Classes of meromorphic harmonic functions and duality principle

We introduce new classes of meromorphic harmonic univalent functions. Using the duality principle, we obtain the duals of such classes of functions leading to coefficient bounds, extreme points and some applications for these functions.


Introduction
A continuous function f = u +iv is a complex-valued harmonic function in a domain D⊂C if both u and v are real harmonic in D. In any simply connected domain D, we write f = h + g, where h and g are analytic in D. A necessary and sufficient condition for f to be locally one-to-one and orientation preserving in D is that |g (z)| < |h (z)| for z ∈ D (see Clunie and Sheil-Small [4]). Functions that are harmonic and univalent in D = {z : |z| > 1} are investigated by Hengartner and Schober [7]. In particular, it was shown in [7] that a complex-valued, harmonic, orientation preserving univalent mapping f , defined on D and satisfying f (∞) = ∞, must admit the representation where h(z) = αz + ∞ n=0 a n z −n , g(z) = βz + ∞ n=0 b n z −n , 0 ≤ |β| < |α|, and ω = f z / f z is analytic and satisfies |ω(z)| < 1 for z ∈ D. We remove the logarithmic singularity in (1) by letting A = 0 and also let α = 1 and β = 0 and focus on the family H of meromorphic harmonic orientation preserving univalent mappings of the form where h(z) = z + ∞ n=1 a n z −n , g(z) = ∞ n=1 b n z −n .
We say that a function f ∈ H is harmonic starlike in D if where z = re it ∈D, 0≤t≤2π , and n a n z −n − b n z −n .
For l = 1, 2 and functions f l ∈ H of the form we definethe convolution of f 1 and f 2 by We note that D 0 In 2008 Mauir [13] considered a weak subordination for complex-valued harmonic functions defined in the open unit disk := {z : |z| < 1}. In [8] Jahangiri (see also [10]) investigated the classes of harmonic meromorphic starlike and convex functions of order γ. To obtain some generalizations of these classes we introduce definition of weak subordination for complex-valued functions in D.
A complex-valued function f in D is said to be weakly subordinate to a complexvalued function F in D, and we write f (z) If f F and F is univalent in D, then we can consider the function F. Thus, we have the following equivalence.

Lemma 1 A complex-valued function f in D is weakly subordinate to a function complex-valued function F in D if and only if there exists a complex-valued function
The following two subclasses of H are the main focus of this paper.
We further let T η be the class of functions f = h+g ∈ H with varying coefficients (e.g. see [9]) for which there exists a real number η so that

Dual sets and coefficient bounds
Here we make use of the "duality principle" defined by Ruscheweyh ([14] , Chapter 1). For the set V ⊂ H we define the dual set of V by Consequently, the second dual of V or the dual of V * is defined as This duality principle indicates that under fairly weak conditions on V which is a subset of H , many linear and other extremal problems in the second dual of V are solved in V. This is a very useful tool since in many cases of interest (such as convex, starlike or close-to-convex harmonic univalent functions), the set V * * is much larger than the set V. This, on its own right, would be a separate endeavor and research topic that can be explored further which is not the focus of the present paper. The following theorem presents a duality condition for the set m H (A, B). . Since the above inequality (5) yields A similar argument can be used to obtain a duality condition for the set V m H (A, B).
Next we determine sufficient coefficient bounds for function in m  H (A, B).
Thus for z ∈ D it suffices to show that Indeed, letting |z| = r (r > 1) we have A)) and β n = m n (n(1 + B) − (1 + A))). Thus, according to the hypothesis of the theorem, f ∈ m H (A, B).
It is clear that the denominator of the ratio in the inequality (7) cannot vanish for r > 1. Moreover, it is positive as r →∞ and consequently for r > 1. Thus, we must have ∞ n=1 (α n |a n | + β n |b n |) r −n−1 < B − A(r > 1) (8) where α n = m n (n (1 + B) + (1 + A)) and β n = m n (n (1 + B) − (1 + A))). The sequence of partial sums {S n } associated with the series ∞ n=1 (α n |a n | + β n |b n |) is a non-decreasing sequence. Moreover, by (8), it is bounded above by B − A. Hence, the sequence {S n } is convergent and A similar argument can be used to prove the following theorem.

Theorem 5 Let f = h + g be of the form (2) and −B ≤
As a consequence of Theorems 4 and 5 we have the following corollary. (A, B).

Extreme points
The Krein-Milman theorem (see [11]) is fundamental in the theory of extreme points. In particular, it implies the following lemma.

Lemma 2 Let F be a non-empty compact convex subset of the set H and J : H → R be a real-valued, continuous and convex functional on F.
Then where EF denotes the set of extreme points of F.
Since H is a complete metric space, Montel's theorem [12] implies the following lemma.

Lemma 3 A set F ⊂ H is compact if and only if F is closed and locally uniformly bounded.
Now we are equipped to state and prove the two main theorems in this section.

Theorem 6 The set m η (A, B) is a convex and compact subset of H .
Proof For l = 1, 2 let f l ∈ m η (A, B) be functions of the form (3), 0 ≤ γ ≤ 1. Since and since for (by Theorem 4), we have (A, B). Hence, the set m η (A, B) is convex. Furthermore, for f ∈ m η (A, B) and |z| = r > 1 we have Thus, the class m η (A, B) is locally uniformly bounded. Now, by Lemma 3, we only need to show that it is closed i.e. if f l ∈ m η (A, B) and f l → f then f ∈ m η (A, B). Let f l and f be given by (3) and (4), respectively. Using Theorem 4 we have ∞ n=1 α n a l,n + β n b l,n ≤ B − A (l ∈ N) .
Since f l → f , we conclude that a l,n → a n and b l,n → b n as l → ∞ (n ∈ N). The sequence of partial sums {S n } associated with the series ∞ n=1 (α n |a n | + β n |b n |) is a non-decreasing sequence. Moreover, by (10), it is bounded above by B − A. Therefore, the sequence{S n } is convergent and This implies that f ∈ m η (A, B) which completes the proof.
In the following theorem we determine the extreme points of m η (A, B).
Proof Suppose that 0 < γ < 1 and g n = γ f 1 + (1 − γ ) f 2 where f 1 , f 2 ∈ m η (A, B) are functions of the form (3) and let α n , β n be defined by (9). Then b 1,n = b 2,n = B−A β n and consequently a 1,n = a 2,n = 0 and b 1,k = b 2,k = 0 for k ∈ N {n} . It follows that g n = f 1 = f 2 , and so g n ∈ m η (A, B). Similarly, we can verify that the functions h n of the form (11) are the extreme points of the class m η (A, B). Now, suppose that the function f belongs to the set E m η (A, B) and f is not of the form (11). Then there exists s ∈ N such that we have that h s = ϕ and so f = γ h s +(1 − γ ) ϕ, 0 < γ < 1. Thus, f / ∈ E m η (A, B).
It follows that f / ∈ E m η (A, B) and so the proof is completed.

Applications
It is clear that if the class is locally uniformly bounded, then Thus, by Theorem 7, we have the following corollary. where h n and g n are given by (11).
For f ∈ H , γ ≥ 1 and r > 1 the real-valued functional is continuous and convex on H .
Moreover, for f ∈ H , z ∈ D and each fixed n ∈ N the real-valued functionals  (1 + A))) .
The result is sharp and the functions h n and g n of the form (11) are the extremal functions.

Corollary 4 Let f ∈ m η (A, B). Then
The result is sharp and the extremal function is given by (11).
The following covering result follows from Corollary 4.