Certain geometric properties of close-to-convex harmonic mappings

Lemma 1.1. (see, [9]). Let p ∈P, where P denotes the class of Carath´eodory functions in D. Then |p′(z)| ≥1 −|z| 1 + |z| and

Lemma 1.2 (see [2]). If the harmonic mapping f = h + g : D →C satisfies |g′(0)| < |h′(0)| and the function Fǫ = h + ǫg is close-to-convex for every |ǫ| = 1, then f is close-to-convex function.

Theorem 2.1. The harmonic mapping f = h + g ∈W0 H(α, β) if and only if Fǫ = h + ǫg ∈ W(α, β) for each |ǫ| = 1.

Lemma 2.1. (Jack’s Lemma [10]) Let ω(z) be analytic in D with ω(0) = 0. If |ω(z)| attains its maximum value on the circle |z| = r < 1 at a point z0 ∈D, then we have z0ω′(z0) = kω(z0) for a real number k ≥1.

Lemma 2.2. If f ∈W(α, β), then ℜ(f ′(z)) > β (0 ≤β < 1), and hence f is close-to-convex in D.

Theorem 2.2. The functions in the class W0 H(α, β) are close-to-convex in D.

Theorem 3.1. Let f = h + g ∈W0 H(α, β) be of the form (1.1) with b1 = 0. Then we have (3.1) |bn| ≤ 1 −β n(1 + α(n −1)). The result is sharp and equality in (3.1) is obtained by f(z) = z + 1 −β n(1 + α(n −1))zn.

Theorem 3.2. Let f = h + g ∈W0 H(α, β) be of the form (1.1) with b1 = 0. Then for n ≥2, we have (i) |an| + |bn| ≤ 2(1 −β) n(1 + α(n −1)), (ii) ||an| −|bn|| ≤ 2(1 −β) n(1 + α(n −1)), (iii) |an| ≤ 2(1 −β) n(1 + α(n −1)). All these results are sharp for the function f(z) = z + P∞ n=2 2(1 −β)

Theorem 3.3. Let f = h + g ∈H0, where h and g are of the form (1.1). If (3.4) ∞ X n=2 n(1 + α(n −1))(|an| + |bn|) ≤1 −β, then f ∈W0 H(α, β).

Theorem 3.4. If f = h + g ∈W0 H(α, β), then (3.5) |z| −2 ∞ X n=2 (−1)n−1(1 −β)|z|n αn2 + n(1 −α) ≤|f(z)| ≤|z| + 2 ∞ X n=2 (1 −β)|z|n αn2 + n(1 −α).

Lemma 4.1. [5] If {cn}∞ n=0 be a convex null sequence, then function q(z) = c0 2 + P∞ n=1 cnzn is analytic and ℜ(q(z)) > 0 in D.

Lemma 4.2. [20] Let the function p be analytic in D with p(0) = 1 and ℜ(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.

Theorem 4.1. The class W0 H(α, β) is closed under convex combinations.

Lemma 4.3. If F ∈W(α, β), then ℜ F(z) z  > 1 2 −β .

Lemma 4.4. Let F1 and F2 belong to W(α, β), then F1 ∗F2 ∈W(α, β).

Theorem 4.2. If functions f1 and f2 belong to W0 H(α, β), then f1 ∗f2 ∈W0 H(α, β).

Theorem 4.3. Let f ∈W0 H(α, β) and φ ∈A be such that ℜ φ(z) z  > 1 2 for z ∈D, then f b∗φ ∈W0 H(α, β).

Corollary 4.1. Suppose f ∈W0 H(α, β) and φ ∈K, then f b∗φ ∈W0 H(α, β).

Theorem 5.1. Let f = h+g ∈W0 H(α, β). If p and q satisfies one of the following conditions: (i) 1 = p < q (ii) 3 ≤p < q (iii) 3 ≤q < p, then sp,q(f)(z) is convex in |z| < 1/4.

Theorem 5.2. Let f = h + g ∈W0 H(α, β). Then (i) For q > 2, s2,q(f)(z) is convex in the disk |z| < R1, where R1 is smallest positive root of the equation (5.7) 1 −4r + (6β −2)r2 −8(1 −β)r3 + (1 −2β)r4 + 4(1 −β)r5 = 0 in (0, 1). (ii) For p > 2, sp,2(f)(z) is convex in the disk |z| < R2, where R2 is smallest positive root of the equation (5.8) 1 −4r + (1 −β)r2 −(8 −3β)r3 −(5 −2β)r4 −(4 −3β)r5 + 3(1 −β)r6 = 0 in (0, 1).

Theorem 5.3. If f = h + g ∈W0 H(α, β), then s2,2(f)(z) is convex in |z| < (1 + α)/4(1 −β).

Lemma 6.1. [18] Let a, b > 0. Then the following holds: (i) For c > a + b + 1, ∞ X n=0 (n + 1)(a)n(b)n (c)nn! = Γ(c)Γ(c −a −b −1) Γ(c −a)Γ(c −b) (ab + c −a −b −1). (ii) For c > a + b + 2, ∞ X n=0 (n + 1)2(a)n(b)n

Theorem 6.1. Let f1(z) = z + z2F(a, b; c; z), f2(z) = z + z(F(a, b; c; z) −1) and f3(z) = z + z R z 0 F(a, b; c; t)dt, where a, b, c are positive real numbers such that c > a + b + 2. Then the following holds: (i) If (6.3) Γ(c)Γ(c −a −b −1) Γ(c −a)Γ(c −b)  α(a)2(b)2 c −a −b −2 + (1 + 4α)ab + 2(1 + α)(c −a −b −1)  ≤1 −β, then f1 ∈W0

Corollary 6.1. Let m ∈N, c be a positive real number and F1(z) = z + m X n=0 m n (m −n + 1)n (c)n zn+2, F2(z) = z + m X n=0 m

Corollary 6.2. If G1(z) = z + z2 + 4z3 + z4, G2(z) = z + 4z2 + z3, and G3(z) = z + z2 + 2z3 + 1 3z4, then the following holds: (i) If 2(19α + 9) ≤1 −β, then G1(z) ∈W0 H(α, β). (ii) If 28α + 13 ≤2(2 −β), then G2(z) ∈W0 H(α, β). (iii) If 108α + 37 ≤6(1 −β), then G3(z) ∈W0 H(α, β). References [1] P. N. Chichra, New subclass of the class of close-to-convex function, Proc. Amer. Math. Soc. 62(1977), 37–43. [2] J. Clunie and T. Sheil-Small, Harmonic univalent functions, Anna. Acad. Sci. Fenn. Ser. A I