Certain spaces of functions over the field of non-Newtonian complex numbers (Q1722395)

Summary: This paper is devoted to investigate some characteristic features of complex numbers and functions in terms of non-Newtonian calculus. Following \textit{M. Grossman} and \textit{R. Katz} [Non-Newtonian calculus. Pigeon Cove, MA: Lee Press. (1972; Zbl 0228.26002)], we construct the field \(\mathbb C^*\) of \(*\)-complex numbers and the concept of \(*\)-metric. Also, we give the definitions and the basic important properties of \(*\)-boundedness and \(*\)-continuity. Later, we define the space \(C_*(\Omega)\) of \(*\)-continuous functions and state that it forms a vector space with respect to the non-Newtonian addition and scalar multiplication and we prove that \(C_*(\Omega)\) is a Banach space. Finally, Multiplicative calculus (MC), which is one of the most popular non-Newtonian calculus and created by the famous exp function, is applied to complex numbers and functions to investigate some advance inner product properties and give inclusion relationship between \(C_*(\Omega)\) and the set of \(C_*'(\Omega)\) \(*\)- differentiable functions.