Some results on the Bazilevic functions B1(α) related to the Lemniscate Bernouli (LB)

This research is concerned with Bazilevic B1(α) on the unit disc D = {z| < 1}, related to the Lemniscate Bernoulli (LB), defined by kind of subordination for some positive alpha. There will be determined the Hankel determinant, especially the third Hankel determinant which B1(α) subordinates to LB, and we start with the case for starlike and convex functions, subset of B1(α). In this article we impeove the result of Kumar and Ravichandran.

Next, the coefficient bounds yield information regarding the geometric properties of some subclass of univalent functions. In 1916, Bieberbach [1] computed an estimate for the second coefficient of normalized univalent analytic function (1) and this bound provides the growth, distortion, and covering theorems.
The qth Hankel determinant of a function f given by (1) is defined for q > 1 and n > 0 as follows, In recent years a great deal of attention has been devoted to finding estimates of Hankel determinants whose elements are the coefficients of univalent (and p-valent) functions. For ∈ , growth results have been established for the general Hankel determinant ( ). The second Hankel determinant 2 (2) = | 2 4 − 3 2 | has received more attention of Janteng [3] and Marjono and Thomas [5], with significant results being obtained for ∈ .
In this paper we will see this for subclass Bazilevic functions 1 ( ) and try to learn about third Hankel determinant which is defined as follows : We will also want to see about starlike and convex functions related to the third Hankle determinant as a subclass of Bazilevic functions 1 ( ).

Known Result
Denote by P, the class of functions p satisfying Re (p(z)) > 0 for ∈ , with Taylor series 1− 2 and also we can define Lemniscate Bernoulli (1 + ) 1/2 .
We will use the following Lemmas : Lemma 2.1. If ∈ with coefficient as above then for some complex valued x with | | ≤ 1 and some complex valued with | | ≤ 1, The following theorem of Kumar and Ravichandran [4] gives an improvement to the existing estimate on the third Hankel determinant related to the starlike and convex functions with respect to the symmetric points. ( 2 2 + 2 4 ).
Using the above we can write Further, by suitable arranging the terms, we have By using Lemma 2.3, we see that Thus using (3) and (4), we have ( 2 2 + 2 4 ).
Using the above, we have

Results
Relate to the above result, we can remind two important classes ( ) and ( ) respectively subclasses of S, which is defined as the following.  The proof is similar to the above proof, first start by equating the class which is functions with positive real part.
Next continued by comparing its coefficients on both sides related to class M first. Then we compute the value of 3,1 ( ) and arrange this into better representation. Finally we simplify and obtained the result for subclass M, i.e. (a) Let ∈ . Then, we can associate a function ( ) = 1 + 1 + 2 2 + 3 3 + ⋯ ∈ such that Next continued by comparing its coefficients on both sides related to class N first. Then we compute the value of 3,1 ( ) and arrange this into better representation. Finally we simplify and obtained the result for subclass N, i.e. We have the following theorem for analytic functions especially convex and starlike functions. Proof. In this case we will substitute ( ) of Kumar and Ravichandran [4] i.e.

Conclusion
In this paper, we are working on the univalent functions based on the result of Marjono and Thomas [5]. We give the boundary for the third Hankle determinant for starlike and convex functions as subset of Bazilevic B1( ) as an improvement of te work of Kumar and Ravichandran. To prove this we use lemma for the functions with positive real part p(z).