Notions of generalized s-convex functions on fractal sets

holds, then f is called a generalized convex function on I []. In α = , we have convex function, convexity is defined only in geometrical terms as being the property of a function whose graph bears tangents only under it []. The convexity of functions plays a significant role in many fields, for example, in biological system, economy, optimization, and so on [–]. In recent years, the fractal theory has received significantly remarkable attention from scientists and engineers. In the sense of Mandelbrot, a fractal set is one whose Hausdorff dimension strictly exceeds the topological dimension [, ]. Many researchers studied the properties of functions on fractal space and constructed many kinds of fractional calculus by using different approaches [–]. Particularly, in [], Yang gave the analysis of local fractional functions on fractal space systematically, which includes local fractional calculus and the monotonicity of function. Let R be the real line numbers on fractal space. Then by using Gao-Yang-Kang’s concept one can explain the definitions of the local fractional derivative and local fractional integral as in [–]. Now if r  , r  and r  ∈Rα ( < α ≤ ), then () r  + r  ∈Rα , r  r  ∈Rα , () r  + r  = r  + r  = (r + r) = (r + r) , () r  + (r  + r  ) = (r  + r  ) + r  , () r  r  = r  r  = (rr) = (rr) , () r  (r  r  ) = (r  r  )r  , () r  (r  + r  ) = (r  r  ) + (r  r  ),