On third Hankel determinants for subclasses of analytic functions and close-to-convex harmonic mappings

Lemma 1. [12] If p ∈P is of the form (2.1), then |pn| ≤2 (n ∈N). (2.2) The inequality (2.2) is sharp and the equality holds for the function φ(z) = 1 + z 1 −z = 1 + 2 ∞ X n=1 zn.

Lemma 2. [27] If p ∈P is of the form (2.1), then holds the sharp estimate |pn −pkpn−k| ≤2 (n, k ∈N, n > k). (2.3)

Lemma 3. [16] If p ∈P is of the form (2.1), then holds the sharp estimate |pn −µpkpn−k| ≤2 (n, k ∈N, n > k; 0 ≤µ ≤1). (2.4)

Lemma 4. [25, 26] If p ∈P is of the form (2.1), then there exist x, z such that |x| ≤1 and |z| ≤1, 2p2 = p2 1 + (4 −p2 1)x, (2.5) and 4p3 = p3 1 + 2p1(4 −p2 1)x −p1(4 −p2 1)x2 + 2(4 −p2 1)(1 −|x|2)z. (2.6) 4

Theorem 1. Let f ∈S∗(α), h ∈K(α) and g ∈R(α) with 0 ≤α < 1, respectively. Then H3,1(f) ≤1 18(1 −α)2(18 −α), (3.1) H3,1(h) ≤ 1 540(1 −α)2(49 −16α), (3.2) and H3,1(g) ≤1 60(1 −α)2(36 −20α + 5|1 −4α|). (3.3)

Theorem 2. The following estimates hold for Jn(f): 1. If f ∈S∗(α) (0 ≤α < 1), then J3(f) ≤1 2(1 −α)(8 −7α). 2. If h ∈K(α) (−1/2 ≤α < 1), then J3(h) ≤ 1 360(1 −α)(127 −109α). 3. If g ∈R(α) (0 ≤α < 1), then Jn(g) ≤ 2 2n−1(1 −α) (n ≥2).

Theorem 2.3]. Furthermore, using the similar argument in Theorem 2, we may obtain the bounds of the Zalcman functional J2(f) and J2(h): If f ∈S∗(α) (0 ≤α < 1), then J2(f) ≤1 −α. If h ∈K(α) (−1/2 ≤α < 1), then J2(h) ≤1 3(1 −α) . 4 Bounds of Hankel determinants for M(α) In this section, we obtain upper bounds for the Hankel determinants |H3,1(h)| and |H3,1(g)| of close-to-convex harmonic mappings f = h + g ∈M(α).

Theorem 3. Let f = h + g ∈M(α) be of the form (1.2). Then H3,1(h) ≤ 1 540(1 −α)2(15α2 −34α + 52), and H3,1(g) ≤1 30(1 −α).

Corollary 1. Let f = h + g ∈M(−1/2) be of the form (1.2). Then H3,1(h) ≤291 960 ≈0.303125, H3,1(g) ≤1 20 = 0.05.