New general integral inequalities for quasi-geometrically convex functions via fractional integrals
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New general integral inequalities for quasi-geometrically convex functions via fractional integrals ˙Imdat ˙Is¸ can* * Correspondence: [email protected] Department of Mathematics, Faculty of Sciences and Arts, Giresun University, Giresun, 28100, Turkey
Abstract In this paper, the author introduces the concept of the quasi-geometrically convex functions, gives Hermite-Hadamard’s inequalities for GA-convex functions in fractional integral forms and defines a new identity for fractional integrals. By using this identity, the author obtains new estimates on generalization of Hadamard et al. type inequalities for quasi-geometrically convex functions via Hadamard fractional integrals. MSC: Primary 26A33; 26A51; secondary 26D15 Keywords: quasi-geometrically convex function; Hermite-Hadamard-type inequalities; Hadamard fractional integrals
1 Introduction Let a real function f be defined on some nonempty interval I of a real line R. The function f is said to be convex on I if inequality f tx + ( – t)y ≤ tf (x) + ( – t)f (y)
()
holds for all x, y ∈ I and t ∈ [, ]. We recall that the notion of quasi-convex function generalizes the notion of convex function. More exactly, a function f : [a, b] ⊂ R → R is said to be quasi-convex on [a, b] if f tx + ( – t)y ≤ max f (x), f (y) for all x, y ∈ [a, b] and t ∈ [, ]. Clearly, any convex function is a quasi-convex function. Furthermore, there exist quasi-convex functions which are not convex (see []). The following inequalities are well known in the literature as the Hermite-Hadamard inequality, the Ostrowski inequality and the Simpson inequality, respectively. Theorem . Let f : I ⊆ R → R be a convex function defined on the interval I of real numbers, and let a, b ∈ I with a < b. The following double inequality holds:
a+b f
≤ b–a
a
b
f (x) dx ≤
f (a) + f (b) .
()
©2013 ˙Is¸ can; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
˙Is¸ can Journal of Inequalities and Applications 2013, 2013:491 http://www.journalofinequalitiesandapplications.com/content/2013/1/491
Page 2 of 15
Theorem . Let f : I ⊆ R → R be a mapping differentiable in I ◦ , the interior of I, and let a, b ∈ I ◦ with a < b. If |f (x)| ≤ M, x ∈ [a, b], then the following inequality holds: f (x) –
b–a
b
a
M (x – a) + (b – x) f (t) dt ≤ b–a
for all x ∈ [a, b]. Theorem . Let f : [a, b]→ R be a four times continuously differentiable mapping on (a, b) and f () ∞ = supx∈(a,b) |f () (x)| < ∞. Then the following inequality holds:
b f (a) + f (b) a+b f () (b – a) . + f – f (x) dx ≤ ∞ b–a a , The following definitions are well known in the literature. Definition . ([, ]) A function f : I ⊆ (, ∞) → R is said to be GA-convex (geometricarithmetically convex) if
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