On the coefficients of $$\mathbf {{\mathcal {B}}_1}(\varvec{\alpha })$$ B
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On the coefficients of B1 (˛) Bazileviˇc functions Khadija Bano1 · Mohsan Raza1
· Derek K. Thomas2
Received: 24 May 2020 / Accepted: 8 October 2020 © The Royal Academy of Sciences, Madrid 2020
Abstract Denote class of functions f , analytic in D = {z : |z| < 1} and given by f (z) = by A, the n z+ ∞ in D. n=2 an z for z ∈ D, and by S the subset of A whose elements are univalent z f (z) f (z) α The class B1 (α) ⊂ S , of Bazileviˇc functions is defined by Re > 0, for f (z) z f (z) n α ≥ 0 and z ∈ D. We give sharp bounds for |γn |, where log =2 ∞ n=1 γn z , when z n = 1, 2, 3, and α ≥ 0, and obtain the sharp bound for |γ4 | when 0 ≤ α ≤ α ∗ (α ∗ ≈ 1.5464), together with another bound for |γ4 | when α ≥ 0. Sharp bounds for some initial coefficients of the inverse function when f ∈ B1 (α) are also found, which augment known results. Keywords Bazileviˇc functions · Logarithmic coefficients · Inverse coefficients Mathematics Subject Classification 30C45
1 Introduction Let A denote the class of functions f of the form f (z) = z +
∞
an z n
(1.1)
n=2
which are analytic in the open unit disk D = {z : |z| < 1}, and denote by S the class of functions in A which are univalent in D. For α ≥ 0, a function f ∈ A belongs to the class B(α) of Bazileviˇc functions [2], if there exists a starlike function g such that
B
Mohsan Raza [email protected] Khadija Bano [email protected] Derek K. Thomas [email protected]
1
Department of Mathematics, Government College University Faisalabad, Faisalabad, Pakistan
2
Swansea University, Bay Campus, Swansea SA1 8EN, UK 0123456789().: V,-vol
123
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K. Bano et al.
Re
z f (z) f (z)
f (z) g(z)
α
> 0, z ∈ D.
It is well-known that B(α) ⊂ S [10] . Clearly B(1) is the familiar class of close-to-convex functions, and B(0) the class of starlike functions. Taking g(z) = z, we obtain the class B1 (α) of Bazileviˇc functions satisfying the relation z f (z) f (z) α Re > 0, z ∈ D. (1.2) f (z) z The class B1 (α) has been the subject of a great many papers in recent times, and a summary of some of the most fundamental and important properties of functions in B1 (α) can be found in [20]. Denote by P , the class of functions h of the form h(z) = 1 +
∞
cn z n
(1.3)
n=1
satisfying Re {h(z)} > 0 in D. Thus from (1.2 ), we can write z f (z) f (z) α = h (z) , z ∈ D, f (z) z where h ∈ P . The logarithmic coefficient γn of f ∈ S are defined as ∞ f (z) =2 γn z n . log z
(1.4)
(1.5)
n=1
The numbers γn are called the logarithmic coefficients of f , and play an important role in the theory of univalent functions. The Köebe function has logarithmic coefficients γn = 1/n, and the inequality |γn | ≤ 1/n holds for starlike functions, but is false for the full class S [6, p. 898]. Next note that for any univalent function f , there exist an inverse function f −1 defined on some disk |w| ≤ 1/4 ≤ r ( f ), with Taylor series expansion f −1 (w) = w + A2 w 2 + A3 w 3 + · · · .
(1.6)
Many recent papers have been devoted to finding sharp boun
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