Hankel inequalities for bounded turning functions in the domain of cosine Hyperbolic function

: In the present article, we define and investigate a new subfamily of holomorphic functions connected with the cosine hyperbolic function with bounded turning. Further some interesting results like sharp coe ffi cients bounds, sharp Fekete-Szeg¨o estimate, sharp 2 nd Hankel determinant and non-sharp 3 rd order Hankel determinant. Moreover, the same estimates have been investigated for 2-fold, 3-fold symmetric functions, the first four initial sharp bounds of logarithmic coe ffi cient and sharp second Hankel determinant of logarithmic coe ffi cients fort his defined function family.


Introduction
The class of all analytic functions u (ε) defined in the open unit disk U = {ε : ε ∈ C and |ε| < 1}, is denoted by A and normalized also by the conditions u (0) = 0 and u ′ (0) = 1.
Thus, the Taylor series expansion of each u (ε) ∈ A is as follows: (1.1) Furthermore, let S denotes a subfamily of A, which are univalent in U. For two functions h 1 , h 2 ∈ A, we say that the function h 1 is subordinate to the function h 2 ( written as h 1 ≺ h 2 ) if there exists an holomorphic function w with the property |w (ε)| ≤ |ε| and w (0) = 0 such that h 1 (ε) = h 2 (w (ε)) for ε ∈ U. Moreover, if h 2 ∈ S, then the above conditions can be written as: The family S * (Ψ), given by is introduced by Ma and Minda [1] in 1992, where Ψ is an univalent function in U having the properties Ψ(0) = 1 and ℜ (Ψ) > 0.
Many useful and intrusting properties of these classes have been obtained by them. If specifically, we take Ψ(ε) = (1+ε)/(1−ε), then we have the family S * (Ψ) of starlike functions. For different choice of Ψ involved in the right hand side of (1.2), one can get a number of known subclasses of starlike functions. Some of them are listed as follows: (1) If we choose Ψ(ε) = 1 + sinh −1 (ε) , then we get the family given by S * pet = S * 1 + sinh −1 (ε) .
The function Ψ(ε) maps open unit disc onto the image domain which is bounded by petal shape and was established by Kumar et al. [2]. (2) If we take then we obtain the follow family this family starlike functions based on modified sigmoid functions was established and investigated by Geol et al. [3].
then we obtain the follow family (see [6]) then the functions family lead to the family which is described as the functions of starlike functions, bounded by lemniscate of Bernoulli (see [7]). (7) Moreover, if we take then we obtain the follow family which was studied by Sharma et al. [8]. (8) Furthermore if we pick Ψ(ε) = e ε we get the family S * exp = S * (e ε ) , which was introduced and studied by Mendiratta et al. [9]. On the other side, if we take Ψ(ε) =ε+ √ 1 + ε 2 , we get the family S * l = S * ε + √ 1 + ε 2 , which maps U to crescent shaped region and was introduced by Raina and Sokól [10].
The Hankel determinant H q,n (u) for function u ∈ S of the form (1.1),was given firstly by Pommerenke [14,15]  For particular values, for example q = 2 and n = 1, we get the first order Hankel determinant is where d 1 = 1. And for q = 2 and n = 2, in (1.3) we get the second order Hankel determinant 3 . For the third order Hankel determinant we take q = 3 and n = 1, and get the following Note that H 2,1 (u) = d 3 − d 2 2 , is the particular case of Fekete-Szegö approximations. The sharp upper bounds for H 2,1 (u) for different subfamilies of holomorphic functions was investigated by different authors (see [16][17][18] for details). Moreover, the second Hankel determinant and the sharp upper bound of this has been studied and investigated by several authors from many different directions and perspectives. For few of them are, Hayman [19], the Ohran et al. [20], Noonan and Thomas [21] and Shi et al. [22]. Furthermore, the bounds for the third Hankel determinant for subfamilies of holomorphic functions was first investigated by Babalola [23]. Some recent and interested works on this topic may be found in [24][25][26] and the reference therein. Recently, Mundalia et al. [27] defined the family of holomorphic starlike functions based on the trigonometric cosine hyperbolic function as follows: For more about this study, we may refer the readers to see [28][29][30]. By taking motivation from above cited work we introduce the following family of holomorphic function: In this paper we evaluate first three initial sharp coefficient bounds, sharp Fekete-Szegö functional, sharp second Hankel determinant non-sharp third Hankel determinant, third Hankel for 2, 3-fold symmetric function and Krushkal inequality for functions belonging to this family. Further, sharp initial four logarithmic coefficients bounds and second Hankel determinant are investigated.

A set of Lemmas
We next denote by P the family of holomorphic functions p which are normalized by p(0) = 1, with Re(p(ε)) > 0, ε ∈ U and have the following form: (2.1) Lemma 2.1. If p ∈ P and has the form (2.1). Then, for x and δ with |x| ≤ 1, |δ| ≤ 1, such that We note that (2.2) and (2.3) are taken from [31].
Lemma 2.2. If p ∈ P and has the form (2.1), then we get following estimates and for complex number η, we have For the inequalities (2.4)-(2.6) see [16] and (2.7) is given in [32].
If p ∈ P and has the form (2.1), then where Λ 1 , Λ 2 and Λ 3 are real numbers.

Main results
Theorem 3.1. If u (ε) ∈ R Cosh and it has the form given in (1.1), then Equalities in these inequalities are obtained for functions defined as follow: respectively.
Proof. From (3.13) and (3.14) , we get Applying (2.7) , to above we get the required results. □ Corollary 3.3. If u (ε) ∈ R Cosh and it has the form given in (1.1), then Equalities of this inequalities is obtained for function u 2 defined in (3.6) .
Theorem 3.4. If u (ε) ∈ R Cosh and it has the form given in (1.1), then Equalities of this inequalities is obtained for function u 3 defined in (3.7) .
Proof. From (3.13)-(3.15) , we get Applications of Lemma 2.3, lead us to required results. □ Theorem 3.5. If u (ε) ∈ R Cosh and it has the form given in (1.1), then . (3.20) Equalities of this inequalities is obtained for function u 2 defined in (3.6) .
Proof. From c.
Obviously Υ ′ (c, 1) ≤ 0, is decreasing function, so maximum value attained at c = 2, that is □ Theorem 3.6. If u (ε) ∈ R Cosh and it has the form given in (1.1), then Putting values of (3.  The family of all m-fold symmetric functions belong to well-known family S, and denoted by S m having the following Taylor series form: The holomorphic functions of the form (4.1) is in the family R m Cosh , if and only if Where p (ε) belong to the family P (m) is defined by: Proof. Let u (ε) ∈ R 2 Cosh . Then, there exists a function p ∈ P (2) , using the series form (4.1) and (4.3), when m = 2 in the above relation (4.2 ), we obtain u ′ (ε) = 1 + 3d 3 ε 2 + 5d 5 ε 4 + · · · . (4.5) Comparing (4.5) and (4.6) we obtained Applying the trigonometric inequality to (2.4) and (2.5), we get Hence, the proof is complete. □ Theorem 4.2. If u (ε) ∈ R Cosh and it has the form given in (4.1), then Equalities of this inequalities is obtained for function defined as: Proof. As u ∈ R 3 Cosh , therefore there exists a function p ∈ P (3) , such that For m = 3 and form (4.1) and (4.3), the above condition become as: 1 + 4d 4 ε 3 + · · · = 1 + c 3 4 ε 3 + · · · . (4.8) Comparing the coefficients of (4.8), we obtained Utilizing (2.4) and triangle inequality, we have Thus the complete the proof. □

Logarithmic coefficients for the family R Cosh
The logarithmic coefficients of u ∈ S denoted by γ n = γ n (u) , are defined by with the following series expansion: For function u given by (1.1) , the logarithmic coefficients are as follow: Theorem 5.1. If u (ε) ∈ R Cosh and it has the form given in (1.1), then Equalities in these inequalities are obtained for function u n (ε) = ε 0 cosh √ t n dt = ε + 1 2n + 2 ε n+1 + · · · , for n = 1, 2, 3, 4. (5.5) Proof. Now from (5.1) to (5.4) and (3.13) to (3.16) , we get Also, using Lemma 2.4 to (5.9) , we get Proof for sharpness: Since it follows that these inequalities are obtained for the functions u n (ε) for n = 1, 2, 3, 4 defined in (5.5). □ Theorem 5.2. If u (ε) ∈ R Cosh and it has the form given in (1.1), then Equalities in this inequalities are obtained for function u 2 in (5.5) .

Conclusions
Recently the investigations of the Hankel determinant got attractions of many researchers, due to its applications in many diverse areas of mathematics and other sciences. Here in this paper, we have defined a new subfamily of holomorphic functions connected with the Tan hyperbolic function with bounded boundary rotation. We have then investigated the upper bound of the third Hankel determinant for this newly defined functions family. On the other hand, we have obtained the same bounds for 2fold, 3-fold symmetric functions,The first four initial sharp bounds of logarithmic coefficient and sharp second Hankel determinant of logarithmic coefficients for this defined function family.
Here, we passing to remark the fact that one can extend the suggested results investigated in this article, for some other subfamilies of holomorphic functions and also the interested can use the D q derivative operator (see for example [35][36][37][38]) and can generalize the work presented here.