On Janowski functions associated with (n,m)-symmetrical functions

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.


Introduction
(1) For f and g are two analytic functions in U, we say that the function f is subordinate to the function g in U, if there exists a function w ∈ , such that f (z) = g(w(z)) for all z ∈ U, and we denote this by f (z) ≺ g(z). Furthermore, if g is univalent in U, then the subordination is equivalent to f (0) = g(0) and f (U) ⊂ g (U). (see, for details, [1]) Using the notation of the subordination, let define the class P of functions with positive real parts (or Carathéodory functions) in U: From the above-mentioned reasons, it follows that any function p ∈ P has the representation p(z) = (1 + w(z))/(1 − w(z)), for some w ∈ .
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 S * , C, S k 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 ∈ P.

Definition 1.4:
Denoted be S (n,m) for the set of all (n, m)-symmetrical functions. Let us observe that the sets S (0,2) , S (1,2) , and S (1,m) are well-known families of odd functions, of even functions and of m−symmetrical functions, respectively.
The following decomposition theorem holds: Al Sarari and Latha [6] introduced the classes S (n,m) [X, Y] and K (n,m) [X, Y] which are the classes of Janowski type functions with respect to (n, m)symmetric points.
By using the theory of (n, m)-symmetrical functions and the generalized Janowski type functions, we will define the following class: where the function f n,m (z) is defined by (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 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 [6].
is the class was introduced by Ohang and Youngjae in [8].
The following lemmas are important to proof our results: for some w ∈ , where was defined by (1).
For some w, w ∈ .
Proof: Supposing that f ∈ S (n,m) [X, Y, γ ], then there exists a function w ∈ , such that Combining the above relation with Theorem 2.1, we have and integrating the above relations we obtain our result. Thus, and according to (3) hence, there exists a Schwarz function w ∈ such that Thus, and using the fact that |w(z)| < 1 for all z ∈ U, we may easily prove that From the above two inequalities, it follows and consequently, from (6) we obtain (ii) For Y = 0, we need to determine the value r * ∈ (0, 1), such that whenever |z| < r * , which is equivalent to For |z| < r * . According to the Remark 1.1 and the definition of the subordination we have that Next, we will prove that implies Since f ∈ S (n,m) [X, Y, γ ], from Definition 1.5 and Theorem 2.1, it follows that there exist the functions w, w ∈ , such that Using the above relations, the assumption (10) is equivalent to that is If we denote the above inequality shows that Reζ > 0, therefore whenever 0 < μ < 1. Dividing this inequality by μ > 0, we obtain which represents (11). From the above reasons, we will determine now the biggest value of r * such that (10) holds for |z| < r * . Since f ∈ S (n,m) [X, Y, γ ], there exists a function w ∈ , such that and from (16) we obtain and from (16) we obtain Now, by combining (17) and (18), we get (ii) If Y = 0, there exists Schwarz functions, w ∈ such that f n,m (z) = z exp[(1 − γ )Xw(z)], and therefore for |z| ≤ r < 1. Since By a similar way as in the previous case, we get Thus, (20) yield to for |z| ≤ r < 1. That is complete the proof of our theorem.

Proof:
Integrated the function f along the close segment connecting the origin with an arbitrary z ∈ U, since any point of this segment is of the form ζ = ρe iθ , with ρ ∈ [0, r], where θ = arg z and r = |z|, we get Using this inequality and the right-hand side inequalities of Theorem 2.4, we need to discuss the next two cases:

Disclosure statement
No potential conflict of interest was reported by the authors.