A subclass of close-to-convex harmonic mappings

Lemma 1.1. Let f ∈FH where h and g are given by (1.1). Then (a) f is close-to-convex in D. (b) ||an| −|bn|| ≤1/n for n ≥2 whenever b1 = 0. (c) P∞ n=2 n2(|an|2 + |bn|2) ≤1 −|b1|2.

Lemma 1.2. Let f = h + ¯g ∈H where h and g are given by (1.1), and satisfy the condition ∞ X n=2 n(|an| + |bn|) ≤1 −|b1|. Then f ∈FH.

Lemma 1.2 gives a sufficient condition for a function f ∈H to be in FH. Under the hypothesis of Lemma 1.2, Jahangiri (see [9]) has proved that f is starlike (of order 0) in D. The subfamily FH ⊂SH is not affine and linear invariant, that is, if f = h + ¯g ∈FH then ǫf(z) + f(z) 1 + ǫg′(0) (|ǫ| < 1) and f((z + z0)/(1 + z0z)) −f(z0) (1 −|z0|2)h′(z0) (|z0| < 1) need not belong to the class FH. Moreover, the class FH is not preserved under passage to locally uniform limits. To see this, consider the sequ

Theorem 2.1. A harmonic function f = h+ ¯g ∈F 0 H if and only if the analytic functions Fǫ = h + ǫg belongs to F for each |ǫ| = 1.

Theorem 2.2. Let f = h+ ¯g ∈F 0 H where h and g are given by (1.1) with b1 = g′(0) = 0. Then (i) |an| ≤1/n for n = 2, 3, . . .. Equality holds for some m ≥2 ifff is analytic in D and f(z) = z + zm m .

Theorem 2.3. Every function f ∈F 0 H satisfies the inequalities |z| −1 2|z|2 ≤|f(z)| ≤|z| + 1 2|z|2, z ∈D. In particular, the range of each function f ∈F 0 H contains the disk |w| < 1/2. Moreover, these results are sharp for the functions z + z2/2 and z + ¯z2/2.

Theorem 2.1 so that [11] 1 −|z| ≤|F ′ ǫ(z)| ≤1 + |z|, z ∈D, or equivalently 1 −|z| ≤|h′(z) + ǫg′(z)| ≤1 + |z|. From this, we may deduce that 1 −|z| ≤|h′(z)| −|g′(z)| and |h′(z)| + |g′(z)| ≤1 + |z|.

Corollary 2.4. The class F 0 H is a compact normal family.

Theorem 2.5. If f ∈F 0 H then the Jacobian of f satisfy Jf(z) ≤(1 + |z|)2, z ∈D. Equality occurs only if f is analytic and is a rotation of the function f(z) = z + z2/2.

Theorem 2.6. The area of the image of each function f in F 0 H is less than or equal to 3π/2 and this is a maximum attained only by the analytic functions f(z) = z + z2/2 and its rotations.

Theorem 3.1. Each function in F 0 H maps D onto a domain bounded by a rectifiable Jordan curve.

Theorem 3.2. The radius of convexity of the class F 0 H is 1/2. Moreover, the bound 1/2 is sharp.

Theorem 3.2 shows that the classes F and F 0 H have the same radius of convexity. A similar statement holds regarding the radius of starlikeness as seen by the following theorem.

Theorem 3.3. The classes F and F 0 H have the same radius of starlikeness.

Theorem 3.4. Suppose that f = h + ¯g ∈F 0 H(λ) (0 < λ ≤1) where h and g are given by (1.1) with b1 = g′(0) = 0. Then

Theorem 3.5. F 0 H(2/ √ 5) ⊂S∗0 H and the bound 2/ √ 5 is best possible.

Theorem 3, p. 10], it follows that f ∈S∗0 H . □ Similar to Lemma 1.2, the next theorem provides a sufficient condition for a harmonic function to be in F 0 H(λ).

Theorem 3.6. Let f = h + ¯g ∈H where h and g are given by (1.1) with b1 = g′(0) = 0. Suppose that λ ∈(0, 1]. If (3.1) ∞ X n=2 n(|an| + |bn|) ≤λ then f ∈F 0 H(λ) and is starlike of order 2(1 −λ)/(2 + λ). The result is sharp.

Corollary 3.7. Let f = h + ¯g ∈H where h and g are given by (1.1) with b1 = g′(0) = 0. Suppose that λ ∈(0, 1]. If ∞ X n=2 n2(|an| + |bn|) ≤λ then f ∈F 0 H(λ/2) and is starlike of order 2(2 −λ)/(4 + λ). Moreover, f is convex of order 2(1 −λ)/(2 + λ). These results are sharp for the function f(z) = z + λ¯z2/4.

Theorem 4.1. Let f, F ∈F 0 H(λ) (0 < λ ≤1). Then (a) f ∗F ∈F 0 H(λ2/2); (b) f ∗F is starlike of order 2(2 −λ2)/(4 + λ2); and

Corollary 4.2. If f and F belong to F 0 H then so does the convolution function f ∗F. Moreover, (i) f ∗F is starlike of order 2/5; and (ii) f ∗F is convex in D. All these results are sharp by considering the function f(z) = F(z) = z + ¯z2/2. In [1], Clunie and Sheil-Small showed that if ϕ ∈K and f ∈KH then the functions (αϕ + ϕ) ∗f ∈CH (|α| ≤1). The result is even true if KH is replaced by F 0 H with a stronger conclusion. This is seen by the following theorem which makes use of the result due t

Theorem 4.3. Let ϕ ∈K and f ∈F 0 H(λ) (0 < λ ≤1). Then the functions (αϕ+ϕ)∗f ∈ F 0 H(λ) for |α| ≤1.

Corollary 4.4. Let ϕ ∈K and f ∈F 0 H. Then (i) ϕ˜∗f is starlike in |z| < r0, r0 being the radius of starlikeness of F; and (ii) ϕ˜∗f is convex in |z| < 1/2.

Corollary 4.5. If ϕ ∈K and f ∈F 0 H(2/ √ 5) then ϕ˜∗f ∈S∗0 H .

Corollary 4.6. Let ϕ ∈F and f ∈F 0 H. Then ϕ˜∗f ∈F 0 H ∩K0 H.

Theorem 2.1. □ Given f, F ∈F 0 H, their integral convolution is defined as (f ⋄F)(z) = z + ∞ X n=2 anAn n zn + ∞ X n=2 bnBn

Theorem 4.7. Let f, F ∈F 0 H(λ) (0 < λ ≤1). Then (a) f ⋄F ∈F 0 H(λ2/4); (b) f ⋄F is starlike of order 2(4 −λ2)/(8 + λ2); and (c) f ⋄F is convex of order 2(2 −λ2)/(4 + λ2). Moreover, all these results are sharp. If λ = 1 in Theorem 4.7, then we obtain the integral convolution properties for the functions in the class F 0 H.

Corollary 4.8. If f, F ∈F 0 H then so does f ⋄F. Moreover (i) f ⋄F is starlike of order 2/3; and (ii) f ⋄F is convex of order 2/5.

Theorem 4.9. The class F 0 H is closed under convex combinations.

Theorem 4.9 immediately gives

Corollary 4.10. The class F 0 H is convex. We end this section with a simple observation regarding the neighborhoods of harmonic mappings. Following [14], if we define the δ-neighborhood (δ ≥0) of a function f = h+¯g ∈ H by Nδ(f) = ( F ∈H : F(z) = z + ∞ X n=2 Anzn + ∞ X n=1