Abstract
In this article we consider the class $\mathcal{A}(p)$ which consists of functions that are meromorphic in the unit disc $\ID$ having a simple pole at $z=p\in (0,1)$ with the normalization $f(0)=0=f'(0)-1 $. First we prove some sufficient conditions for univalence of such functions in $\ID$. One of these conditions enable us to consider the class $\mathcal{V}_{p}(λ)$ that consists of functions satisfying certain differential inequality which forces univalence of such functions. Next we establish
Results & Lemmas (6)
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Theorem 1.
Theorem 1. Let f ∈A(p). If |Uf(z)| < 1 holds for all z ∈D then f ∈Σ(p).
Theorem 1. Let f ∈A(p). If |Uf(z)| < 1 holds for all z ∈D then f ∈Σ(p).
Theorem 2.
Theorem 2. Let f ∈A(p) and f/z is non-vanishing in D 0. If |(z/f(z))′′| ≤2 for z ∈D, then f is univalent in D. This condition is only…
Theorem 2. Let f ∈A(p) and f/z is non-vanishing in D \ {0}. If |(z/f(z))′′| ≤2 for z ∈D, then f is univalent in D. This condition is only sufficient for univalence but not necessary.
Theorem 3.
Theorem 3. Let f ∈A(p) and f(z) ̸= 0 for D 0. If for n ≥3, n−3 X k=0 k + 1 (k + 2)!|αk| + n −1 n!
Theorem 3. Let f ∈A(p) and f(z) ̸= 0 for D \ {0}. If for n ≥3, n−3 X k=0 k + 1 (k + 2)!|αk| + n −1 n!
Theorem 4.
Theorem 4. Let f ∈A(p) and each z/f has the expansion of the form (1.2). If f satisfies any one of the following three conditions namely (i)…
Theorem 4. Let f ∈A(p) and each z/f has the expansion of the form (1.2). If f satisfies any one of the following three conditions namely (i) P∞ n=2(n −1)|bn| ≤λ (ii) P∞ n=2 n(n −1)|bn| ≤2λ
Theorem 5.
Theorem 5. Let each f ∈Vp(λ) has the Taylor expansion f(z) = z+P∞ n=2 an(f)zn, in the disc z: |z| < p. Then the exact region of variability…
Theorem 5. Let each f ∈Vp(λ) has the Taylor expansion f(z) = z+P∞ n=2 an(f)zn, in the disc {z : |z| < p}. Then the exact region of variability of the second Taylor coefficient a2(f) is the disc determined by the inequality |a2(f) −1/p| ≤λp. (2.4)
Corollary 1.
Corollary 1. Let for some λ ∈(0, 1], f ∈Vp(λ) and has the form f(z) = z + P∞ n=2 an(f)zn, in |z| < p. Then |a2(f)| ≤1/p + λp and equality…
Corollary 1. Let for some λ ∈(0, 1], f ∈Vp(λ) and has the form f(z) = z + P∞ n=2 an(f)zn, in |z| < p. Then |a2(f)| ≤1/p + λp and equality holds in this inequality for the function kλ p.
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