Coefficient conditions for harmonic close-to-convex functions

New sufficient conditions, concerned with the coefficients of harmonic functions $f(z)=h(z)+\bar{g(z)}$ in the open unit disk $\mathbb{U}$ normalized by $f(0)=h(0)=h'(0)-1=0$, for $f(z)$ to be harmonic close-to-convex functions are discussed. Furthermore, several illustrative examples and the image domains of harmonic close-to-convex functions satisfying the obtained conditions are enumerated.


Introduction
For a continuous complex-valued function f (z) = u(x, y) + iv(x, y) (z = x + iy), we say that f (z) is harmonic in the open unit disk U = {z ∈ C : |z| < 1} if both u(x, y) and v(x, y) are real harmonic in U, that is, u(x, y) and v(x, y) satisfy the Laplace's equations ∆u = u xx + u yy = 0 and ∆v = v xx + v yy = 0.
A complex-valued harmonic function f (z) in U is given by f (z) = h(z) + g(z) where h(z) and g(z) are analytic in U. We call h(z) and g(z) the analytic part and the co-analytic part of f (z), respectively. A necessary and sufficient condition for f (z) to be locally univalent and sense-preserving in U is |h ′ (z)| > |g ′ (z)| in U (see, [2] or [8]). Let H denote the class of harmonic functions f (z) in U with f (0) = h(0) = 0 and h ′ (0) = 1. Thus, every normalized harmonic function f (z) can be written by f (z) = h(z) + g(z) = z + ∞ n=2 a n z n + ∞ n=1 b n z n ∈ H where a 1 = 1 and b 0 = 0, for convenience.
We next denote by S H the class of functions f (z) ∈ H which are univalent and sense-preserving in U. Since the sense-preserving property of f (z), we see that |b 1 | = |g ′ (0)| < |h ′ (0)| = 1. If g(z) ≡ 0, then S H reduces to the class S consisting of normalized analytic univalent functions. Furthermore, for every function f (z) ∈ S H , the function is also a member of S H . Therefore, we consider the subclass S 0 We say that a domain D is a close-to-convex domain if the complement of D can be written as a union of non-intersecting half-lines (except that the origin of one half-line may lie on one of the other half-lines). Let C, C H and C 0 H be the respective subclasses of S, S H and S 0 H consisting of all functions f (z) which map U onto a certain close-to-convex domain.
Bshouty and Lyzzaik [1] have stated the following result. and A simple and interesting example is below.
Remark 1.1. Let M be the class of all functions satisfying the conditions of Theorem 1.1. Then, it was earlier conjectured by Mocanu [9,10] that M ⊂ S 0 H . Furthermore, we can immediately see that the function f (z) in Example 1.1 is a member of the class M and it shows that f (z) ∈ M is not necessarily starlike with respect to the origin in U (f (z) is starlike with respect to the origin in U if and only if tw ∈ f (U) for all w ∈ f (U) and t (0 ≦ t ≦ 1)).
which means that f (z) maps the unit circle ∂U = {z ∈ C : |z| = 1} onto a union of several concave curves (see, [6, Theorem 2.1]). Jahangiri and Silverman [7] have given the following coefficient inequality for f (z) ∈ H to be in the class C H .
belongs to the class C 0 H ⊂ C H and satisfies the condition of Theorem 1.2. Indeed, f (z) maps U onto the following hypocycloid of six cusps (cf. [3] or [6]). The object of this paper is to find some sufficient conditions for functions f (z) ∈ H to be in the class C H . In order to establish our results, we have to recall here the following lemmas due to Clunie and Sheil-small [2]. Lemma 1.1. If h(z) and g(z) are analytic in U with |h ′ (0)| > |g ′ (0)| and h(z) + εg(z) is close-to-convex for each ε (|ε| = 1), then f (z) = h(z) + g(z) is harmonic close-to-convex.
We also need the following result due to Hayami, Owa and Srivastava [5].
for some real numbers α and β, then F (z) is convex in U.
Example 2.1. The function satisfies the condition of Theorem 2.1 with ϕ = 0 and belongs to the class C H . In particular, putting m = 1, we obtain the following.  Figure 3. The image of f (z) = −z − 2 log |1 − z|.
By making use of Lemma 1.2 with ε = 0 and applying Lemma 1.3, we readily obtain the next theorem.
for some real numbers α and β, then f (z) ∈ C H .
Putting α = β = 0 in the above theorem, we arrive at the following result due to Jahangiri and Silverman [7].
The next lemma was obtained by Fejér [4].
Lemma 3.1. Let {c n } ∞ k=0 be a convex null sequence. Then, the function c n z n is analytic and satisfies Re(p(z)) > 0 in U.
Applying the above lemma, we deduce Theorem 3.1. For some b (|b| < 1) and some convex null sequence {c n } ∞ n=0 with c 0 = 2, the function c n−1 n z n belongs to the class C H .
Proof. Let us define F (z) by c n−1 n z n for each ε (|ε| = 1). Then, we know that c n z n (c 0 = 2).
In the same manner, we also have Theorem 3.2. For some b (|b| < 1) and some convex null sequence {c n } ∞ n=0 with c 0 = 2, the function belongs to the class C H .
Setting b = 1 4 in Theorem 3.1 with the above sequence {c n } ∞ n=0 , we derive  Figure 6. The image of f (z) in Example 3.2.