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Coefficient inequalities for a subclass of Bazilevič functions

Fitri Sa’adatul, Marjono, Thomas Derek K., Wibowo R. B. E.

Demonstratio Mathematica

2020

Vol. 53, nr 1

27--37

Let f be analytic in D={z:|z| < 1} with f(z)=z+∑∞n=2anzn, and for α ≥ 0 and 0 < λ ≤ 1, let B1(α,λ) denote the subclass of Bazilevič functions satisfying (…) <λ for 0 < λ ≤ 1. We give sharp bounds for various coefficient problems when f ∈ B1(α,λ), thus extending recent work in the case λ = 1.

An upper bound for third Hankel determinant of starlike functions related to shell-like curves connected with Fibonacci numbers

Sokół J., İlhan S., Güney H. Ö.

Journal of Mathematics and Applications

2018

Vol. 41

195--206

We investigate the third Hankel determinant problem for some starlike functions in the open unit disc, that are related to shell-like curves and connected with Fibonacci numbers. For this, firstly, we prove a conjecture, posed in [17], for sharp upper bound of second Hankel determinant. In the sequel, we obtain another sharp coefficient bound which we apply in solving the problem of the third Hankel determinant for these functions.

On the conjecture of Hayami and Owa concerning the class R(α)

Zaprawa P.

Journal of Applied Analysis

2017

Vol. 23, nr 1

25--32

In the paperwe discuss the functional Φf(μ) ≡ a2a4 − μa23 for functions in the class R(α), α ϵ [0, 1). This class consists of analytic functions which satisfy the condition Re f’ (z) > α for all z in the unit disk Δ.We show that the conjecture of Hayami and Owa [1], that is, |Φf(μ)| ≤ (1 − α)2 · max{ 1/2 – 4/9μ, 4/9μ} for all f ϵ R(α) and μ ϵ R, is false. Moreover, we find estimates of |Φf(μ)| that improve the results obtained by Hayami and Owa.