Continue from this paper
Indexing is not verification. Extracted statements should be checked against the source.
Abstract
In this paper, we first find an estimate for the range of polyharmonic mappings in the class $HC_{p}^{0}$. Then, we obtain two characterizations in terms of the convolution for polyharmonic mappings to be starlike of order $α$, and convex of order $β$, respectively. Finally, we study the radii of starlikeness and convexity for polyharmonic mappings, under certain coefficient conditions.
Results & Lemmas (9)
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.
Proposition 1.
Proposition 1. ([22]) If F is univalent, F(0) = 0 and ∂ ∂θ arg F(reiθ) > 0 for z = reiθ ̸= 0, then F is starlike with respect to the…
Proposition 1. ([22]) If F is univalent, F(0) = 0 and ∂ ∂θ arg F(reiθ) > 0 for z = reiθ ̸= 0, then F is starlike with respect to the origin. Definition 2. ([21]) A univalent polyharmonic mapping F with F(0) = 0 and ∂ ∂θF(reiθ) ̸= 0 whenever r ∈(0, 1), is said to be convex if the curve F(reiθ) is convex for each r ∈(0, 1).
Proposition 2.
Proposition 2. ([22]) If F is univalent, F(0) = 0, ∂ ∂θF(reiθ) ̸= 0 whenever r ∈ (0, 1), and ∂ ∂θ arg ∂ ∂θF(reiθ) > 0 for z = reiθ…
Proposition 2. ([22]) If F is univalent, F(0) = 0, ∂ ∂θF(reiθ) ̸= 0 whenever r ∈ (0, 1), and ∂ ∂θ arg ∂ ∂θF(reiθ) > 0 for z = reiθ ̸= 0, then F is convex. In [21], J. Qiao and X. Wang introduced the subclass of H0 p denoted by HS0 p of
Theorem 1.
Theorem 1. Let F ∈HC0 p of the form (2.1). Then the range F(U) contains the full disk |w| < 1/2.
Theorem 1. Let F ∈HC0 p of the form (2.1). Then the range F(U) contains the full disk |w| < 1/2.
Theorem 2.
Theorem 2. Let F = Pp k=1 |z|2(k−1)(hk(z) + gk(z)) ∈H0 p be univalent. Then F is starlike of order α if and only if p X k=1 |z|2(k−1) …
Theorem 2. Let F = Pp k=1 |z|2(k−1)(hk(z) + gk(z)) ∈H0 p be univalent. Then F is starlike of order α if and only if p X k=1 |z|2(k−1) hk(z) ∗ z + ((αξ + α + ξ −1)/(2 −α −αξ))z2 (1 −z)2 −gk(z) ∗ (2ξ + α + αξ)/(2 −α −αξ)z −((αξ + α + ξ −1)/(2 −α −αξ))z2
Theorem 3.
Theorem 3. Let F = Pp k=1 |z|2(k−1) hk(z) + gk(z) ∈H0 p be univalent such that ∂ ∂θF(reiθ) ̸= 0 for all r ∈(0, 1). Then F is convex of…
Theorem 3. Let F = Pp k=1 |z|2(k−1) hk(z) + gk(z) ∈H0 p be univalent such that ∂ ∂θF(reiθ) ̸= 0 for all r ∈(0, 1). Then F is convex of order β if and only if p X k=1 |z|2(k−1) hk(z) ∗ (2 −βξ −β)z + (2ξ + βξ + β)z2 (1 −z)3
Theorem 4.
Theorem 4. Let F ∈H0 p of the form (2.1) and the coefficients of the series satisfy the conditions |ak,j| ≤1 6(2j + 1)(j + 1) and |bk,j| ≤1…
Theorem 4. Let F ∈H0 p of the form (2.1) and the coefficients of the series satisfy the conditions |ak,j| ≤1 6(2j + 1)(j + 1) and |bk,j| ≤1 6(2j −1)(j −1). 8
Theorem 5.
Theorem 5. Under the hypothesis of Theorem 4, F ∈H0 p is univalent and convex of order β in the disk |z| < r1(β), where r1(β) is the…
Theorem 5. Under the hypothesis of Theorem 4, F ∈H0 p is univalent and convex of order β in the disk |z| < r1(β), where r1(β) is the smallest positive root of the equation 0 =6(1 −β)(1 −r)5 − p X k=1 r2(k−1) (8k −6 −6β)(1 + r)(1 −r)2 + 4(k −1)(1 −r)4 + 4(1 + r)(1 + 10r + r2) −6(2k −1 −β)(1 −r)5 (5.4) in the interval (0, 1). The result is sharp.
Theorem 6.
Theorem 6. Let F ∈H0 p of the form (2.1) and the coefficients of the series satisfy the conditions |ak,j| + |bk,j| ≤C for all j ≥2. Then F is…
Theorem 6. Let F ∈H0 p of the form (2.1) and the coefficients of the series satisfy the conditions |ak,j| + |bk,j| ≤C for all j ≥2. Then F is univalent and starlike of order α in |z| < r2(α), where r2(α) is the smallest positive root of the equation (5.6) (1−α)(1−r)2− p X k=1 Cr2(k−1) (2k−2+α)(1−r)+1−(2k+α−1)(1−r)2 = 0 in the interval (0,1). The result is sharp.
Theorem 7.
Theorem 7. Under the hypothesis of Theorem 6, F ∈H0 p is univalent and convex of order β in the disk |z| < r3(β), where r3(β) is the…
Theorem 7. Under the hypothesis of Theorem 6, F ∈H0 p is univalent and convex of order β in the disk |z| < r3(β), where r3(β) is the smallest positive real root of the equation 0 =(1 −β)(1 −r)3 − p X k=1 Cr2k−2 (2k −2)(1 −r)2 + 1 + r + β −βr) −(2k + β −1)(1 −r)3 (5.8) in the interval (0, 1). The result is sharp.
Function classes studied:
Related Papers