# Convolutions of Harmonic Functions with Certain Dilatations.

1. Introduction

Let A denote the class of functions that are analytic in the open unit disc D := {z [member of] C : [absolute value of z] < 1} and let A' be the subclass of A consisting of functions h with the normalization h(0) = h' (0) - 1 = 0. We let K ([alpha]) denote the class of functions h [member of] A' so that

[mathematical expression not reproducible]. (1)

Consider the family of complex-valued harmonic functions f = u + iv defined in D, where u and v are real harmonic in D. Such functions can be expressed as f = h + [bar.g], where h [member of] A and g [member of] A. Clunie and Sheil-Small in their remarkable paper [1] explored the functions of the form f = h + [bar.g] that are locally one-to-one, sense-preserving, and harmonic in D. By Lewy's Theorem (see [2] or [1]), a necessary and sufficient condition for the harmonic function f = h + [bar.g] to be locally one-to-one and sense-preserving in D is that its Jacobian [J.sub.f] = [[absolute value of h'].sup.2] - [[absolute value of g'].sup.2] is positive or equivalently, if and only if h' (z) [not equal to] 0 in D and the second complex dilatation [omega] of f satisfies [[absolute value of [omega]] = [absolute value of g'/h'] < 1 in D.

In an interesting article, Bshouty and Lyzzaik [3] proved the following.

Theorem 1. Let f = h + [bar.g] be a harmonic mapping of D, with h'(0) [not equal to] 0, that satisfies g'(z) = zh'(z) and h [member of] K (-1/2) for all z [member of] D. Then f is a univalent close-to-convex mapping.

A simply connected proper subdomain of C is said to be close-to-convex if its complement in C is the union of closed half-lines with pairwise disjoint interiors. Consequently, a univalent analytic or harmonic function f: D [right arrow] C is said to be close-to-convex if f (D) is close-to-convex (e.g., see Clunie and Sheil-Small [1] or Bshouty and Lyzzaik [3]).

Ruscheweyh and Sheil-Small in a striking article [4] proved that the Hadamard product or convolution of two analytic convex functions is also convex analytic and that the convolution of an analytic convex function and an analytic close-to-convex function is close-to-convex analytic in the unit disk D. Ironically, these results could not be extended to the harmonic case, since the convolution of harmonic functions, unlike the analytic case, proved to be very challenging. The purpose of the present paper is to introduce dilatation conditions that guarantee the convolution of two harmonic functions to be locally one-to-one, sense-preserving, and close-to-convex harmonic in the unit disk D. In other words, we extend Theorem 1 to the convolution of two harmonic functions [f.sub.1] = [h.sub.1] + [bar.[g.sub.1]] and [f.sub.2] = [h.sub.2] + [bar.[g.sub.2]] with certain dilatations, where [h.sub.1] * [h.sub.2] [member of] K ([alpha]).

The operator * stands for the convolution or Hadamard product of two power series [h.sub.1] (z) = [[summation].sup.[infinity].sub.n=1] [a.sub.n] [z.sup.n] and [h.sub.2] (z) = [[summation].sup.[infinity].sub.n = 1] [b.sub.n] [z.sub.n] given by [h.sub.1] (z) * [h.sub.2] (z) = ([h.sub.1] * [h.sub.2])(z) = [[summation].sup.[infinity].sub.n=1] [a.sub.n] [b.sub.n] [z.sub.n]. Similarly, the convolution of two harmonic functions [f.sub.1] = [h.sub.1] + [bar.[g.sub.1]] and [f.sub.2] = [h.sub.2] + [bar.[g.sub.2]] is given by [f.sub.1] * [f.sub.2] = [h.sub.1] * [h.sub.2] + [bar.[g.sub.1]] * [g.sub.2]].

In regard to the convolution of harmonic univalent functions, Clunie and Sheil-Small [1] proved the following.

Theorem 2. If [phi] [member of] K(0) and if f is convex harmonic in D, then their convolution ([phi] + [epsilon] [bar.[phi]]) * f ([absolute value of [epsilon]] [less than or equal to] 1) is close-to-convex harmonic in D.

A mapping f : D [right arrow] C is called convex harmonic if f (D) is a convex domain.

The convexity condition for the function [phi] in Theorem 2 cannot be compromised as it is demonstrated in the following.

Example 3. Set

[mathematical expression not reproducible] (2)

and consider the starlike analytic function [phi] (z) = z + [z.sup.n]/n; n [greater than or equal to] 2 in D. Letting [epsilon] = 0 in Theorem 2, we observe that the harmonic convolution

[mathematical expression not reproducible] (3)

is not even univalent in D.

In an attempt to investigate the possibilities of improving the required convexity condition for [phi], the authors in [5] proved the following.

Theorem 4. Let [phi] [member of] K(0) and h [member of] K(0). Also let w be a Schwarz function. Then the convolution function

[mathematical expression not reproducible], (4)

is close-to-convex harmonic in D.

Theorem 4 for [phi] (z) = z/(1 - z) and [epsilon] = 1 yields a theorem given by Bshouty et al. ([6], Theorem 2). From what is said above, especially Example 3, one wonders if there are other conditions that guarantee the close-to-convexity of the convolution of two harmonic functions. In the following theorem, we find such conditions.

Theorem 5. Let [h.sub.1] [member of] A' and [h.sub.2] [member of] A' so that [h.sub.1] * [h.sub.2] [member of] K ([alpha]), where K([alpha]) is given by inequality (1). If one of the following conditions hold

(i) [alpha] [greater than or equal to] -1/2; [g.sub.1](z) = [zh.sub.1](z) and [g'.sub.2](z) = [zh'.sub.2](z),

(ii) [alpha] [greater than or equal to] 0; [g'.sub.1][(z) = [z.sub.n] [h'.sub.1] [(z) and [g'.sub.2] (z) = [z.sub.n] [h'.sub.2] (z), where n [member of] N,

then the convolution function F(z) = [h.sub.1] (z) * [h.sub.2] (z) + [bar.[g.sub.1](z) * [g.sub.2](z)] is locally one-to-one, sense-preserving, and close-to-convex harmonic in D.

Since the convolution of two convex analytic functions is also convex (see Ruscheweyh and Sheil-Small [4]), an obvious consequence of the above theorem would be as follows.

Corollary 6. Let [h.sub.1] [member of] K(0) and [h.sub.2] [member of] K(0) and set [g.sub.1] (z) = [z.sup.n] [h'.sub.1] (z) and [g'.sub.2] (z) = [z.sup.n] [h'.sub.2](z). Then the convolution function F(z) = [h.sub.1] (z)* [h.sub.2] (z)+ [bar.[[g.sub.1] (z) * [g.sub.2](z)] is locally one-to-one, sense-preserving, and close-to-convex harmonic in D.

2. Preliminary Lemmas and Proof of Theorem 5

To prove our Theorem 5, we shall need the following three lemmas, the first of which is a celebrated result by Clunie and Sheil-Small [1] and the second one is given by Kaplan [7]. The third lemma which is on subordination is a modification of a result given by Miller and Mocanu (e.g., see [8] Lemma 1 or [9]). For functions p and q, where p(0) = q(0) = 0, we write p [??] q (i.e., p is subordinate to q) if there exists an analytic function [omega] with [omega] (0) = 0 and [absolute value of ([omega](z))] < 1 so that p(z) = q([omega](z)) in D.

Lemma 7. (i) If g and h are analytic in D so that [absolute value of g'(0)] < [absolute value of h'(0)] and if h + [epsilon]g is close-to-convex analytic in D for each [epsilon] ([absolute value of [epsilon]] = 1), then the function f = h + [bar.g] is close-to-convex harmonic in D.

(ii) If h and g are analytic in D so that h [member of] K(0) and if f = h + [bar.g] is locally univalent in D, then the function f = h + [bar.g] is close-to-convex harmonic in D.

Lemma 8. A necessary and sufficient condition for the analytic function h : D [right arrow] C to be close-to-convex is that h' is nonvanishing in D and

[mathematical expression not reproducible]. (5)

Lemma 9. If R p(z) > 0 and q(z) is analytic in D, then q(z) + zp(z)q'(z) < z implies q(z) < z.

Proof of Theorem 5.

Proof of Part (i). The convolution function F(z) = [h.sub.1](z) * [h.sub.2](z) + [bar.[g.sub.1](z) * [g.sub.2](z)] = H(z) + [bar.G(z)] is locally univalent and sense-preserving since

[mathematical expression not reproducible]. (6)

Obviously [absolute value of G'(0)] < [absolute value of H'(0)]; therefore, in view of Lemma 7, it suffices to prove that [T.sub.[lambda]](z) = H(z) - [lambda]G(z) for [absolute value of [lambda]] = 1 is close-to-convex analytic in D.

We note that

[mathematical expression not reproducible] (7)

and [T".sub.[lambda]](z) = -[lambda]H'(z) + (1 - [lambda]z) H" (z).

We also observe that [T.sub.[lambda]] is nonvanishing in D since H'(0) [not equal to] 0. Therefore,

[mathematical expression not reproducible]. (8)

Now, by Lemma 8 and inequality (5) for [[theta].sub.1] < [[theta].sub.2] < [[theta].sub.1 + 2[pi] and 0< r < 1, it suffices to show that

[mathematical expression not reproducible]. (9)

For z [member of] D, one may verify (also see Bshouty and Lyzzaik [3] p. 770) that

[mathematical expression not reproducible]. (10)

For z = [re.sup.i[theta]], replacing z by [lambda]z and letting [zeta] = [bar.[lambda]]r yield

[mathematical expression not reproducible], (11)

where [P.sub.[zeta]]([theta]) is the Poisson Kernal. It then follows that

[mathematical expression not reproducible]. (12)

On the other hand, since H(z) = [h.sub.l](z) * [h.sub.2](z) [member of] K (-1/2), we obtain

[mathematical expression not reproducible]. (13)

Therefore, in view of the required condition (9), we get

[mathematical expression not reproducible]. (14)

Proof of Part (ii). In view of Lemma 7, it suffices to show that the convolution function F(z) = [h.sub.1] (z) * [h.sub.2] (z) + [bar.[g.sub.i](z) * [g.sub.2](z)] = H(z) + [bar.G(z)] is locally univalent and sense-preserving in D. In other words, we need to show that

[mathematical expression not reproducible]. (15)

Using the Hadamard product properties of power series, we have

[mathematical expression not reproducible]. (16)

Therefore,

[mathematical expression not reproducible]. (17)

On the other hand, since [h.sub.1] * [h.sub.2] [member of] K(0), z([h.sub.1] * [h.sub.2])' is starlike or

[mathematical expression not reproducible]. (18)

Thus, in view of Lemma 9, [mathematical expression not reproducible].

Remark 10. It is left as an open problem whether Theorem 5(i) can be extended to the case [g.sub.1](z) = [z.sup.n] [h.sub.1](z) and [g'.sub.2](z) = [z.sup.n] [h'.sub.2](z) if n > 1.

https://doi.org/10.1155/2017/4015268

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

References

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[2] H. Lewy, "On the non-vanishing of the Jacobian in certain one-to-one mappings," Bulletin (New Series) of the American Mathematical Society, vol. 42, no. 10, pp. 689-692, 1936.

[3] D. Bshouty and A. Lyzzaik, "Close-to-convexity criteria for planar harmonic mappings," Complex Analysis and Operator Theory, vol. 5, no. 3, pp. 767-774, 2011.

[4] S. Ruscheweyh and T. Sheil-Small, "Hadamard products of SCHlicht functions and the Polya-SCHoenberg conjecture," Commentarii Mathematici Helvetici, vol. 48, pp. 119-135, 1973.

[5] O. P. Ahuja, J. M. Jahangiri, and H. Silverman, "Convolutions for special classes of harmonic univalent functions," Applied Mathematics Letters, vol. 16, no. 6, pp. 905-909, 2003.

[6] D. Bshouty, S. S. Joshi, and S. B. Joshi, "On close-to-convex harmonic mappings," Complex Variables and Elliptic Equations, vol. 58, no. 9, pp. 1195-1199, 2013.

[7] W. Kaplan, "Close-to-convex schlicht functions," Michigan Mathematical Journal, vol. 1, pp. 169-185, 1952.

[8] S. S. Miller and P. T. Mocanu, "Univalent solutions of Briot-Bouquet differential equations," Journal of Differential Equations, vol. 56, no. 3, pp. 297-309, 1985.

[9] S. S. Miller and P. T. Mocanu, Differential Subordination--Theory and Applications, vol. 225 of Monorgraphs and Textbooks in Pure and Applied Mathematics, Marcel Dekker, New York, NY, USA, 2000.

Om P. Ahuja and Jay M. Jahangiri

Mathematical Sciences, Kent State University, Burton, OH 44021-9500, USA

Correspondence should be addressed to Jay M. Jahangiri; jjahangi@kent.edu

Received 1 October 2017; Accepted 13 November 2017; Published 29 November 2017