Abstract
In 1984, Clunie and Sheil-Small proved that a sense-preserving harmonic function whose analytic part is convex, is univalent and close-to-convex. In this paper, certain cases are discussed under which the conclusion of this result can be strengthened and extended to fully starlike and fully convex harmonic mappings. In addition, we investgate the properties of functions in the class $\mathcal{M}(α)$ $(|α|\leq 1)$ consisting of harmonic functions $f=h+\overline{g}$ with $g'(z)=αzh'(z)$, $\RE (1+{
Results & Lemmas (12)
Each result is auto-extracted from the paper via OCR - the math is shown verbatim and may carry occasional transcription artifacts. Click to expand the full statement.
Lemma 1.1.
Lemma 1.1. [4, Lemma 5.15, p. 19] Suppose that H, G are analytic in D with |G′(0)| < |H′(0)| and that H +ϵG is close-to-convex for each |ϵ|…
Lemma 1.1. [4, Lemma 5.15, p. 19] Suppose that H, G are analytic in D with |G′(0)| < |H′(0)| and that H +ϵG is close-to-convex for each |ϵ| = 1. Then F = H +G is harmonic univalent and close-to-convex in D. Making use of Lemma 1.1, Clunie and Sheil-Small [4] proved that if f = h + g is sense- preserving in D and h + ϵg is convex for some ϵ (|ϵ| ≤1), then f is harmonic univalent and close-to-convex in D. A particular case of this result is the following.
Lemma 1.2.
Lemma 1.2. Let f = h + g ∈H be sense-preserving and h ∈K. Then f ∈C0 H. The conditions in the hypothesis of Lemma 1.2 can’t be relaxed,…
Lemma 1.2. Let f = h + g ∈H be sense-preserving and h ∈K. Then f ∈C0 H. The conditions in the hypothesis of Lemma 1.2 can’t be relaxed, that is, if f = h+g ∈H is sense-preserving and h is non-convex, then f need not be even univalent. Similarly the conclusion of Lemma 1.2 can’t be strengthened, that is, if f = h + g ∈H is sense- preserving and h ∈K, then f need not map D onto a starlike or convex domain. These two statements are illustrated by examples in Section 2 of the paper. In addition, we
Lemma 1.3.
Lemma 1.3. [11] Let f = h+¯g ∈H where h and g are given by (1.1) with b1 = g′(0) = 0. Suppose that λ ∈(0, 1]. (i) If P∞ n=2 n(|an| + |bn|)…
Lemma 1.3. [11] Let f = h+¯g ∈H where h and g are given by (1.1) with b1 = g′(0) = 0. Suppose that λ ∈(0, 1]. (i) If P∞ n=2 n(|an| + |bn|) ≤λ then f is fully starlike of order 2(1 −λ)/(2 + λ). (ii) If P∞ n=2 n2(|an| + |bn|) ≤λ then f is fully starlike of order 2(2 −λ)/(4 + λ). Moreover, f is fully convex of order 2(1 −λ)/(2 + λ). All these results are sharp. For α ∈C with |α| ≤1, let M(α) denote the set of all harmonic functions f = h+¯g ∈H that satisfy g′(z) = αzh′(z) and Re 1 + zh′′(z)
Theorem 2.3.
Theorem 2.3. Let f = h + g ∈H where h and g are given by (1.1), and let α ∈C. Further, assume that g′(z) = αzh′(z) (z ∈D) and ∞ X n=2…
Theorem 2.3. Let f = h + g ∈H where h and g are given by (1.1), and let α ∈C. Further, assume that g′(z) = αzh′(z) (z ∈D) and ∞ X n=2 n2|an| ≤1. If |α| ≤1 then f is univalent close-to-convex. If |α| ≤1/3 then f is fully starlike of order 2(1 −3|α|)/(5 + 3|α|).
Theorem 2.4.
Theorem 2.4. Let f = h + g ∈H where h and g are given by (1.1), and let α ∈C. Further, assume that g′(z) = αzh′(z) (z ∈D) and ∞ X n=2…
Theorem 2.4. Let f = h + g ∈H where h and g are given by (1.1), and let α ∈C. Further, assume that g′(z) = αzh′(z) (z ∈D) and ∞ X n=2 n3|an| ≤1. If |α| ≤2/11 then f is fully starlike of order 2(6 −11|α|)/(18 + 11|α|). Moreover, f is fully convex of order 2(2 −11|α|)/(10 + 11|α|).
Theorem 3.1.
Theorem 3.1. Let α ∈C with |α| ≤1. Then we have the following. (i) M(α) ⊂C0 H.
Theorem 3.1. Let α ∈C with |α| ≤1. Then we have the following. (i) M(α) ⊂C0 H.
Lemma 1.1.
Lemma 1.1. It is easy to verify that Re 1 + zF ′′ ϵ (z) F ′ ϵ(z) = Re αϵz 1 + αϵz + Re 1 + zh′′(z) h′(z)
Lemma 1.1. It is easy to verify that Re 1 + zF ′′ ϵ (z) F ′ ϵ(z) = Re αϵz 1 + αϵz + Re 1 + zh′′(z) h′(z)
Theorem 3.3.
Theorem 3.3. Let ϕ ∈K and f ∈M(α) (|α| ≤1). Then the functions (βϕ+ϕ)∗f ∈C0 H for |β| ≤1.
Theorem 3.3. Let ϕ ∈K and f ∈M(α) (|α| ≤1). Then the functions (βϕ+ϕ)∗f ∈C0 H for |β| ≤1.
Theorem 3.5.
Theorem 3.5. Let α ∈C with |α| ≤1. (a) Each function in M(α) maps the disk |z| < 2 − √ 3 onto a convex domain. (b) Each function in M(α)…
Theorem 3.5. Let α ∈C with |α| ≤1. (a) Each function in M(α) maps the disk |z| < 2 − √ 3 onto a convex domain. (b) Each function in M(α) maps the disk |z| < 4 √ 2 −5 onto a starlike domain.
Theorem 3.7. · radius
Theorem 3.7. The radius of convexity of the class M(1) is 2− √ 3. Moreover, the bound 2 − √ 3 is sharp. The next example determines the…
Theorem 3.7. The radius of convexity of the class M(1) is 2− √ 3. Moreover, the bound 2 − √ 3 is sharp. The next example determines the radius of starlikeness of the mapping F given by (3.3). Example 3.8. The harmonic mapping F given by (3.1) sends each disk |z| < r ≤r0 to a starlike domain, but the image is not starlike when r0 < r < 1, where r0 is given by (3.4) r0 = 1 3 r 1
Theorem 3.9. · radius
Theorem 3.9. If rS is the radius of starlikeness of M(1), then 4 √ 2 −5 ≤rS ≤1 3 r 1 3(37 −8 √ 10). By Remark 3.4, F ∗F is univalent and…
Theorem 3.9. If rS is the radius of starlikeness of M(1), then 4 √ 2 −5 ≤rS ≤1 3 r 1 3(37 −8 √ 10). By Remark 3.4, F ∗F is univalent and starlike in D. However, the product L∗F where L is the harmonic half-plane mapping given by (3.4) is not even univalent, although it is sense-preserving in D. In fact, the convolution of F with certain right-half plane mappings is sense-preserving in D. This is seen by the following theorem.
Theorem 3.10.
Theorem 3.10. Let f = h+¯g ∈K0 H with h(z)+g(z) = z/(1−z) and w(z) = g′(z)/h′(z) = eiθzn, where θ ∈R. If n = 1, 2 then F ∗f is locally…
Theorem 3.10. Let f = h+¯g ∈K0 H with h(z)+g(z) = z/(1−z) and w(z) = g′(z)/h′(z) = eiθzn, where θ ∈R. If n = 1, 2 then F ∗f is locally univalent in D, F being given by (3.1).
Function classes studied:
Related Papers