Coefficient Inequalities For A Class Of Analytic Functions Associated With The Lemniscate Of Bernoulli

abstract: In this paper, a new subclass of analytic functions ML λ associated with the right half of the lemniscate of Bernoulli is introduced. The sharp upper bound for the Fekete-Szegö functional |a3 − μa22| for both real and complex μ are considered. Further, the sharp upper bound to the second Hankel determinant |H2(1)| for the function f in the class ML∗λ using Toeplitz determinant is studied. Relevances of the main results are also briefly indicated.


Introduction and Motivation
Let A be the class of functions of the form which are analytic in U := {z ∈ C : |z| < 1}.
Let S be the subclass of A consisting of univalent functions in U. A function f ∈ A is said to be starlike of order α, (0 ≤ α < 1), denoted by S(α) if and only if It may be noted that for α = 0, the class S(α) = S * , the familiar subclass of starlike functions in U. Similarly, a function f ∈ A is said to be in the class R(α), α > 0, if it satisfies the inequality 3) The class R(1) = R was considered by Sahoo and Patel [28].
Trailokya Panigrahi and Janusz Sokó l Let f and g be two analytic functions in U. We say f is subordinate to g, written f (z) ≺ g(z) (z ∈ U), if and only if there exists an analytic function w in U such that w(0) = 0 and |w(z)| < 1 for |z| < 1 and f (z) = g(w(z)).In particular, if g is univalent in U, we have the following (see [19]): f (z) ≺ g(z) ⇐⇒ f (0) = g(0) and f (U) ⊂ g(U).
In 1966, Pommerenke [26] defined the q th Hankel determinant of f for q ≥ 1 and n ≥ 1 as A good amount of literature is available about the importance of Hankel determinant.It is useful in the study of power series with integral coefficients (see [5]), meromorphic functions (see [32]) and also singularities (see [7]).Noonan and Thomas [22] studied about the second Hankel determinant of a really mean pvalent functions.Noor [23] determined the rate of growth of H q (n) as n −→ ∞ for the functions in S with a bounded boundary.Ehrenborg [9] studied the Hankel determinant of exponential polynomials.

Sokó l and Stankiewicz
. We called such function as Sokó l-Stankiewicz starlike function.Recently, Raza and Malik [27] determined the upper bound of third Hankel determinant H 3 (1) for the class SL * .Further, Sahoo and Patel [28] obtained the upper Coefficient Inequalities For A Class Of Analytic Functions...

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bound to the second Hankel determinant for the class R = {f ∈ A : Motivated by the above mentioned works obtained by earlier researchers, we introduce the following subclass of analytic function as below: (1.4) Note that for λ = 0, the class M L * 0 reduces to the class SL * , studied by Raza and Malik [27] and while for λ = 1, the class M L * 1 reduces to R studied by Sahoo and Patel [28].In term of subordination, relation (1.4) can be written as.
where q(0) = 1.To state the geometrical significance of the class M L * λ , consider It follows from (1.6) that w 2 −1 = e iθ , which implies |w 2 −1| = 1.Taking w = u+iv and simplifying we get Therefore, q(U) is the region bounded by the right half of the lemniscate of Bernoulli given by ( In this paper, following the techniques devised by Libera and Z lotkiewicz [16,17], we solve the Fekete-Szegö problem and also determine the upper bounds of the Hankel determinant |H 2 (1)| for a subclass M L * λ .

Preliminaries
Let P be the class of analytic functions p normalized by ) Each of the following results will be required in our present investigation.
Lemma 2.1.[18] Let p ∈ P and of the form (2.1).Then When ν < 0 or ν > 1, the equality holds if and only if p(z) = 1+z 1−z or one of its rotations.If 0 < ν < 1, then the equality holds if and only if p(z) = 1+z 2 1−z 2 or one of its rotations.If ν = 0, the equality holds if and only if , or one of its rotations.If ν = 1, the equality holds if and only if p is the reciprocal of one of the functions such that the equality holds in the case of ν = 0.Although the above upper bound is sharp, when 0 < ν < 1, it can be improved as follows: and Lemma 2.2.[18] Let p ∈ P be of the form (2.1), then for any complex number ν, This result is sharp for the functions for some complex numbers x, z satisfying |x| ≤ 1 and |z| ≤ 1.

Main Results
The first two theorems give the results related to Fekete-Szegö functional, which is a special case of the Hankel determinant.Theorem 3.1.Let the function f given by (1.1) be in the class M L * λ .Then for real µ, we have (3.1) and for δ 1 + β < µ < δ 1 + 2β, where ) and These results are sharp.
Proof.Let f ∈ M L * λ .In view of Definition 1.1, there exists an analytic function w(z) satisfying the condition of Schwarz lemma such that Define a function Clearly p ∈ P. From (3.8), we get From (3.7) and (3.9), we have Now, by substituting the series expansion of p(z) from (3.8) in (3.10), it follows that (3.11) Using series expansions for f (z) and f ′ (z) from (1.1) give Making use of (3.11) and (3.12) in (3.10) and equating the coefficients of z, z 2 and z 3 in the resulting equation, we deduce that and For real µ, it follows from (3.13) and (3.14) that where In view of (3.16) and by an application of Lemma 2.1, we obtain the desired assertion.
The results are sharp for the functions ψ i (z), i = 1, 2, 3, 4 such that where Thus, the proof of Theorem 3.1 is completed.✷ Theorem 3.4.Let the function f given by (1.1) be in the class M L * λ .Then, for a complex number µ, we have (3.17) The estimate in (3.17) is sharp.
Proof.From (3.16), we have Therefore, by virtue of Lemma 2.2, we obtain the desired assertion.The result is sharp for the function

✷
The estimate in (3.19) is sharp.
Trailokya Panigrahi and Janusz Sokó l Proof.From (3.13), (3.14) and (3.15), we have For convenience of notation, we write p 1 = p (0 ≤ p ≤ 2).Putting the values of p 2 and p 3 from Lemma 2.3 in (3.20), we obtain for some x (|x| ≤ 1) and for some z (|z| ≤ 1).An application of triangle inequality in (3.21) and replacing |x| by y in the resulting equation with assumption that Coefficient Inequalities For A Class Of Analytic Functions...
The result is sharp for the functions The proof of Theorem 3.8 is thus completed.✷ Remark 3.9.Putting λ = 0 and λ = 1 in Theorem 3.8, we get the result of Raza and Malik (see [27]) and Sahoo and Patel (see [28]).
The sharp upper bound for the fourth coefficient of the function f ∈ M L * λ is given by the following theorem.
It is clear that ∂T ∂y < 0 for 0 < p < 2 and 0 < y < 1.Thus, T (p, y, λ) is an decreasing function of y, contradicting our assumption.Therefore, Hence by second derivative test, S(p) has minimum value at p, where p is given by (3.33).
From the above discussion, it is clear that S(p) attains its maximum at p = 0. Thus, the upper bound in (3.27) corresponds to p = 0 and y = 0 from which we get the required estimate (3.26).The estimate in (3.26) is sharp for the function f ∈ A given by This complete the prove of Theorem 3.10.✷ Remark 3.11.Taking λ = 0 and λ = 1 in Theorem 3.10, we get the upper bounds for |a 4 | for the class of SL * and R respectively studied by Raza and Malik [27] and Sahoo and Patel [28].

Theorem 3 . 10 .
Let the function f given by (1.1) be in the class M L * L Proceeding similarly as in the proof of Theorem 3.8 and making use of Lemma 2.2 in (3.15) assuming that (1 − 4λ − 3λ 2 ) > 0, it follows that