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)
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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:
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