On Certain Subclasses of Analytic Functions Involving Carlson-Shaffer Operator and Related to Lemniscate of Bernoulli

such that Re{φ(z)} > 0 in U. For functions f and g, analytic in U, we say that f is subordinate to g, written as f ≺ g or f(z) ≺ g(z) (z ∈ U), if there exists a Schwarz function ω, which (by definition) is analytic in U with ω(0) = 0, |ω(z)| < 1, and f(z) = g(ω(z)), z ∈ U. Furthermore, if the function g is univalent in U, then we have the following equivalence relation (cf., e.g., [1]; see also [2]):


Introduction and Preliminaries
Let A be the class of functions  of the form Similarly, a function  ∈ A is said to be convex of order , if Re {1 +   ()   () } >  (0 ≤  < 1;  ∈ U) .
By usual notations, we write these classes of functions by S ⋆ () and K(), respectively.We denote S ⋆ (0) = S ⋆ and K(0) = K, the familiar subclasses of starlike, convex functions in U.
Furthermore, let P denote the class of analytic functions  normalized by such that Re{()} > 0 in U.
It follows from (12) and the definition of subordination that a function  ∈ R(, ) satisfies the following subordination relation: We further note that if  ∈ R(, ), then the function L(, )()/ lies in the region bounded by the right half of the lemniscate of Bernoulli given by Noonan and Thomas [6] defined the th Hankel determinant of a sequence   ,  +1 ,  +2 , . . . of real or complex numbers by This determinant has been studied by several authors including Noor [7] with the subject of inquiry ranging from the rate of growth of   () (as  → ∞) to the determination of precise bounds with specific values of  and  for certain subclasses of analytic functions in the unit disc U.
For  = 1,  = 2,  1 = 1, and  =  = 2, the Hankel determinant simplifies to The Hankel determinant  2 (1) was considered by Fekete and Szegö [8] and we refer to  2 (2) as the second Hankel determinant.It is known [9] that if  given by ( 1) is analytic and univalent in U, then the sharp inequality  2 (1) = | 3 −  2  2 | ≤ 1 holds.For a family F of functions in A of the form (1), the more general problem of finding the sharp upper bounds for the functionals ) is popularly known as Fekete-Szegö problem for the class F. The Fekete-Szegö problem for various known subclasses of univalent functions (i.e., starlike, convex, close-to-convex, etc.) has been completely settled [8,[10][11][12].Recently, Janteng et al. [13,14] have obtained the sharp upper bounds to the second Hankel determinant  2 (2) for the family of functions in A whose derivatives have positive real part in U.For initial work on this class of functions, one may refer to the work of MacGregor [15].
In our present investigation, we follow the techniques adopted by Libera and Złotkiewicz [16,17] to solve the Fekete-Szegö problem and also determine the sharp upper bound to the second Hankel determinant for the class R(, ).
To establish our main results, we will need the following lemma for functions belonging to the class P. Lemma 2. Let the function , given by ( 4), be a member of the class P. Then           ≤ 2 ( ≥ 1) , (17 for some complex numbers ,  satisfying || ≤ 1 and || ≤ 1. The estimates in (17) and ( 18) are sharp for the functions defined in U by We note that the estimate ( 17) is contained in [9]; the estimate ( 18) is due to Ma and Minda [18], whereas the results in (19) are obtained by Libera and Złotkiewicz [17] (see also [16]).

Main Results
Unless otherwise mentioned, we assume throughout the sequel that  ≥  > 0. Now, we solve the Fekete-Szegö problem for the class R(, ).( The estimate is sharp. Proof.From (13), it follows that where  is analytic and satisfies the conditions (0) = 0 and we see that  ∈ P. From the above expression, we get so that, by (22), we get Now, it is easily seen that ( 2() 1 + () ) and the estimate for | 3 | is sharp for the function  defined by For the case  ∈ R, Theorem 3 reduces to the following result.
Corollary 5. Let  ∈ R. If the function , given by ( 1), belongs to the class R(, ), then ( The estimate is sharp for the function  defined in U by ( Putting  = 2 and  = 1 in Corollary 5, we get the following.

Corollary 6. If the function 𝑓, given by (1), satisfies the subordination relation
then (37 The estimate is sharp for the function  defined in U by In the following theorem, we find the sharp upper bound to the second Hankel determinant for the class R(, ).Theorem 7. If  ≥  ≥ 1/2 and the function , given by ( 1), belongs to the class R(, ), then The estimate in ( 39) is sharp for the function , given by (33).
Proof.From ( 27), (28), and (29), we deduce that       2  4 − where Setting we note that either  = 0 or Since  < 2 + 7 + 4, we further observe that F  (0) < 0. Thus, the maximum value of F is attained at  = 0 so that the upper bound in (42) corresponds to  = 0 and  = 1 from which we get the assertion of the theorem.
Letting  = 2 and  = 1 in Theorem 7, we get the following.

Corollary 8. If the function 𝑓, given by (1), satisfies the condition (36), then
and the estimate is sharp for the function  defined by Next, we find the sharp upper bound for the fourth coefficient of functions belonging to the class R(, ).Theorem 9.If the function , given by (1) ) which are analytic in the open unit disk U = { ∈ C : || < 1}.A function  ∈ A is said to be starlike of order , if Re {   ()  () } >  (0 ≤  < 1;  ∈ U) .