Some special families of holomorphic and Al-Oboudi type bi-univalent functions related to k-Fibonacci numbers involving modified Sigmoid activation function

• Published in: Afrika mathematica Vol. 32; no. 3-4; pp. 631 - 643
• Main Authors: Frasin, B. A., Swamy, S. R., Nirmala, J.
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer Berlin Heidelberg 01.06.2021; Springer Nature B.V
• Subjects: Applications of Mathematics; Fibonacci numbers; History of Mathematical Sciences; Mathematical analysis; Mathematics; Mathematics and Statistics; Mathematics Education; Upper bounds; Primary 11B39; 30C45; 33C45; Secondary 30C50; Bi-univalent function; Holomorphic function; Modified sigmoid function; Fekete–Szegö inequality; Fibonacci numbers; 33C05
• ISSN: 1012-9405; 2190-7668
• DOI: 10.1007/s13370-020-00850-w

The aim of the present paper is to introduce some special families of holomorphic and Al-Oboudi type bi-univalent functions related to k -Fibonacci numbers involving modified Sigmoid activation function ϕ ( s ) = 2 1 + e - s , s ≥ 0 in the open unit disc D . We investigate the upper bounds on initial coefficients for functions of the form g ϕ ( z ) = z + ∑ j = 2 ∞ ϕ ( s ) d j z j , in these newly introduced special families and also discuss the Fekete–Szegö problem. Some interesting consequences of the results established here are also indicated.