Classes of harmonic functions defined by convolution

The object of the present paper is to investigate classes of harmonic functions defined by convolution. Some necessary and sufficient conditions, topological properties, radii of convexity and starlikeness, as well as extreme points for the classes are considered.


Introduction
Let H denote the class of complex harmonic functions in U := U (1) , where U (r ) := {z ∈ C : |z| < r }, and let H 0 denote the class of functions f ∈ H of the form f (z) = z + ∞ n=2 a n z n + b n z n (z ∈ U) . (1) By S H we denote the class of functions f ∈ H 0 , which are univalent and sense preserving in U.
We say that a function f ∈ S H is harmonic starlike in U (r ) if f maps the circle ∂U (r ) onto a closed curve that is starlike with respect to the origin, i.e., where Let us consider the function ϕ ∈ H of the form with |u n | ≥ 1, |v n | ≥ 1 (n = 2, 3, . . .) .
We say that a function f ∈ H of the form (1) is correlated with the function ϕ, if u n a n = − |u n | |a n | , v n b n = |v n | |b n | (n = 2, 3, . . .) .
Moreover, let us define In the paper, we obtain some analytic criteria, radii of convexity and starlikeness for defined classes of functions. Using extreme points theory, we obtain coefficients estimates, distortion theorems and integral mean inequalities. Some applications of the obtained results are also considered.

Analytic criteria
Let V ⊂ H, U 0 := U {0} . Motivated by Ruscheweyh [31] we define the dual set of V by First, we show that the defined classes of functions can be presented as dual sets.
Since f = h + g for some analytic functions h, g, and Similarly as Theorem 1 we prove the following theorem.
Theorem 3 Let f ∈ H be of the form (1) and where .
Proof Let f ∈ H be a function of the form (1). By a theorem of Lewy [25], a necessary and sufficient condition for f to be locally univalent and sense preserving in U is that By (3) we obtain and, by (11) we get ∞ n=2 (n |a n | + n |b n |) ≤ 1. Moreover, Thus, by (13) the function f is locally univalent and sense preserving in U. Also, for Hence, by (15) we get and, in consequence, f ∈ S H . Thus, we need only prove that the function f satisfies (4), i.e., there exists a complex-valued function or equivalently Therefore, it is suffice to prove that Indeed, letting |z| = r (0 < r < 1) we have Now, we show that the condition (11) is also a sufficient condition for a function f ∈ T η (ϕ) to be in the class W  (7) and satisfies (16) It is clear that the denominator of the left-hand side cannot vanish for r ∈ 0, 1) . Moreover, it is positive for r = 0, and in consequence for r ∈ 0, 1) . Thus, by (17) we have ∞ n=2 (|γ n a n | + |δ n b n |) The sequence of partial sums {S n } associated with the series ∞ n=2 (|γ n a n | + |δ n b n |) is a nondecreasing sequence. Moreover, by (18) which yields the assertion (11).
The following result may be proved in the same way as Theorem 4.
By Theorems 4 and 5 we have the following corollary.
In particular,

Extreme points
implies f 1 = f 2 = f . We shall use the notation EF to denote the set of all extreme points of F. It is clear that EF ⊂ F.
We say that F is locally uniformly bounded if for each r , 0 < r < 1, there is a real constant M = M (r ) so that We say that a class F is convex if Moreover, we define the closed convex hull of F as the intersection of all closed convex subsets of H that contain F. We denote the closed convex hull of F by coF.
The Krein-Milman theorem (see [24]) is fundamental in the theory of extreme points. In particular, it implies the following lemma.
Since H is a complete metric space, Montel's theorem (see [26]) implies the following lemma.

Theorem 6 The class W
Let f 1 , f 2 ∈ W η T (ϕ; A, B) be functions of the form (19), 0 ≤ γ ≤ 1. Since γ a 1,n + (1 − γ ) a 2,n z n + γ b 1,n + (1 − γ ) b 2,n z n , and by Theorem 4 we have Thus, we conclude that the class W η T (ϕ; A, B) is locally uniformly bounded. By Lemma 2, we only need to show that it is closed, i.e., if ϕ; A, B) . Let f k and f be given by (19) and (1), respectively. Using Theorem 4 we have Since f k → f , we conclude that a k,n → |a n | and b k,n → |b n | as k → ∞ (n ∈ N). The sequence of partial sums {S n } associated with the series ∞ n=2 (|γ n a n | + |δ n b n |) is a nondecreasing sequence. Moreover, by (21) it is bounded by B − A. Therefore, the sequence {S n } is convergent and ∞ n=2 (|γ n a n | + |δ n b n |) = lim This gives the condition (11), and, in consequence, f ∈ W η T (ϕ; A, B) , which completes the proof.
Proof Suppose that 0 < γ < 1 and ϕ; A, B) are functions of the form (19). Then, by (11) we have b 1,n = b 2,n = B−A |δ n | , and, in consequence, a 1,k = a 2,k = 0 for k ∈ {2, 3 . . .} and b 1,k = b 2,k = 0 for k ∈ {2, 3 . . .} {n} . It follows that g n = f 1 = f 2 , and consequently g n ∈ EW η T (ϕ; A, B) . Similarly, we verify that the functions h n of the form (22) are the extreme points of the class W η H (ϕ; A, B). Now, suppose that a function f belongs to the set EW η H (ϕ; A, B) and f is not of the form (22). Then there exists m ∈ {2, 3, . . .} such that It follows that f / ∈ EW η H (ϕ; A, B), and the proof is completed.

Radii of starlikeness and convexity
We say that a function f ∈ H 0 is starlike of order α in U(r ) if Analogously, we say that a function f ∈ H 0 is convex of order α in U(r ) if It is easy to verify that for a function f ∈ T (ϕ) the condition (23) is equivalent to the following Re or equivalently Let B be a subclass of the class H 0 . We define the radius of starlikeness R * α (B) and the radius of convexity R c α (B) for the class B by

Theorem 8 The radius of starlikeness of order α for the class W η H (ϕ; A, B) is given by
where γ n and δ n are defined by (12).

Thus, the condition (24) is true if and only if
By Theorem 1, we have where γ n and δ n are defined by (12). Thus,the condition (26) is true if It follows that the function f is starlike of order α in the disk U (r * ), where r * The functions h n , g n of the form (22) realize equality in ( 27), and the radius r * cannot be larger. Thus we have (25).
The following result may be proved in much the same way as Theorem 8 .

Theorem 9
The radius of convexity of order α for the class W η H (ϕ; A, B) is given by where γ n and δ n are defined by (12).

Applications
It is clear that if the class F = { f n ∈ H : n ∈ N} is locally uniformly bounded, then Thus, by Theorem 8 we have the following corollary. where h n , g n are defined by (22).
For each fixed value of n ∈ N, z ∈ U, the following real-valued functionals are continuous and convex on H: (29) Moreover, for γ ≥ 1, 0 < r < 1, the real-valued functional is also continuous and convex on H.
Therefore, by Lemma 1 and Theorem 8 we have the following corollaries.
where γ n , δ n are defined by (12). The result is sharp. The functions h n , g n of the form (22) are the extremal functions.
where u 2 is defined by (2). The result is sharp. The function h 2 of the form (22) is the extremal function.
where h 2 is the function defined by (22).
The following covering result follows from Corollary 4.
where u n and v n are defined by (2). The results are sharp. The functions h n , g n of the form (32) are the extremal functions.

Concluding remarks
We conclude the paper by observing that, in view of the subordination relation (5) and (6), choosing the function ϕ and the parameters A, B we can consider new and also well-known classes of functions. In particular, the classes contain functions f ∈ S H which satisfy the conditions respectively. The considered classes are defined using the convolution ϕ * f or equivalently by the linear operator J ϕ : H → H, J ϕ ( f ) = ϕ * f .