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COEFFICIENT INEQUALITIES FOR ANALYTIC FUNCTIONS CONNECTED WITH k-FIBONACCI NUMBERS
( Serap Bulut ),( Janusz Sokół ) 호남수학회 2022 호남수학학술지 Vol.44 No.4
In this paper, we introduce a new class R<sup>k</sup><sub>λ</sub> (λ ≥ 1, k is any positive real number) of univalent complex functions, which consists of functions f of the form f(z) = z + ∑<sup>∞</sup><sub>n=2</sub> a<sub>n</sub>z<sup>n</sup> (|z| < 1) satisfying the subordination condition (1 - λ) f (z)/z + λf′ (z) ≺ 1 + τ<sup>2</sup><sub>k</sub>z<sup>2</sup>/1 - kτ<sub>k</sub>z - τ<sup>2</sup><sub>k</sub>z<sup>2</sup>, τk = k - √ k<sup>2</sup> + 4/2, and investigate the Fekete-Szegö problem for the coefficients of f ∈ Rk λ which are connected with k-Fibonacci numbers Fk,n = (k - τ<sub>k</sub>)<sup>n</sup> - τ<sup>n</sup><sub>k</sub>/√k<sup>2</sup> + 4 (n ∈ N ∪ {0}) . We obtain sharp upper bound for the Fekete-Szegö functional |a3 - μa<sup>2</sup><sub>2</sub>| when μ ∈ R. We also generalize our result for μ ∈ C.