Abstract
The aim in the present work is to introduce and study new subclasses of analytic functions that are defined by using the generalized classes of Janowski functions combined with the -symmetrical functions, that generalize many others defined by different authors. We gave a representation theorem for these classes, certain inherently properties, while covering and distortion properties are also pointed out.
2010 Mathematics Subject Classification:
1. Introduction
Let be the space of all analytic functions in the open unit disk and let denote the class of functions normalized with . Denote by Ω for the class of Schwarz functions, that is (1) (1) For f and g are two analytic functions in , we say that the function f is subordinate to the function g in , if there exists a function , such that for all , and we denote this by . Furthermore, if g is univalent in , then the subordination is equivalent to and . (see, for details, [Citation1])
Using the notation of the subordination, let define the class of functions with positive real parts (or Carathéodory functions) in :
Definition 1.1
[Citation2]
Let denote the class of functions satisfying and for all .
From the above-mentioned reasons, it follows that any function has the representation , for some .
The class of functions with positive real part plays a significant role in complex function theory. Its significance can be seen from the fact that all simple subclasses of the class of univalent functions have been defined by using the concept of the class of functions with positive real part like the classes , , which are respectively the class of starlike, convex functions and the class of starlike functions with respect to symmetric points, etc., have been defined by using the class .
Definition 1.2
Like in [Citation3], let , with , denote the class functions that satisfy the subordination condition .
The class of generalized Janowski type functions was introduced in [Citation4] as follows: for arbitrary fixed numbers , with , ,
Like in [Citation4], a function , then
Definition 1.3
For the positive integer number m, a domain is said to be m-fold symmetric domain, if a rotation of about the origin through and angle carries onto itself.
A function , where is a m-fold symmetric domain, is said to be m-fold symmetrical function if, The set of all m-fold symmetrical denoted by ,
The theory of -symmetrical functions for and () is a generalization of the notion of odd, even, m-symmetrical functions.
Definition 1.4
Let , where and .
A function , where is a m-fold symmetric domain, is called -symmetrical function if .
Denoted be for the set of all -symmetrical functions. Let us observe that the sets , , and are well-known families of odd functions, of even functions and of symmetrical functions, respectively.
The following decomposition theorem holds:
Theorem 1.1
[Citation5]
Let be a m-fold symmetric domain, the for every function , can be written in the form and this partition is unique sequence of -symmetrical functions. (2) (2)
Al Sarari and Latha [Citation6] introduced the classes and which are the classes of Janowski type functions with respect to -symmetric points.
By using the theory of -symmetrical functions and the generalized Janowski type functions, we will define the following class:
Definition 1.5
A function is said to belongs to the class , with and , if where the function is defined by (Equation2(2) (2) ).
Remark 1.1
By applying the definition of the subordination we can easily obtain that the equivalent condition for a function f belonging to the class , with and , is
Note that special values of n, m, γ, X and Y yield the following classes that have been previously introduced by different authors:
is the class studied by Al Sarari and Latha [Citation6].
is the class is introduced by Sakaguchi [Citation7].
is the class was introduced by Ohang and Youngjae in [Citation8].
The following lemmas are important to proof our results:
Lemma 1.1
[Citation4, Corollary 1]
For a function then for some where Ω was defined by (Equation1(1) (1) ).
Lemma 1.2
[Citation9, Lemma 2.3]
For then
2. Main results
Theorem 2.1
If , then (3) (3) where is defined by (Equation2(2) (2) ), and for some .
Proof.
Let , we can get (4) (4) Replacing z by in (Equation4(4) (4) ), hence (5) (5) Letting in (Equation5(5) (5) ), and since is a convex set, we deduce that or equivalently that is , and by Lemma 1.1 we finally obtain our result.
Theorem 2.2
For with and . Then, For some .
Proof.
Supposing that , then there exists a function , such that Combining the above relation with Theorem 2.1, we have and integrating the above relations we obtain our result.
Theorem 2.3
Let and with . Then,
, if Y = 0;
, for , if Y >0;
for , if Y <0.
Proof.
Since , then Thus, hence (6) (6)
(i) For Y = 0 it is sufficient to show that (7) (7) From we have which implies and according to (Equation3(3) (3) ) hence, there exists a Schwarz function such that Thus, and using the fact that for all , we may easily prove that From the above two inequalities, it follows and consequently, from (Equation6(6) (6) ) we obtain that is .
(ii) For , we need to determine the value , such that (8) (8) whenever , which is equivalent to
For . According to the Remark 1.1 and the definition of the subordination we have that is equivalent to (9) (9) Next, we will prove that (10) (10) implies (11) (11) Since , from Definition 1.5 and Theorem 2.1, it follows that there exist the functions , such that Using the above relations, the assumption (Equation10(10) (10) ) is equivalent to that is or If we denote the above inequality shows that , therefore whenever .
Dividing this inequality by , we obtain that is which represents (Equation11(11) (11) ).
From the above reasons, we will determine now the biggest value of such that (Equation10(10) (10) ) holds for . Since , there exists a function , such that therefore (12) (12)
Next, we will prove that (13) (13) Thus, from Theorem 2.1, we have for some , and we will split this proof in the next two cases:
Case 1. If Y >0, since X − Y >0 it follows that
Case 2. If Y <0, denoting , then C + X>0, and it follows that
Combining the above two cases, we get
From (Equation12(12) (12) ) and (Equation13(13) (13) ) we easily deduce that where is given like in the assumptions (ii) and (iii) of Theorem 1.1.
The next distortion and covering theorems for the class holds:
Theorem 2.4
If then where (14) (14) (15) (15) and .
Proof.
Let , according to Theorem 2.1 we need to distinguish the next two cases:
(i) If , then there exists Schwarz functions, such that , and by Lemma 1.2 we get (16) (16) Since , we have
Case 1. For Y >0. By using the fact that and , we have and from (Equation16(16) (16) ) we obtain (17) (17)
Case 2. If Y <0, from the fact that and , we have and from (Equation16(16) (16) ) we obtain (18) (18) Now, by combining (Equation17(17) (17) ) and (Equation18(18) (18) ), we get (19) (19)
(ii) If Y = 0, there exists Schwarz functions, such that , and therefore (20) (20) for . Since By a similar way as in the previous case, we get Thus, (Equation20(20) (20) ) yield to for . That is complete the proof of our theorem.
Theorem 2.5
If then where .
Proof.
Integrated the function along the close segment connecting the origin with an arbitrary , since any point of this segment is of the form , with , where and , we get hence Using this inequality and the right-hand side inequalities of Theorem 2.4, we need to discuss the next two cases:
If , then that is
If Y = 0, then that is
Disclosure statement
No potential conflict of interest was reported by the authors.
ORCID
Fuad Alsarari http://orcid.org/0000-0002-4759-5449
Satyanarayana Latha http://orcid.org/0000-0002-1513-8163
Teodor Bulboacă http://orcid.org/0000-0001-8026-218X
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