Sections of univalent harmonic mappings

Theorem 1. Let f = h + g ∈S0 H with series representation as in (3). Suppose that f belongs to any one of the following geometric subclasses of S0 H : S∗0 H , C0 H, S0 H(S) or the class of harmonic mappings convex in one direction. Then the section sn,m(f) is univalent in the disk |z| < rn,m. Here rn,m is the unique positive root of the equation ψ(n, m, r) = 0, where (4) ψ(n, m, r) = 1 12r 1 −r

Theorem 2. Let f = h + g ∈K0 H with series representation as in (3). Then the section sn,m(f) is univalent in the disk |z| < rn,m, where rn,m is the unique positive root of the equation µ(n, m, r) = 0. Here (6) µ(n, m, r) = 1 −r (1 + r)3 − ∞ X k=n+1 k(k + 1) 2 rk−1 

Lemma 1. The following identities are true for 0 < r < 1: (i) ∞ X k=n+1 krk−1 = rn (1 −r)2[1 + n(1 −r)]. (ii) ∞ X k=n+1 k2rk−1 = rn (1 −r)3[2 + (2n −1)(1 −r) + n2(1 −r)2].

Corollary 1. Let f ∈S0 H satisfies the hypothesis of Theorem 1. Then sn,n(f)(z) is univalent in the disk (i) |z| < 1/4, whenever n ≥7, (ii) |z| < 1/2, whenever n ≥22, (ii) |z| < 3/4, whenever n ≥78. The bound for the radius of univalence rn,n of sn,n(f) for certain values of n are listed in Table 1. The following shearing theorem due to Clunie and Sheil-Small is needed for the proof of Theorem 2. Theorem D. [5, Theorem 5.3] A locally univalent harmonic function f = h + g in D is a univalent mappi

Lemma 2. If f = h + g ∈K0 H, r ∈(0, 1), t, ψ ∈R, then

Corollary 2. Let f ∈K0 H satisfy the hypothesis of Theorem 2. Then sn,n(f; θ)(z) is univalent in the disk (i) |z| < 1/4, whenever n ≥2, (ii) |z| < 1/2, whenever n ≥17, (ii) |z| < 3/4, whenever n ≥46.