Abstract
In this paper, we study the family ${\mathcal C}_{H}^0$ of sense-preserving complex-valued harmonic functions $f$ that are normalized close-to-convex functions on the open unit disk $\mathbb{D}$ with $f_{\bar{z}}(0)=0$. We derive a sufficient condition for $f$ to belong to the class $\CC_{H}^0$. We take the analytic part of $f$ to be $zF(a,b;c;z)$ or $zF(a,b;c;z^2)$ and for a suitable choice of co-analytic part of $f$, the second complex dilatation $w(z)=\bar{f_{\bar{z}}}/f_z$ turns out to be a
Results & Lemmas (11)
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Corollary 1.
Corollary 1. Suppose that f = h+g ∈H satisfies the condition (2). Then f ∈F1, where F1 = f ∈H: |(1 −z)h′(z) −1| < 1 −|(1 −z)g′(z)|, z ∈D,…
Corollary 1. Suppose that f = h+g ∈H satisfies the condition (2). Then f ∈F1, where F1 = {f ∈H : |(1 −z)h′(z) −1| < 1 −|(1 −z)g′(z)|, z ∈D}, and F1 ⊂F. If we choose G(z) = (1/2) log((1 + z)/(1 −z)) in Lemma 4, it leads to the family F2 = {f ∈H : Re {(1 −z2)fz(z)} > |(1 −z2)fz(z)|, z ∈D}. According to Lemma 4, functions in F2 are harmonic and close-to-convex in D. To state a stronger version of the conclusion, it is necessary to recall the following definitions. Definition 1. A domain D ⊂C is called
Lemma 1.
Lemma 1. Functions in F2 are convex in the vertical direction.
Lemma 1. Functions in F2 are convex in the vertical direction.
Lemma 2.
Lemma 2. Suppose that f = h + g ∈H satisfies the following condition (3) ∞ X n=1 |(n + 1)an+1 −(n −1)an−1| + ∞ X n=1 |(n + 1)bn+1 −(n…
Lemma 2. Suppose that f = h + g ∈H satisfies the following condition (3) ∞ X n=1 |(n + 1)an+1 −(n −1)an−1| + ∞ X n=1 |(n + 1)bn+1 −(n −1)bn−1| ≤1 −|b1| (a1 = 1). Then f ∈F2. In particular, f is convex in the vertical direction in D and hence close-to-convex in D.
Lemma 4
Lemma 4, it suffices to show that (1) holds for some convex function G. Now, we set G(z) = (1/2) log((1 + z)/(1 −z)). Then using (3) we find…
Lemma 4, it suffices to show that (1) holds for some convex function G. Now, we set G(z) = (1/2) log((1 + z)/(1 −z)). Then using (3) we find that Re fz(z) G′(z) = Re {(1 −z2)h′(z)} = Re
Theorem 1.
Theorem 1. Let a > 0, b > 0, or a ∈C 0 with b = a, Re a > 0. Suppose that a, b, m ∈N and α ∈C are related by any one of the following…
Theorem 1. Let a > 0, b > 0, or a ∈C \ {0} with b = a, Re a > 0. Suppose that a, b, m ∈N and α ∈C are related by any one of the following conditions: (6) ab ≤1 and |α|(2B(a, b) −1) ≤1, (7) ab ≥max n 1, a + b 2 o and |α| ≤2B(a, b) −1. Then the harmonic function f given by (8) f(z) = zF(a, b; a + b; z) + αzm+1 m + 1
Corollary 2.
Corollary 2. Assume the hypotheses of Theorem 1 on a, b, m and α. In addition, if m = 2k, then the formula (Re (f(z)), Im (f(z)), t(z))…
Corollary 2. Assume the hypotheses of Theorem 1 on a, b, m and α. In addition, if m = 2k, then the formula (Re (f(z)), Im (f(z)), t(z)) defines a minimal surface, where f(z) = zF(a, b; a + b; z) + αz2k+1 2k + 1 n F(a, b; a + b; z) ∗F(2, 2k + 1; 2k + 2; z) o and t(z) = 2Im n√αzk+1 k + 1 [F(a, b; a + b; z) ∗F(2, k + 1; k + 2; z)] o + c, c ∈R,
Corollary 3.
Corollary 3. Let a > 0, b > 0, m ∈N and α ∈C satisfies any one of the following conditions a ∈(0, ∞), b ∈ 0, 1 a i and |α|(2B(a, b) −1)…
Corollary 3. Let a > 0, b > 0, m ∈N and α ∈C satisfies any one of the following conditions a ∈(0, ∞), b ∈ 0, 1 a i and |α|(2B(a, b) −1) ≤1, a ∈ 1 2, ∞ , b ∈ h a
Theorem 2.
Theorem 2. Let a > 0, b > 0, or a ∈C 0 with b = a, Re a > 0. Suppose that a, b, m ∈N and α ∈C are related by any one of the following…
Theorem 2. Let a > 0, b > 0, or a ∈C \ {0} with b = a, Re a > 0. Suppose that a, b, m ∈N and α ∈C are related by any one of the following conditions: (12) ab ≤min n1 2, a + b 3 o and |α|(B(a, b) −1) ≤1, (13) ab ≥max n1 2, a + b 3 o
Corollary 4.
Corollary 4. Suppose a, b, m and α satisfies the hypothesis of Theorem 1 and further m = 2k. Then the formula (Re (f(z)), Im (f(z)), t(z))…
Corollary 4. Suppose a, b, m and α satisfies the hypothesis of Theorem 1 and further m = 2k. Then the formula (Re (f(z)), Im (f(z)), t(z)) defines a minimal surface, where f(z) = zF(a, b; a + b; z2) + αz2k+1 2k + 1 n F(a, b; a + b; z2) ∗F(2, 2k + 1; 2k + 2; z) o and t(z) = 2Im n√αzk+1 k + 1 [F(a, b; a + b; z2) ∗F(2, k + 1; k + 2; z)] o + c, c ∈R, whose projection is f(z).
Corollary 5.
Corollary 5. Let a > 0, b > 0 be such that b ∈ 0, 1 2a if a ∈(0, 1 2] ∪[1, ∞),
Corollary 5. Let a > 0, b > 0 be such that b ∈ 0, 1 2a if a ∈(0, 1 2] ∪[1, ∞),
Corollary 6.
Corollary 6. Let a > 0, b > 0 be such that b ∈ a 3a −1, ∞ if a ∈(1 3, 1
Corollary 6. Let a > 0, b > 0 be such that b ∈ a 3a −1, ∞ if a ∈(1 3, 1
Function classes studied:
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