Classes of Harmonic Functions Defined by Subordination

and Applied Analysis 3 Thus, conditions (17) and (19) are equivalent, and the proof is completed. Putting B = −A = 1 inTheorem 1 we obtain the following corollary. Corollary 2. Let f ∈ SH. Then f ∈ S∗H if and only if f (z) ∗ φ (z; ξ) ̸ = 0, (ξ ∈ C, 󵄨󵄨󵄨ξ 󵄨󵄨󵄨 = 1) , (22) where φ (z; ξ) = 2ξz + (1 − ξ) z (1 − z) − 2z − (1 − ξ) z (1 − z) ,


Introduction
A continuous complex-valued function  = +V defined in a simply connected domain  ⊂ C is said to be harmonic in  if  and V are real harmonic in .There is a close interrelation between analytic functions and harmonic functions.For real harmonic functions  and V there exist analytic functions  and  so that  = Re() and V = Im().Therefore, every complex-valued function  harmonic in  with 0 ∈  can be uniquely represented as where ℎ = (+)/2 and  = (−)/2 are analytic functions in  with (0) = 0. Then we call ℎ the analytic part and  the coanalytic part of .It is easy to verify that the Jacobian of  is given by   () =      ℎ  ()      2 −        ()      2 , ( ∈ ) . ( The mapping  is locally univalent if   () ̸ = 0 in .A result of Lewy [1] shows that the converse is true for harmonic mappings.Therefore,  is locally univalent and sense-preserving if and only if      ℎ  ()      >        ()      , ( ∈ ) .
Let H denote the class of harmonic functions in the unit disc U fl U (1), where U() fl { ∈ C : || < }, and by H 0 we denote the class of function  ∈ H normalized by (0) =    (0) − 1 = 0. Then we may express the analytic functions ℎ and  defined by (1) as that is, (    +     ) , ( 1 = 1,      1     < 1,  ∈ U) . ( By S H we denote the class of functions  ∈ H 0 which are univalent and sense-preserving in U, and by A we denote the class of functions  ∈ H for which the coanalytic part vanishes. We say that a function  : U → C is subordinate to a function  : U → C and write () ≺ () (or simply  ≺ ), if there exists a complex-valued function  which maps U into oneself with (0) = 0, such that  () =  ( ()) , ( ∈ U) . (6) Abstract and Applied Analysis In particular, if  is univalent in U, we have the following equivalence: () ≺  () ⇐⇒ [ (0) =  (0) ,  (U) ⊂  (U)] .(7) For functions  1 ,  2 ∈ H of the form by  1 *  2 we denote the Hadamard product or convolution of  1 and  2 , defined by Firstly, Clunie and Sheil-Small [2] and Sheil-Small [3] studied S H together with some of its geometric subclasses.In particular, they investigated harmonic starlike functions in U and harmonic convex functions in U, which are defined as follows.We say that  ∈ S H is said to be harmonic starlike functions in U() if (U()) is a starlike domain with respect to the origin.Likewise  ∈ S H is said to be harmonic convex functions in U() if (U()) is a convex domain.
In particular, we have that  ∈ H 0 is harmonic starlike function if Re where Let − ≤  <  ≤ 1, 0 ≤  < 1. Motivated by Janowski [4] we define the following classes of functions.
Let S * H (, ) denote the class of functions  ∈ S H such that Also, by R H (, ) we denote the class of functions  ∈ S H such that Moreover, let us define We should notice here that Janowski [4] introduced the classes S * (, ) fl S * H (, ) ∩ A and R(, ) fl R H (, ) ∩ A. The classes S * H () and S  H () were investigated by Jahangiri [5,6].The classes S * H fl S * H (0) and S  H fl S  H (0) are the classes of functions  ∈ S H which are starlike in U() or convex in U(), respectively, for all  ∈ (0, 1⟩.It is easy to verify (e.g., see [5]) that In the paper we obtain some necessary and sufficient conditions for defined classes of functions.Some topological properties, radii of convexity and starlikeness, and extreme points of the classes are also considered.By using extreme points theory we obtain coefficients estimates, distortion theorems, and integral mean inequalities for the classes of functions.

Necessary and Sufficient Conditions
Since we have where then  ∈ S * H (, ).
Proof.It is clear that the theorem is true for the function () ≡ .Let  ∈ H 0 be a function of form (5) and let there exist  ∈ N such that   ̸ = 0 ( ≥ 2) or   ̸ = 0. Since by (24) we have Thus, by (3) function  is locally univalent and sense-preserving in U.
Therefore, by (27) we have This leads to the univalence of ; that is,  ∈ S H . Therefore,  ∈ S * H (, ) if and only if there exists a complex-valued function , (0 or equivalently Thus, it is sufficient to prove that Indeed, letting || =  (0 <  < 1) we have whence  ∈ S * H (, ).
Motivated by Silverman [7] we denote by T  ( ∈ {0, 1}) the class of functions  ∈ H 0 of form (5) Moreover, let us define Now, we show that condition (24) is also the sufficient condition for a function  ∈ T 0 to be in class S * T (, ).
Therefore, putting  =  (0 ≤  < 1) we obtain It is clear that the denominator of the left hand side cannot vanish for  ∈ (0, 1).Moreover, it is positive for  = 0, and in consequence for  ∈ (0, 1).
Thus we have the following.

Radii of Starlikeness and Convexity
We say that a function  ∈ H 0 is starlike of order  in U() if Analogously, we say that a function  ∈ H 0 is convex of order . (58) Therefore, the radius  * given by ( 57) cannot be larger.Thus we have (51).
The following result may be proved in much the same way as Theorem 11.

Topological Properties
We consider the usual topology on H defined by a metric in which a sequence {  } in H converges to  if and only if it converges to  uniformly on each compact subset of U. It follows from the theorems of Weierstrass and Montel that this topological space is complete.Let F be a subclass of the class H.A function  ∈ F is called an extreme point of F if the condition implies  1 =  2 = .We will use the notation F to denote the set of all extreme points of F. It is clear that F ⊂ F. We say that F is locally uniformly bounded if for each , 0 <  < 1, there is a real constant  = () so that      ()     ≤ , ( ∈ F, || ≤ ) .
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 co F.
A real-valued functional J : The Krein-Milman theorem (see [8]) is fundamental in the theory of extreme points.In particular, it implies the following lemma.
Lemma 13.If F is a nonempty compact subclass of the class H, then F is nonempty and co F = co F. [9, pp. 45] (see also [10]) we prove the following result.Lemma 14.Let F be a nonempty compact convex subclass of the class H and let J : H → R be a real-valued, continuous, and convex functional on F. Then max {J () :  ∈ F} = max {J () :  ∈ F} .

Due to Hallenbeck and MacGregor
(64) Proof.Since the functional J is continuous on the compact set F there exists max {J () :  ∈ F} =: .
Therefore, the set  fl { ∈ F : J() = } is nonempty compact subclass of F. Hence, by Lemma 13 we find that  has an extreme point  0 .Suppose that where  1 ,  2 ∈ F and 0 <  < 1.Then and we must have Since  0 is an extreme point of  we have  1 =  2 =  0 and, in consequence,  0 ∈ F.Thus, we conclude that there exists max{J() :  ∈ F} = , and the proof is complete.
Since H is a complete metric space, Montel's theorem (see [11]) implies the following lemma.

Lemma 15. A class F ⊂ H is compact if and only if F is closed and locally uniformly bounded.
Theorem 16.The class S * T (, ) is convex and compact subset of H 0 .

Applications
It is clear that for the locally uniformly bounded class Thus, by Theorems 16, 17, and 18 we have the following two corollaries.
For each fixed value of  ∈ N,  ∈ U, the following realvalued functionals are continuous and convex on H: ( The result is sharp.The function ℎ 2 of form ( 83) is the extremal function.
Due to Littlewood [12] we obtain the integral means inequalities for functions from the classes S * T (, ), S  T (, ).