Coefficient inequalities for a subclass of Bazilevič functions

Abstract Let f be analytic in D = { z : | z | < 1} {\mathbb{D}}=\{z:|z\mathrm{|\hspace{0.17em}\lt \hspace{0.17em}1\}} with f ( z ) = z + ∑ n =2 ∞ a n z n f(z)=z+{\sum }_{n\mathrm{=2}}^{\infty }{a}_{n}{z}^{n} , and for α ≥ 0 and 0 < λ ≤ 1, let ℬ 1 ( α , λ ) { {\mathcal B} }_{1}(\alpha ,\lambda ) denote the subclass of Bazilevič functions satisfying | f ′ ( z ) ( z f ( z ) ) 1 − α − 1 | < λ \left|f^{\prime} (z){\left(\frac{z}{f(z)}\right)}^{1-\alpha }-1\right|\lt \lambda for 0 < λ ≤ 1. We give sharp bounds for various coefficient problems when f ∈ ℬ 1 ( α , λ ) f\in { {\mathcal B} }_{1}(\alpha ,\lambda ) , thus extending recent work in the case λ = 1.

Then, for α > 0, it was shown by Bazilevič [1] that if ∈ f and is given by eq. (1), then there exists starlike functions g such that it follows that ∈ f . We denote this class of Bazilevič functions by ( ) α , so that ( ) ⊂ α when α > 0. The case α = 0 was subsequently considered by Sheil-Small [2], who showed that ( ) ⊂ α when α ≥ 0. Taking g(z) ≡ z gives the class ( ) α 1 of Bazilevič functions, which has been the subject of much recent research. We note that ( ) 0 1 is the class ⁎ of starlike functions, and ( ) 1 Various properties have been obtained for functions in ( ) α 1 . Among other results, Singh [3] found sharp estimates for the moduli of the first four coefficients and obtained the solution to the Fekete-Szegö problem. Sharp bounds for the second Hankel determinant, the initial coefficients of the function log(f(z)/z), and the initial coefficients of the inverse function f −1 were obtained in [4], and distortion theorems and some length-area results were also obtained in [5][6][7].
We now define the subclass ( ) α λ , 1 of ( ) α 1 , which was introduced in 1996 by Ponnusamy and Singh [8,Theorem 3]. In this article, the authors determine condition on λ so that functions in ( ) α λ , 1 are starlike in (see also [9,Theorem 3] for an extension of this result). Later in [10, Theorems 1 and 2], the authors considered complex values of α and obtained condition on λ such as the functions in ( ) α λ , 1 are spirallike in . Two of the present authors in [11] studied the class ( ) α λ , 1 in the case λ = 1. Therefore, it is natural to consider the investigation of the problems discussed in this article for the complex values of α in the context of the investigation from [10].
and be given by eq. (1). Then, for α ≥ 0 and 0 < λ ≤ 1, ∈ ( ) f α λ , 1 , if and only if, for ∈ z We note that ∈ ( ) f 0, 1 1 reduces to the class of bounded starlike functions considered by Singh [12]. Although the aforementioned definition requires that α ≥ 0, choosing α = −1 gives the class ( ) λ of univalent functions defined for ∈ z by The class ( ) λ has been the focus of a great deal of research in recent years (see e.g. [13,14], and for a summary of some known results, see [15]). Although the classes ( ) α λ , 1 for α ≥ 0 and ( ) λ have similar structural representations, they are fundamentally different in many ways, and we shall see in the following analysis that the methods used in this study cannot be applied to the class ( ) λ . It is also interesting to note that the only known negative value of α which gives a subset of appears to be α = −1. See [16,17] and references therein for recent investigation, which also deals with the case α = −1 for meromorphic functions.
In this study, we give sharp bounds for the modulus of the coefficients a n for ∈ ( ) f α λ , 1 when 2 ≤ n ≤ 5, together with other related results, noting that when ∈ ( ) f λ , sharp bounds have been found only for some initial coefficients.
First note that from eq. (2), we can write where ω is the Schwarz function. Next, recall the class of functions with positive real part in , so that ∈ h , if, and only if, Re h(z) > 0 for ∈ z . We write Thus, as we can write

eq. (3) can be written as
We shall use the following results concerning the coefficients of ∈ h .

Initial coefficients
We first give sharp bounds for some initial coefficients for ∈ ( ) f α λ , 1 , extending those given in [11].
For a 5 , we apply Lemma 1.5 with α 1 , α 2 , β 1 and β 2 the respective coefficients of a 5 in eq. (7). Since 0 < α 1 < 1 and 0 < α 2 < 1, for α ≥ 0 and 0 < λ ≤ 1, then by expanding both sides and subtracting, it is easily seen that the conditions (6) of Lemma 1.5 are satisfied (the detailed proof of this step can be found in [21]), and so the inequality for |a 5 | follows. The inequality of |a i | is sharp on choosing c i = 2 when 2 ≤ i ≤ 5, and c j = 0 when i ≠ j. □

Inverse coefficients
Since ( ) ⊂ α λ , 1 , inverse functions f −1 exist, and so we can write We first prove the following.
The inequalities for |A 4 | for ∈ ( ) f α λ , 1 are complicated, and in the interest of brevity, we omit many of the detailed calculations. Also, to simplify the analysis and presentation of the results, we define Γ i (α) for i = 1, 2, 3 as follows: , with inverse coefficients given by eq. (8), then when either (i) > α α 1 All the inequalities are sharp.
Proof. Again on substituting eq. (7) in eq. (9)   Using the same techniques as in the proof of Theorem 2.1, it is possible to prove the following (proofs can be found in [21]). for α ≥ 0 and 0 < λ ≤ 1 with logarithmic coefficients given by eq. (16). Then, for 1 ≤ n ≤ 4, All the inequalities are sharp.

The second Hankel determinant
The qth Hankel determinant of f is defined for q ≥ 1 and n ≥ 1 as follows and has been extensively studied (see e.g. We prove the following, noting that the result is valid for α ≥ 0.