Starlikness Associated with Cosine Hyperbolic Function

: The main contribution of this article is to deﬁne a family of starlike functions associated with a cosine hyperbolic function. We investigate convolution conditions, integral preserving properties, and coefﬁcient sufﬁciency criteria for this family. We also study the differential subordinations problems which relate the Janowski and cosine hyperbolic functions. Furthermore, we use these results to obtain sufﬁcient conditions for starlike functions connected with cosine hyperbolic function.


Introduction and Definitions
The aims of this particular section is to include some basic notions about the Geometric Function Theory that will help to understand our key findings in a clear way. In this regards, first we start to define the most basic family A which consists of holomorphic (or analytic) functions in D = {z ∈ C : |z| < 1} by: A = q : q is holomorphic in D with q(z) = z + ∞ ∑ k=2 a k z k .
Also the set S ⊂ A describes the family of all univalent functions which is define here by the following set builder form: S = {q ∈ A : q is univalent in D} .
Next we consider defining the idea of subordinations between holomorphic functions q 1 and q 2 , indicated by q 1 ≺ q 2 , as; the functions q 1 , q 2 ∈ A are connected by the relation of subordination, if there exists a holomorphic function v with the restrictions v(0) = 0 and |v (z)| < |z| such that q 1 (z) = q 2 (v(z)). Moreover, if the function q 2 ∈ S in D, then we obtain: Image domains are of primary significance in the analysis of analytical functions. Analytic functions are classified into various families based on geometry of image domains. In 1992, Ma and Minda [1] considered a holomorphic function ∆ normalized by the conditions ∆(0) = 1 and ∆ (0) > 0 with Re∆ > 0 in D. The function ∆ transforms the D disc into a region that is star-shaped about 1 and is symmetric on the real axis. In particular, if we take ∆(z) = 1+Lz 1+Mz with −1 ≤ M < L ≤ 1, then it maps D to a disc which lies in the right-half plan with center on the real axis while 1−L 1−M and 1+L 1+M are its different end points of the diameter. This familiar function is recognized as Janowski function [2]. Some interesting problems such as convolution properties, coefficients inequalities, sufficient conditions, subordinates results, and integral preserving were discussed recently in [3][4][5][6][7] for some of the generalized families associated with circular domain. The image of the function ∆(z) = √ 1 + z shows that the image domain is bounded by right-half plan of the Bernoullis lemniscate given by v 2 − 1 < 1, see [8]. The function ∆(z) = 1 + 4 3 z + 2 3 z 2 maps D into the image set bounded by the cardioid which was examined in [9] and further studied in [10]. The function ∆(z) = 1 + sin z was established by Cho and his coauthors in [11] while ∆(z) = e z is recently studied in [12] and [13]. Furthermore, many subfamilies of starlike functions have also been introduced recently in [14-18] by choosing some particular functions such as functions associated with Bell numbers, functions related with shell-like curve connected with Fibonacci numbers, functions connected with conic domains and rational functions instead of the function ∆. Differential subordinations are natural generalizations in complex plane of differential inequalities on real line. Information obtained from derivative plays important role in studying properties of real valued functions. In complex plane, there are various differential implications, in which a function is characterized by using differential conditions. Noshiro-Warschawski theorem is an example of such differential implication which gives the univalency criterion for analytic functions. In numerous cases, properties of function are determined from the range of the combination of the derivatives of the function. For more details about differential subordinations, see [19].
Let h be a holomorphic function defined on D with h(0) = 1. Recently, Ali et al. have obtained sufficient conditions on λ such that Similar type implications have been investigated in some of the recent papers by different researchers, for example see the articles contributed by Haq et al. [20], Kumar et al. [21,22], Paprocki and Sokół [23], Raza et al. [24], Sharma et al. [25] and Tuneski [26]. Now we establish the family S * cosh of starlike functions connected with cosine hyperbolic function that are defined by: Geometrically, the function . It is interesting to see that the cosine and cosine hyperbolic functions have the same image domain in D. For detail see [14]. Also, since cosh (z) maps the region D onto the image which is bounded by Thus, the class S * cosh can also be defined in a different way as; a function q ∈ A belongs to the class S * cosh if and only if the following inequality will be true We need to get the foregoing Lemma to establish our principal results. Lemma 1. [27] Let v be a holomorphic function in D with v (0) = 0. If |v (z 0 )| = max {|v (z)| for |z| ≤ |z 0 |} , then a number l (l ≥ 1) occurs in such a way that z 0 v (z 0 ) = lv (z 0 ) .
To avoid repetitions, we assume the following restrictions otherwise we will state it where different.

Sufficient Conditions Associated with Cosh
Theorem 1. Let an analytic function h (with h(0) = 1) satisfying the relation of subordination with the following limitation Then Proof. Let us assume that Then the function p is holomorphic in D with p(0) = 1. Also consider where we selected the principle branches of the functions that are logarithmic and square root. Then v is clearly a holomorphic function in D with v (0) = 0. Also since To complete the proof of this result, we just need to prove |v (z)| < 1 in D. By virtue of (7) , we have Therefore Also, by Lemma 1, a number l ≥ 1 exists with z 0 v (z 0 ) = lv (z 0 ). In addition, we also suppose that v (z 0 ) = e iθ for θ ∈ [−π, π] . Then we have If |z| = r, −π ≤ θ ≤ π, then simple calculation illustrates that A routine simplification ensures that 0, ±π, ± π 2 are the roots of φ (θ) = 0 and µ (θ) = 0 in [−π, π] . Also, since it is enough to conclude that θ ∈ [0, π] and thus we achieve Thus, we have Therefore, using (9) , (10) , (8) , we attain Then This confirms that the function ς is increasing and therefore ς (l) ≥ ς (1) , so and this contradicts the hypothesis Hence the proof is completed. (3) , we achieve the below Corollary.

Corollary 1. Let q ∈ A and justifying
Then q ∈ S * cosh .
If we choose L = 1, M = 0 in (11) , we get the following result.

Corollary 2.
If q ∈ A and obeying the subordination Then q ∈ S * cosh .

Theorem 2.
Let an analytic function h (h(0) = 1) satisfying the relation of subordination with the following restriction Then Proof. Let us suppose Then the function p is holomorphic in D with p(0) = 1. Inserting (7) , we have By virtue of Lemma 1 along with (9) and (10) , we have Then A contradiction to the hypothesis occurs and hence the proof is completed.
If we take h (z) = zq (z) q(z) in (12), we obtain the below result.

Corollary 3.
If q ∈ A and obeying the subordination then the function q ∈ S * cosh .
If we choose L = 1, M = 0 in (14) , we get the following result.

Corollary 4.
If q ∈ A and obeying the subordination then q ∈ S * cosh .
If h is a holomorphc function defined on D with h (0) = 1 and satisfying then h (z) ≺ cosh (z) .

Proof. Let us choose a function
Then the function p is holomorphic in D with p(0) = 1. Applying some simple computation, we get and so By using Lemma 1, we have Then, which shows that ς is an increasing function and it has its minimum value at l = 1, so Now by using (15) , we have which yields a contradiction to our assumption. This completes the proof.
If we put h (z) = zq (z) q(z) in (16) , we obtain the following result.

Corollary 5. If q ∈ A and obeying the subordination
then q ∈ S * cosh .
If we choose L = 1, M = 0 in (17) , we get the following result.
Corollary 6. If q ∈ A and obeying the subordination Then q ∈ S * cosh .

Bernardi Integral Operator and Its Relationships
The role of operators in the field of functions theory is very crucial in exploring the nature of the geometry of analytic functions. Several differential and integral operators were introduced by using convolution of certain analytic functions. It is found that this formalism gives ease in more mathematical study and also allows explaining the geometrical properties of analytical and univalent functions. Alexander was the first, who started studying the operator back in 1916. Later Libera [28] and Bernardi [29] added several integral operators to study the classes of starlike, convex, and close-to-convex functions. Also, the mapping properties of these operators was discussed in [30]. The Bernardi [29] integral operator is defined by; In this part of the article, we analyze the mapping properties of functions belonging to the class S * cosh under the integral operator described in (18) above. Some similar findings of this type are also discussed here. If where the operator J is given by (18) .

Proof. Let a function v be defined by
where we have chosed the principle branches of the square root and logarithmic functions. Since cosh −1 z function is defined by therefore v is an analytic function in D with v (0) = 0. To prove our result, we need only to show that |v (z)| < 1 in D. From (21) , we have Logarithmic differentiation of above relation yields Differentiating logarithmically, we have Now, we define a function where p is analytic in D with p (0) = 1. Also Suppose that there exists a point z 0 ∈ D such that max |z|≤|z 0 | |v (z)| = |v (z 0 )| = 1.
where J is the Bernardi integral operator defined in (18) .

Proof. Let a function v be defined by
where we have chosed the principle branches of the square root and logarithmic functions. Then v is analytic in D with v (0) = 0. We need only to show that |v (z)| < 1 in D. From (23) , we have Also we define a function where p is analytic in D with p (0) = 1. Now by using (18), (24) and (25), we have Suppose that there exists a point z 0 ∈ D such that max |z|≤|z 0 | |v (z)| = |v (z 0 )| = 1.
By using Lemma 5, there exists a number l ≥ 1 such that z 0 v (z 0 ) = lv (z 0 ). We also suppose that v (z 0 ) = e iθ . Then we have Then which shows that Θ is an increasing function and it has its minimum value at l = 1, so A contraduction to the hypothesis Hence we have the required result. If where J is the Bernardi integral operator defined in (18) .
Proof. Using the same steps as used in the last result, one can easily complete this proof.

Convolution Conditions and Its Consequences
The technique of convolution (or Hadamard product) is extremely important in the solution of various function theory problems and due to this facts this concept becomes the major part of this field. The main goal of this portion is to analyze the properties of convolution and its implications Proof. In the last theorem, we have proved that q ∈ S * cosh if and only if the relation (29) held. We can rewrite (29) as ((δ − 1) n − δ) a n z n−1 n − cosh e iθ cosh e iθ − 1 a n z n−1 , and this completes the proof.

Conclusions
In the present research article, we examined some interesting properties of starlike functions associated with the cosine hyperbolic function which is symmetric about the real axis. These results included convolutions properties, Bernardi integral preserving problems and coefficient sufficiency criteria. In addition to that we also calculated some conditions on λ so that; if for each j ∈ N, k ∈ N 0 = N ∪ {0} 1 + λ (zh (z)) j h k (z) Furthermore, these results are used to find sufficiency criterion for the function belongs to the newly defined family S * cosh . Moreover, some other problems like coefficient bounds, Hankel determinant, partial sum inequalities, and many more can be discussed for this class as a future work.