On Classes of Non-Carathéodory Functions Associated with a Family of Functions Starlike in the Direction of the Real Axis

: In this paper, we introduce a new class of analytic functions subordinated by functions which is not Carathéodory. We have obtained some interesting subordination properties, inclusion and integral representation of the deﬁned function class. Several corollaries are presented to highlight the applications of our main results.


Introduction
In a study related to analytic functions starlike in one direction, Robertson in [1] defined the following integral and established that g(z) is univalent in |z| < 1 if α, β are in [0, π] and Re p(z) ≥ 0. Here, in this paper, we study the geometrical implications of the integrand defined in (1) and its applications to certain class of analytic functions defined in the unit disc.Let A denote the class of functions analytic in the unit disc U = {z : |z| < 1} and having an expansion of the form In addition, let N P denote the class of functions that are analytic in the unit disc and equals 1 at z = 0. We call P the class of functions p ∈ N P which satisfies Re(p(z)) > 0, z ∈ U .
Very well-known subclasses of A are the so-called family of starlike and convex functions, which we denote here by S * and C, respectively.Using the principal of subordination [2], Ma and Minda [3] defined the classes S * (ψ) and C(ψ) as follows.
where ψ(z) ∈ P maps U onto a starlike region with respect to 1 with ψ (0) > 0 and symmetric with respect to the real axis.The classes S * (ψ) and C(ψ) consolidated the study of several generalizations of starlike and convex functions.Setting ψ to be a conic region, several authors studied the classes of analytic functions associated with the conic regions.Most popular among those studies are S * ( √ 1 + z) defined by Sokół [4] and followed by S * (z + √ 1 + z 2 ) defined by Raina and Sokół [5].For studies related to the conic region, refer to [6][7][8][9] and references provided therein.

Convex and Starlike in One Direction
A domain D is convex in the direction of the line L if each line parallel to L either misses D, or is contained entirely in D, or intersection with D is either a segment or a ray.Note that such a domain need not be convex or starlike with respect to any point.A function f ∈ A is said to be convex in the direction of the line L if it maps the unit disc onto a domain which is convex in the direction of the line L. Here, we denote such a set of functions as CV (r), if L is the real axis.Similarly, ST (r) denotes the class of functions starlike in the direction of the real axis, refer to [1] for its formal definition.Now, we define the function with α, β ∈ [0, π] and p(z) ∈ P. The function Λ[α, β; p(z)] is related to the class of functions starlike with respect to the real axis (see page 210 in [10]).To be precise, the function f (z) ∈ A is said to be in ST (r) if and only if there is a α, β ∈ [0, π] and p(z ≯ 0 (see Figure 1).Hence, we observe that in general func- tion Λ[α, β; p(z)] does not belong to class P, but belongs to N P. Further, to illustrate the fact that impact of Λ[α, β; p(z)] is not same on all conic region.We let p(z) = z + √ 1 + z 2 in (3), then the function Λ α, β; z + is convex in U does not belong to P. However, the function Λ α, β; z + √ 1 + z 2 will be convex and in P if |z| < 0.7 (see Figure 2).From Figures 1 and 2, we can see that Λ[α, β; p(z)] ∈ N P and maps the unit disc onto a domain which is symmetric with respect to the real axis irrespective of the choice of p(z).

Inclusion Relations and Integral Representations
Now, we state some results which we use to establish our main results.

Lemma 1 ([25]
).Let g be convex in U , with g(0) = a, γ = 0 and Re(γ) > 0. Suppose that ϑ(z) is analytic U , which is given by If The function q is convex and is the best (a, n)-dominant.
In order to further broaden our study, we drop the necessity of p(z) in (3) to satisfy the condition Re p(z) > 0. So, hereafter, throughout this paper, we denote where ψ(z) ∈ N P is defined as in (5).
and q(z) is the best dominant.
Proof.Let h(z) be defined by Then the function h(z) is of the form h(z) = 1 + c 1 z + c 2 z 2 + • • • and is analytic in U .Differentiating both sides of ( 9) and by simplifying, we have By hypothesis f ∈ MS ν (α, β; γ; ψ(z)), so from Definition 1, we have Applying Lemma 1 to (10) with δ = ν γ and n = 1, we get Hence, the proof of the Theorem 1 Remark 1. From (10), it can be easily seen that if γ = 0, we can get and q(z) is the best dominant.
On replacing the superordinate function in Theorem 1, we get the desired result.
and q(z) is the best dominant.
If we let γ = 1 = ν in Corollary 1, we get and q(z) is the best dominant.
If we let γ = 1 = ν in Corollary 2, we get and q(z) is the best dominant.
As a consequence of Theorem 1, we have the following integral representation of the class MS ν (α, β; γ; ψ(z)).

Initial Coefficients' Bounds
The Fekete-Szegö problem possesses various geometric quantities which are helpful in establishing univalence and norm estimates.Most of all recent papers establish the Fekete-Szegö inequalities for the defined function classes.
We need the following well-known coefficient estimates for functions belonging to the class P.

Lemma 2 ([3]
).Let p ∈ P and also let v be a complex number, then The result is sharp for functions given by for z ∈ U .Also, let α, β ∈ [0, π] and p(z) ∈ N P satisfy the condition for all z ∈ U Then, the bounds of the initial coefficients of f are given by and Further, the Fekete-Szegö inequality for µ ∈ C is given by Proof.The function Λ[α, β; ψ(z)] defined in (7) belongs to N P. The hypothesis ( 13) is equivalent to Im < 1, which implies the function Λ[α, β; ψ(z)] ∈ P (see Theorem 2 in [27]).Now, f ∈ MS ν (α, β; γ; ψ(z)) (z ∈ U ) implies that there is a Schwarz function w(z) such that (16)   Define the function h(z) by We can note that h(0) = 1 and h ∈ P (see Lemma 3).Using (17), it is easy to see that On applying the above expression in (16), after a long and tedious computation, we get The left-hand side of ( 16) is equivalent to From ( 18) and ( 19), we have and Hence, applying Lemma 3 in (20), we get (14).To obtain (15), we apply Lemma 2 in (21).
In view of the Equations ( 20) and ( 21), for µ ∈ C, we have On simplifying (22), we get (16).Hence, the proof of Theorem 3 is completed.