Abstract
Let f(z) be a normalized analytic function in ∆= {z|z ∈C and |z| < 1} satisfying
f(0) = 0 and f ′(0) = 1.
Let Φ be an analytic function in a domain containing f(∆), with
Φ(0) = 0, Φ
′(0) = 1 and Φ(ω) ̸= 0 for ω ∈f(∆) −{0}. Let q(z) be a fixed analytic function in
∆, q(0) = 1. The function f is called Φ-like with respect to q if
zf
′(z)
Φ(f(z)) ≺q(z)
(z ∈∆).
In this paper, we obtain some sufficient conditions for functions to be Φ-like with respect to q(z).
Results & Lemmas (4)
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Lemma 2.1.
Lemma 2.1.(cf. Miller and Mocanu [5, p.132, Theorem 3.4h]) Let q(z) be univalent in the unit disk ∆and θ and φ be analytic in a domain D…
Lemma 2.1.(cf. Miller and Mocanu [5, p.132, Theorem 3.4h]) Let q(z) be univalent in the unit disk ∆and θ and φ be analytic in a domain D containing q(∆) with φ(w) ̸= 0 when w ∈q(∆). Set Q(z) := zq′(z)φ(q(z)) and h(z) := θ(q(z)) + Q(z). Suppose that 1. Q(z) is starlike univalent in ∆and 2. ℜzh′(z) Q(z) > 0 for z ∈∆. If p(z) is analytic with p(0) = q(0), p(∆) ⊆D and θ(p(z)) + zp′(z)φ(p(z)) ≺θ(q(z)) + zq′(z)φ(q(z)), (2.2.1)
Theorem 2.2.
Theorem 2.2. Let α ̸= 0 be a complex number and q(z) be convex univalent in ∆. Define h(z) by h(z):= αq2(z) + (1 −α)q(z) + αzq′(z). (2.2.2)…
Theorem 2.2. Let α ̸= 0 be a complex number and q(z) be convex univalent in ∆. Define h(z) by h(z) := αq2(z) + (1 −α)q(z) + αzq′(z). (2.2.2) Further assume that ℜ 1 −α α + 2q(z) + 1 + zq′′(z) q′(z) > 0 (z ∈∆).
Lemma 3.1.
Lemma 3.1. Let α ̸= 0 be any complex number and β:= max 0, −ℜ1 α. Let q(z) ̸= 0 be analytic in ∆and Q(z):= zq′(z)q 1 α −1(z) be starlike of…
Lemma 3.1. Let α ̸= 0 be any complex number and β := max{0, −ℜ1 α}. Let q(z) ̸= 0 be analytic in ∆and Q(z) := zq′(z)q 1 α −1(z) be starlike of order β in ∆. If p(z) is analytic in ∆and p(z) 1 + zp′(z) p(z) α ≺q(z) 1 + zq′(z)
Theorem 3.2.
Theorem 3.2. Let α ̸= 0 be any complex number and β = max 0, −ℜ1 α. Let q(z) ̸= 0 be analytic in ∆and Q(z) = zq′(z)q 1 α −1(z)
Theorem 3.2. Let α ̸= 0 be any complex number and β = max{0, −ℜ1 α}. Let q(z) ̸= 0 be analytic in ∆and Q(z) = zq′(z)q 1 α −1(z)
Function classes studied:
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