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Abstract and Applied Analysis
Volume 2014 (2014), Article ID 236124, 12 pages
http://dx.doi.org/10.1155/2014/236124
Research Article

Certain Spaces of Functions over the Field of Non-Newtonian Complex Numbers

1Department of Mathematical Engineering, Yıldız Technical University, Davutpaşa Campus, Esenler, 80750 Istanbul, Turkey
2Department of Mathematics, Faculty of Arts and Sciences, Fatih University, Hadimköy Campus, Büyükçekmece, 34500 Istanbul, Turkey

Received 3 December 2013; Accepted 22 January 2014; Published 15 April 2014

Academic Editor: S.A. Mohiuddine

Copyright © 2014 Ahmet Faruk Çakmak and Feyzi Başar. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

This paper is devoted to investigate some characteristic features of complex numbers and functions in terms of non-Newtonian calculus. Following Grossman and Katz, (Non-Newtonian Calculus, Lee Press, Piegon Cove, Massachusetts, 1972), we construct the field 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 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 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 and the set of -differentiable functions.