Abstract
The concept of generalized h-preinvex function on real linear fractal sets R^{beta } (0 < beta le 1) is introduced, which extends generalized preinvex, generalized s-preinvex, generalized Godunova–Levin preinvex, and generalized P-preinvex functions. In addition, some Hermite–Hadamard type inequalities for these classes of functions involving local fractional integrals are established. Lastly, the upper bounds for generalized expectation, generalized rth moment, and generalized variance of a continuous random variable are given to illustrate the applications of the obtained results.
Highlights
We recall the well-known Hermite–Hadamard inequality for convex function.Theorem 1 ([1], Hermite–Hadamard’s inequality) Let g : I ⊆ R → R be a convex function and c, d ∈ I with c < d, c+d g ≤ d g(x) dx g(c) + g(d) . (1.1) d–c cMost of the research on this class of inequalities is related to convexity
Combining the definitions of generalized preinvex function and generalized h-convex function, the main purpose of this paper is to introduce the concept of generalized hpreinvex function on fractal sets, which extends generalized h-convex function, generalized preinvex function, and generalized s-preinvex function
3 Main results In order to establish some new Hermite–Hadamard type inequalities, we firstly introduce the definition of generalized h-preinvex function on fractal sets
Summary
We recall the well-known Hermite–Hadamard inequality for convex function. Theorem 1 ([1], Hermite–Hadamard’s inequality) Let g : I ⊆ R → R be a convex function and c, d ∈ I with c < d, c+d g. Proof Since g is a nonnegative generalized h1-preinvex function and ψ is a nonnegative generalized h2-preinvex function, for all τ ∈ [0, 1], we have g c + τ η(b, c) ψ c + τ η(b, c) ≤ hβ (τ )hβ (τ )g(b)ψ(b) + hβ (τ )hβ (1 – τ )g(b)ψ(c) + hβ (τ )hβ (1 – τ )g(c)ψ(b) + hβ (1 – τ )hβ (1 – τ )g(c)ψ(c) Integrating both sides of the above inequality with respect to τ over [0, 1], letting c + τ η(b, c) = x, we get ηβ g c + τ η(b, c) ψ c + τ η(b, c) (dτ )β. B c and our results reduce to some results for generalized convex, generalized s-convex, generalized Godunova–Levin convex, and generalized P-convex, respectively
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
