The Coefficients of Powers of Bazilević Functions

In the present work, a sharp bound on the modulus of the initial coefficients for powers of strongly Bazilević functions is obtained. As an application of these results, certain conditions are investigated under which the Littlewood-Paley conjecture holds for strongly Bazilević functions for large values of the parameters involved therein. Further, sharp estimate on the generalized Fekete-Szegö functional is also derived. Relevant connections of our results with the existing ones are also made.


Introduction
Let A be the class of analytic functions f defined in the unit disk D = {z : |z| < 1} having Taylor series expansion f (z) = z + ∞ ∑ n=2 a n z n .
(1) Let S be the subclass of A consisting of univalent functions in D. The famous Bieberbach conjecture (now de Branges's theorem [1]) states that the coefficient of functions in the class S satisfy |a n | ≤ n with equality in case of the Koebe function k(z) = z/(1 − z) 2 . Denote, by S * , the subclass of S consisting of starlike functions, so that Re(z f (z)/ f (z)) > 0 for z ∈ D).
The class of strongly starlike functions of order β (0 < β ≤ 1) is defined by For f ∈ B 1 (α), Singh gave sharp estimates for the first four coefficients, together with other results, and in 2017 Marjono et al. [6] obtained sharp estimates for the fifth and sixth coefficients for some values of α and conjectured that when α ≥ 1 |a n | ≤ 2 n − 1 + α for n ≥ 2.
We now consider the validity of the inequality whenever f ∈ S. First note that when γ = 1, (6) becomes de Branges theorem, and when γ = 2, (6) reduces to the Littlewood-Paley conjecture [19], which was shown to be false by Fekete and Szegö [20].
In the case of close-to-convex functions, Jahangiri [22] showed that (6) is valid when n = 2 provided 0 < γ ≤ 3, but is false when γ > 1. Similar problems were considered by Darus and Thomas in [24].
We now introduce the class of strongly Bazilević functions as follows.

Definition 1.
A function f defined by (1) belongs to the class B(α, β) if there exists a normalized analytic function g ∈ S * such that All powers in the above definition are understood to be the principal ones. Clearly B(0, 1) = S * and B(1, 1) = C. Also B(α, β) is a subclass of B α and hence contains only univalent functions.

Powers of Bazilević Functions
First we prove the following theorem, which gives sharp estimates of |a 1 (γ)| and |a 2 (γ)|. These estimates will be later used to discuss the Littlewood-Paley conjecture for functions in the class B(α, β).
We note that the above theorem generalizes many existing results in literature. For example, for α = 1 = β, the above theorem gives the following result due to Jahangiri [22]. Corollary 1. [22] (Theorem 1, p. 1141) Let f ∈ C and a n (γ) (n = 1, 2) be as given in (4). Then the following sharp bounds hold.

Remark 1.
First note from (5) that b 1 (γ) = 2/γ and b 2 (γ) = (2 + γ)/γ 2 . Obviously |a 1 (γ)| ≤ b 1 (γ) holds for all 0 < α ≤ 1, 0 < β ≤ 1 and γ > 0. This verifies the Littlewood-Paley conjecture for n = 1 and all γ > 0. We now consider the case when n = 2 and γ = 1. For this case we have and It is easy to verify that Consider the case when n = 2 and γ = 2. For this case we have where Ω is the set of (α, β) such that either of holds, where S 2 and T 2 are given by and It is a simple matter to check that if 0 < α < 2 and (α 2 . Thus under certain conditions the Littlewood-Paley conjecture is also true for n = 2 = γ. Next assume that n = 2 and γ = 3. In this case we see that |a 2 (3)| ≤ 5/9 = b 2 (3) holds if either of the conditions 0 < α < 1 and is true. In a similar way we can check that for n = 2 and γ = 4, the inequality |a 2 (4)| ≤ 3/8 = b 2 (4) holds if either of the following conditions is true.
It should be noted that inequality (6) in many cases is not true for large values of γ, for example the case of close-to-convex functions it does not hold for value of γ > 3. Another example can be found in [29] (Equation (23), p. 93) due to Farahmand and Jahangiri. They proved that for a subclass of close-to-convex function the inequality |a 2 (γ)| ≤ b 2 (γ) not even true for γ = 4. It is therefore interesting to investigate the cases for which this inequality holds for large values of γ. The following corollary describes certain conditions under which inequality (6) holds for large values of γ for functions f ∈ B(α, β).
To obtain the estimate on |a 2 (γ)|, we rewrite a 2 (γ) as Applying triangle inequality in (15) and using |q 1 | ≤ 2, we get It is clear that the coefficient of q 2 1 in the above expression is positive for all 0 < γ, 0 < α and 0 < β ≤ 1. Also since β(α + γ + 2) > 0, it follows that the coefficient of p 2 1 is also greater than −1/2. Using the inequalities in Lemma 1 along with and using |q i | ≤ 2, we obtain where s := |p 1 | ∈ [0, 2] and M is given by For α > 0 and 0 < β ≤ 1, we have We now consider the case when s ∈ (0, 2). In this case, we see that the unique root of the equation is given by .
To prove the result we now consider the two cases: (a) It is easy to see that s 0 ∈ (0, 2) if α > 0, 0 < β ≤ 1 and γ > γ * . Further Thus h has a maximum at the point s 0 and so h has maximum at s 0 . Now from (18), we see that . (19) (b) Next when 0 < γ ≤ γ * , the critical point of h does not belongs to (0, 2) and so we consider the end point for the maxima and minima. Since γ * ≤ γ 0 , it follows that Thus (19) and (20) together with (18) give the desired estimate.
Thus the inequalities are sharp, which complete the proof of the theorem.
The following theorem provides sharp upper bound for the Fekete-Szegö functional M(γ) for function f ∈ B(α, β).
Equality occurs only if p 1 = q 1 = 0, p 2 = q 2 = 2, and the corresponding function f is defined by This completes the proof.

Conclusions
In this paper, we have investigated the sharp upper bound for |a 1 (γ)|, |a 2 (γ)| and M(γ) for f ∈ B(α, β). In general, it is not easy to verify |a n (γ)| ≤ b n (γ) to hold for many subclasses of normalized univalent functions. However, in this work it has been verified that the inequality |a n (γ)| ≤ b n (γ) holds for larger values of γ, which is rare for many subclasses of normalized univalent functions. The sharp estimate on the generalized Fekete-Szegö functional is also derived. Special cases are also discussed.

Conflicts of Interest:
The authors declare no conflict of interest.