In this work, bi-univalent functions, which are one of the most important research areas of Geometric Function Theory and which are still very popular in recent years, have been investigated. For the studies of functions, it is customary to determine the bound. In this direction, firstly, we introduce a new subclass of bi-univalent functions in the open unit disk D = {z ∈ C: /z/ < 1}. For this purpose, the Komatu integral operator developed for complex functions and subordination princible are used. Afterwards, the relation between the Fibonacci number sequence and the functions with the positive real part is given. This relation is a fundamental role for Results section. The upper bounds of the first two Taylor-Maclaurin coefficients are investigated. Finally, we derive Fekete-Szegö inequalities for functions belonging to this newly-defined class. The obtained results are compared with studies in the literature. [ABSTRACT FROM AUTHOR]