Harmonic Close-to-convex Functions and Minimal Surfaces

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. Functions in F2 are convex in the vertical direction.

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, 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. 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. 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. 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. 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. 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. Let a > 0, b > 0 be such that b ∈         0, 1 2a  if a ∈(0, 1 2] ∪[1, ∞),

Corollary 6. Let a > 0, b > 0 be such that b ∈         a 3a −1, ∞  if a ∈(1 3, 1