Abstract
Let $\mathcal{H}$ denote the class of harmonic functions $f$ in $\mathbb{D}:= \{z\in \mathbb{C}:|z| < 1\}$ normalized by $f(0) = 0 = f_z(0) -1$. For $α\geq 0$, we consider the following class $$\mathcal{W}^0_{\mathcal{H}}(α):= \{f = h + \overline{g}\in\mathcal{H}: {\rm Re\,}(h'(z) + αz h''(z)) >|g'(z) + αz g''(z)|, \quad z\in \mathbb{D}\}. $$ In this paper, we first prove the coefficient conjecture of Clunie and Sheil-Small for functions in the class $\mathcal{W}^0_{\mathcal{H}}(α)$. We also pro
Results & Lemmas (19)
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Lemma 1.3.
Lemma 1.3. [6] Suppose h and g are analytic in D with |g′(0)| < |h′(0)| and h + ǫg is close-to-convex analytic function for each ǫ (|ǫ| =…
Lemma 1.3. [6] Suppose h and g are analytic in D with |g′(0)| < |h′(0)| and h + ǫg is close-to-convex analytic function for each ǫ (|ǫ| = 1). Then f = h+g is harmonic close-to-convex in D. A classical problem for functions in the class S where functions are of the form f(z) = z + ∞ X n=0 anzn is to find the sharp upper bound for the absolute value of the coefficients an for n ≥2. In 1916, Bieberbach [2] proved that if f ∈S then |a2| ≤2 and conjectured that |an| ≤n for n ≥2. In 1985, de Branges [3]
Theorem 2.1.
Theorem 2.1. A harmonic mapping f = h + g is in W0 H(α) if and only if the analytic function F = h + ǫg belongs to W(α) for each |ǫ| = 1.
Theorem 2.1. A harmonic mapping f = h + g is in W0 H(α) if and only if the analytic function F = h + ǫg belongs to W(α) for each |ǫ| = 1.
Lemma 1.3
Lemma 1.3, W0 H(α) is a subclass of C0 H for α ≥0. In 1977, Chichra [7] proved that if 0 ≤β < α then W(α) ⊂W(β). Thus W0 H(α) ⊂W0 H(β) if 0…
Lemma 1.3, W0 H(α) is a subclass of C0 H for α ≥0. In 1977, Chichra [7] proved that if 0 ≤β < α then W(α) ⊂W(β). Thus W0 H(α) ⊂W0 H(β) if 0 ≤β < α. In view of this, W(α) is starlike for α ≥1 because the class W(1) is starlike. For α ≥1, if f ∈W0 H(α) then h + ǫg ∈W(α) is starlike function in D for each ǫ (|ǫ| = 1). Hence by Theorem A, W0 H(α) ⊂S∗ H 0 for all α ≥1. In particular, for α ≥1 each member of W0 H(α) is fully starlike.
Theorem 2.2.
Theorem 2.2. Let f = h + g ∈W0 H(α) for α ≥0 be of the form (1.2). Then for n ≥2, (2.3) |bn| ≤ 1 αn2 + n(1 −α). The result is sharp for the…
Theorem 2.2. Let f = h + g ∈W0 H(α) for α ≥0 be of the form (1.2). Then for n ≥2, (2.3) |bn| ≤ 1 αn2 + n(1 −α). The result is sharp for the function f(z) which is given by f(z) = z + 1 αn2+n(1−α)zn.
Theorem 2.5.
Theorem 2.5. Let f = h + g ∈W0 H(α) for α ≥0 be of the form (1.2). Then for any n ≥2, (i) |an| + |bn| ≤ 2 αn2 + n(1 −α); (ii) ||an| −|bn||…
Theorem 2.5. Let f = h + g ∈W0 H(α) for α ≥0 be of the form (1.2). Then for any n ≥2, (i) |an| + |bn| ≤ 2 αn2 + n(1 −α); (ii) ||an| −|bn|| ≤ 2 αn2 + n(1 −α); (iii) |an| ≤ 2 αn2 + n(1 −α). All these results are sharp for the function f(z) which is given by f(z) = z + P∞ n=2
Theorem 2.9.
Theorem 2.9. Let f = h + g ∈W0 H(α) be as in (1.2) with 0 < α ≤1. Then (2.10) |z| + 2 ∞ X n=2 (−1)n−1|z|n αn2 + n(1 −α) ≤|f(z)| ≤|z| + 2 ∞…
Theorem 2.9. Let f = h + g ∈W0 H(α) be as in (1.2) with 0 < α ≤1. Then (2.10) |z| + 2 ∞ X n=2 (−1)n−1|z|n αn2 + n(1 −α) ≤|f(z)| ≤|z| + 2 ∞ X n=2 |z|n αn2 + n(1 −α). This result is sharp for the function f(z) = z +P∞
Theorem 2.14.
Theorem 2.14. Let f = h+ g ∈S0 H be of the form (1.2) and satisfies the condition (2.15) ∞ X n=2 (αn2 + (1 −α)n)(|an| + |bn|)) < 1. Then f…
Theorem 2.14. Let f = h+ g ∈S0 H be of the form (1.2) and satisfies the condition (2.15) ∞ X n=2 (αn2 + (1 −α)n)(|an| + |bn|)) < 1. Then f ∈W0 H(α).
Theorem 3.1.
Theorem 3.1. The class W0 H(α) is closed under convex combinations.
Theorem 3.1. The class W0 H(α) is closed under convex combinations.
Lemma 3.2.
Lemma 3.2. [29] Let cn ∞ n=0 be a convex null sequence. Then the function q(z) = c0 2 + ∞ X n=1 cnzn is analytic and Re q(z) > 0 in D.
Lemma 3.2. [29] Let {cn}∞ n=0 be a convex null sequence. Then the function q(z) = c0 2 + ∞ X n=1 cnzn is analytic and Re q(z) > 0 in D.
Lemma 3.3.
Lemma 3.3. [29] Let p(z) be an analytic function in the unit disk D with p(0) = 1 and Re (p(z)) > 1/2 in D. Then for any analytic function…
Lemma 3.3. [29] Let p(z) be an analytic function in the unit disk D with p(0) = 1 and Re (p(z)) > 1/2 in D. Then for any analytic function f in D, the function p ∗f takes values in the convex hull of the image of D under f. Using Lemma 3.2 and Lemma 3.3, we prove the following interesting lemma.
Lemma 3.4.
Lemma 3.4. Let F be in the class W(α). Then Re( F (z) z ) > 1/2.
Lemma 3.4. Let F be in the class W(α). Then Re( F (z) z ) > 1/2.
Lemma 3.5.
Lemma 3.5. Let F1 and F2 be in W(α). Then the Hadamard product F1 ∗F2 is in W(α).
Lemma 3.5. Let F1 and F2 be in W(α). Then the Hadamard product F1 ∗F2 is in W(α).
Theorem 3.7.
Theorem 3.7. If f1 and f2 are in W0 H(α) then f1 ∗f2 is in W0 H(α).
Theorem 3.7. If f1 and f2 are in W0 H(α) then f1 ∗f2 is in W0 H(α).
Theorem 3.8.
Theorem 3.8. Let f ∈W0 H(α) and φ ∈A be such that Re φ(z) z > 1/2 for z ∈D. Then f˜∗φ ∈W0 H(α).
Theorem 3.8. Let f ∈W0 H(α) and φ ∈A be such that Re φ(z) z > 1/2 for z ∈D. Then f˜∗φ ∈W0 H(α).
Corollary 3.9.
Corollary 3.9. Suppose f belongs to W0 H(α) and φ ∈K. Then f˜∗φ ∈W0 H(α).
Corollary 3.9. Suppose f belongs to W0 H(α) and φ ∈K. Then f˜∗φ ∈W0 H(α).
Lemma 4.1.
Lemma 4.1. Let f = h + g be in the class W0 H(α) with α ≥0. Then for |ǫ| = 1 and |z| < 1/2, we have Re ((s3(h) + ǫs3(g))′ + αz(s3(h) +…
Lemma 4.1. Let f = h + g be in the class W0 H(α) with α ≥0. Then for |ǫ| = 1 and |z| < 1/2, we have Re ((s3(h) + ǫs3(g))′ + αz(s3(h) + ǫs3(g))′′) > 1/4.
Theorem 4.3.
Theorem 4.3. Let f ∈W0 H(α). Then for each q ≥2, s1,q(f) ∈W0 H(α) for |z| < 1/2.
Theorem 4.3. Let f ∈W0 H(α). Then for each q ≥2, s1,q(f) ∈W0 H(α) for |z| < 1/2.
Theorem 4.4.
Theorem 4.4. Let f ∈W0 H(α) and p and q satisfy one of the following conditions: (i) 3 ≤p < q, (ii) p = q ≥2, (iii) p > q ≥3, (iv) p= 3 and…
Theorem 4.4. Let f ∈W0 H(α) and p and q satisfy one of the following conditions : (i) 3 ≤p < q, (ii) p = q ≥2, (iii) p > q ≥3, (iv) p= 3 and q = 2. Then sp,q(f) ∈W0 H(α) in |z| < 1/2.
Theorem 4.11.
Theorem 4.11. Let f ∈W0 H(α). If p = 2 < q, then s2,q(f) ∈W0 H(α) in |z| < 3−√5 2. If p ≥4 and q = 2, then sp,2(f) ∈W0 H(α) in |z| < r0…
Theorem 4.11. Let f ∈W0 H(α). If p = 2 < q, then s2,q(f) ∈W0 H(α) in |z| < 3−√5 2 . If p ≥4 and q = 2, then sp,2(f) ∈W0 H(α) in |z| < r0 where r0 ≈0.433797 is the unique real root of the equation 1 −2r −r3 −r4 −r5 = 0.
Function classes studied:
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