q(δ) is the class of q-bounded turning…
Theorem 3.1. Let β ∈(−π, π] and δ ∈[0, 1), if condition (6) holds, then Eq(β, δ) ⊂BT q(δ). BT q(δ) is the class of q-bounded turning function of order δ.
Corollary 3.2
Corollary 3.2 ( [1]). Since class BT q(δ) is well-known to consist of univalent functions, then Eq(β, δ) ⊂BT q(δ) consists of univalent…
Corollary 3.2 ( [1]). Since class BT q(δ) is well-known to consist of univalent functions, then Eq(β, δ) ⊂BT q(δ) consists of univalent functions.
Corollary 3.3.
Corollary 3.3. lim q↑1 Eq(β, δ) ⊂BT (δ), z ∈UD.
Corollary 3.3. lim q↑1 Eq(β, δ) ⊂BT (δ), z ∈UD.
Theorem 3.4.
Theorem 3.4. If f ∈A is such that Re
Theorem 3.4. If f ∈A is such that Re
Corollary 3.5.
Corollary 3.5. If f ∈A satisfies condition (8), then f ∈Eq(β, 2(δ−1)/δ).
Corollary 3.5. If f ∈A satisfies condition (8), then f ∈Eq(β, 2(δ−1)/δ).
Corollary 3.6.
Corollary 3.6. If f ∈lim q↑1 Eq(β, 1/2) is such that Re z(1 + κ)f ′′(z) + κz2f ′′′(z) f ′(z) + κzf ′′(z) > −1 2, then Re(f ′(z) + κzf…
Corollary 3.6. If f ∈lim q↑1 Eq(β, 1/2) is such that Re z(1 + κ)f ′′(z) + κz2f ′′′(z) f ′(z) + κzf ′′(z) > −1 2, then Re(f ′(z) + κzf ′′(z)) > 1/2, z ∈UD.
Corollary 3.7.
Corollary 3.7. If f ∈Eq(π, 1/2) is such that Re zDq(Dqf (z)) Dqf (z) > −1 2, (9) then Re(Dqf (z)) > 1 2. This means that if condition…
Corollary 3.7. If f ∈Eq(π, 1/2) is such that Re zDq(Dqf (z)) Dqf (z) > −1 2, (9) then Re(Dqf (z)) > 1 2. This means that if condition (9) holds, then f is a q-bounded turning function of order 1/2. Now if q ↑1, then Re zf ′′(z)
Corollary 3.8.
Corollary 3.8. If f ∈Eq(0, 1/2) is such that Re
Corollary 3.8. If f ∈Eq(0, 1/2) is such that Re
Theorem 3.9.
Theorem 3.9. Let β ∈(−π, π] and δ ∈[0, 1), then the function f (z) = z + amzm ∈Eq(β, δ), m = 2, 3,... (12) if |am| ≤ 2 [m]q |Xm| −((2 +…
Theorem 3.9. Let β ∈(−π, π] and δ ∈[0, 1), then the function f (z) = z + amzm ∈Eq(β, δ), m = {2, 3, . . .} (12) if |am| ≤ 2 [m]q |Xm| −((2 + [m −1]q) cos θ + [m −1]q cos(β + θ0))
Corollary 3.10.
Corollary 3.10. Let f (z) = z + amzm ∈Eq(0, δ) and m = 2, 3,..., then |am| ≤ 1 [m]q nq 1 + 2[m −1]q + [m −1]2q + 1 + [m −1]q o and if q ↑1,…
Corollary 3.10. Let f (z) = z + amzm ∈Eq(0, δ) and m = {2, 3, . . .}, then |am| ≤ 1 [m]q nq 1 + 2[m −1]q + [m −1]2q + 1 + [m −1]q o and if q ↑1, then |am| ≤ 1 2m2 .
Corollary 3.11.
Corollary 3.11. Let f (z) = z + amzm ∈Eq(π, δ) and m = 2, 3,..., then |am| ≦ 1 2[m]q and if q ↑1, then |am| ≤ 1 2m.
Corollary 3.11. Let f (z) = z + amzm ∈Eq(π, δ) and m = {2, 3, . . .}, then |am| ≦ 1 2[m]q and if q ↑1, then |am| ≤ 1 2m.
Theorem 3.13 · coeff
Theorem 3.13 (Coefficient Estimates). Let β ∈(−π, π], δ ∈[0, 1) and let G(z) = 1 + b1z + b2z2 + · · · ∈CV(δ). If f ∈A belongs to Eq(β, δ),…
Theorem 3.13 (Coefficient Estimates). Let β ∈(−π, π], δ ∈[0, 1) and let G(z) = 1 + b1z + b2z2 + · · · ∈CV(δ). If f ∈A belongs to Eq(β, δ), then |am| ≤2(1 −δ)|b1| [m]q|Xm| , m = {2, 3, . . .} (21) where |Xm| is defined in (14).
Corollary 3.14.
Corollary 3.14. Let f (z) ∈Eq(0, δ), then |am| ≤ (1 −δ)|b1| q 1 + 2[m −1]q + [m −1]2q and if q ↑1, then |am| ≤(1 −δ)|b1| m, m = 2, 3,....
Corollary 3.14. Let f (z) ∈Eq(0, δ), then |am| ≤ (1 −δ)|b1| q 1 + 2[m −1]q + [m −1]2q and if q ↑1, then |am| ≤(1 −δ)|b1| m , m = {2, 3, . . .}.
Corollary 3.15.
Corollary 3.15. Let f ∈Eq(π, δ), then |am| ≤(1 −δ)|b1| [m]q and if q ↑1, then |am| ≤(1 −δ)|b1| m, m = 2, 3,...
Corollary 3.15. Let f ∈Eq(π, δ), then |am| ≤(1 −δ)|b1| [m]q and if q ↑1, then |am| ≤(1 −δ)|b1| m , m = {2, 3, . . .}
Function classes studied:
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