Abstract
In this paper, we introduce a class of complex-valued polyharmonic mappings, denoted by $HS_{p}(λ)$, and its subclass $HS_{p}^{0}(λ)$, where $λ\in [0,1]$ is a constant. These classes are natural generalizations of a class of mappings studied by Goodman in 1950's. We generalize the main results of Avci and Złotkiewicz from 1990's to the classes $HS_{p}(λ)$ and $HS_{p}^{0}(λ)$, showing that the mappings in $HS_{p}(λ)$ are univalent and sense preserving. We also prove that the mappings in $HS_{p}^{
Results & Lemmas (9)
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Proposition 1.
Proposition 1. ([18]) If F is univalent, F(0) = 0 and d dθ arg F(reiθ) > 0 for z = reiθ ̸= 0, then F is starlike with respect to the…
Proposition 1. ([18]) If F is univalent, F(0) = 0 and d dθ arg F(reiθ) > 0 for z = reiθ ̸= 0, then F is starlike with respect to the origin. A univalent polyharmonic mapping F with F(0) = 0 and d dθF(reiθ) ̸= 0 whenever 0 < r < 1, is said to be convex if the curve F(reiθ) is convex for each 0 < r < 1.
Proposition 2.
Proposition 2. ([18]) If F is univalent, F(0) = 0 and ∂ ∂θ arg ∂ ∂θF(reiθ) > 0 for z = reiθ ̸= 0, then F is convex. Let X be a…
Proposition 2. ([18]) If F is univalent, F(0) = 0 and ∂ ∂θ arg ∂ ∂θF(reiθ) > 0 for z = reiθ ̸= 0, then F is convex. Let X be a topological vector space over the field of complex numbers, and let D be a set of X. A point x ∈D is called an extremal point of D if it has no representation of the form x = ty + (1 −t)z (0 < t < 1) as a proper convex combination of two distinct points y and z in D. Now we are ready to prove results concerning the geometric properties of mappings
Theorem 1.
Theorem 1. Each mapping in HS0 p(λ) maps the disk Dr, where r ≤max 1 2, λ, onto a convex domain. 4
Theorem 1. Each mapping in HS0 p(λ) maps the disk Dr, where r ≤max{1 2, λ}, onto a convex domain. 4
Corollary 1.
Corollary 1. Let F ∈HSp(λ). Then F is a univalent, sense preserving polyhar- monic mapping. In particular, if F ∈HS0 p(λ), then F maps D…
Corollary 1. Let F ∈HSp(λ). Then F is a univalent, sense preserving polyhar- monic mapping. In particular, if F ∈HS0 p(λ), then F maps D onto a domain starlike w.r.t. the origin. Example 1. Let F1(z) = z + 1 10z2 + 1 5z2. Then F1 ∈HS0 1(2 3) is a univalent, sense preserving polyharmonic mapping. In particular, F1 maps D onto a domain starlike w.r.t. the origin, and it maps the disk Dr, where r ≤2 3, onto a convex domain. See Figure 1. This example shows that the class HS0
Lemma 1.
Lemma 1. Suppose that F ∈HSp(λ). Then the following statements hold: (1) For 0 ≤λ ≤1 2, (1 −|b1,1|)|z| −1 −|b1,1| 2(1 + λ) |z|2 ≤|F(z)| ≤(1…
Lemma 1. Suppose that F ∈HSp(λ). Then the following statements hold: (1) For 0 ≤λ ≤1 2, (1 −|b1,1|)|z| −1 −|b1,1| 2(1 + λ) |z|2 ≤|F(z)| ≤(1 + |b1,1|)|z| + 1 −|b1,1| 2(1 + λ) |z|2. Equalities are obtained by the mappings F(z) = z + |b1,1|eiµz + 1 −|b1,1| 2(1 + λ) eiνz2, for properly chosen real µ and ν; (2) For 1 2 < λ ≤1, |F(z) ≤(1 + |b1,1|)|z| + 1 −|b1,1| −3(|a1,2| + |b1,2|) 2(1 + λ) |z|2 + (|a1,2| + |b1,2|)|z|3
Lemma 2.
Lemma 2. The family HSp(λ) is closed under convex combinations.
Lemma 2. The family HSp(λ) is closed under convex combinations.
Theorem 2.
Theorem 2. The extremal points of HS0 p(λ) are the mappings of the following form: Fk(z) = z + |z|2(k−1)an,kzn or F ∗ k (z) = z +…
Theorem 2. The extremal points of HS0 p(λ) are the mappings of the following form: Fk(z) = z + |z|2(k−1)an,kzn or F ∗ k (z) = z + |z|2(k−1)bm,kzm, where |an,k| = 1 2(k −1) + n(λn + 1 −λ), for n ≥2, k ∈{1, · · · , p}, and |bm,k| = 1 2(k −1) + m(λm + 1 −λ), for m ≥2, k ∈{1, · · · , p}.
Theorem 3.
Theorem 3. Suppose that H(z) = z +P∞ n=2(Anzn +Bnzn) ∈C0 H and F ∈HS0 1(λ). Then for 1 2 ≤λ ≤1, the convolution F ∗H is univalent and…
Theorem 3. Suppose that H(z) = z +P∞ n=2(Anzn +Bnzn) ∈C0 H and F ∈HS0 1(λ). Then for 1 2 ≤λ ≤1, the convolution F ∗H is univalent and starlike, and the integral convolution F ⋄H is convex.
Theorem 4.
Theorem 4. Assume that λ ∈(0, 1] and F ∈HSp(λ). If δ ≤ λ p + λ
Theorem 4. Assume that λ ∈(0, 1] and F ∈HSp(λ). If δ ≤ λ p + λ
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