HERMITE HADAMARD TYPE INEQUALITIES FOR FRACTIONAL INTEGRALS

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1 HERMITE HADAMARD TYPE INEQUALITIES FOR FRACTIONAL INTEGRALS MARIAN MATŁOKA Abstract: In the present note, we have established an integral identity some Hermite-Hadamard type integral ineualities for the fractional integrals. Keywords: Hermite-Hadamard s ineualities, Riemann-Liouville fractional integral, integral ineualities, h - preinvex function. Mathematics Subject Classification: 6A5; 6D; 6A5.

2 . Introduction The f: I R R be a convex function defined on the interval I of real numbers a < b. The following double ineuality: ) fx) dx a b is well known in the literature as Hermite Hadamard s ineuality. Recently, many others [ 3 developed discussed Hermite Hadamard s ineuality in terms of refinements, counterparts, generalizations new Hermite - Hadamard s type ineualities. In 7, Varošanec [ introduced a large class of non-negative functions, the socalled h - convex functions. This class contains several well-known classes of functions such as non-negative convex functions if ht) = t) s - convex functions in the second sense if ht) = t s ). This class is defined in the following way: a non-negative function f: I R, I R, is an interval, is called h convex if ftx + t)y) ht) fx) + h t)fy) holds for all x, y I t [,, where h: J R is a non-negative function, h J is an interval,, ) J. In the following, we will give some necessary definitions mathematical preliminaries of fractional calculus theory which are used further in this paper. For more details, one can consult [4, 5, 6. Let f L[a, b). The Riemann-Liouville integrals Ι a + f Ι b f of order > with a are defined by Ι a + fx) = Γ) x t) ft)dt a x x > a),

3 b Ι b fx) = Γ) t x) ft)dt x x < b) respectively. Here Γ) is the Gamma function Ι a + fx) = Ι b fx) = fx). For some recent results connected with fractional integral ineualities, see [5, 9, 9,. The aim of this paper is to establish Hermite-Hadamard s type ineualities involving Riemann-Liouville fractional integral for functions whose derivatives are h convex using the identity is obtained for fractional integrals.. Main results In order to prove our main theorems, we need the following lemma: Lemma.. Let f: I R R be differentiable on I a, b I, with a < b. If f L[a, b), then ) Γ + ) b, a) = 4 [t f t)a + t [Ι a+b ) Γ + ) ) [Ι a + = 4 [t f t) fa) + Ι a+b + fb) ) ) t) f t) ) + Ι b ) + t b) dt + tb) t) f t)a + t ) Proof. Integrating by part changing variables of integration yields dt 3

4 [t f t)a + t = [t t)a + t ) ) t) f t) + t b) dt t t)a + t ) dt [ t) t) + t b) = 4 [t f t) + b a ) + Γ + ) ) + = [t t) + t b) [Ι a+b ) + t) t) fa) + Ι a+b + fb) ) + t b) t) f t)a + t ) t t) dt + t b) dt + t b) dt [ t) t)a + t ) ) + [Ι a + = [ + Γ + ) This completes the proof. of Lemma.. + t) t)a + t ) dt ) + Ι b ). Using the Lemma., we can obtain the following fractional integral ineualities. Theorem.. Let f: I R R be differentiable on I a, b I, with a < b, f L[a, b). If f is h convex on [a, b, then ) Γ + ) ) [Ι a+b ) fa) + Ι a+b + fb) ) 4

5 4 [ f ) t ht)dt Γ + ) ) [Ι a + + f a) + f b) ) t h t)dt ) + Ι b ) 4 [ f ) t h t)dt + f a) + f b) ) t ht)dt. Proof. By Lemma. since f is h convex, then we have ) Γ + ) ) [Ι a+b ) ) 4 [ t f t)a + t ) dt + t) f t) + tb) dt 4 [ t h t) f a) + ht) f ) ) dt + t) h t) f ) + ht) f b) ) dt = 4 [ f ) t ht)dt analogously Γ + ) ) [Ι a + + f a) + f b) ) t h t)dt ) + Ι b ) 5

6 4 [ t f t) + tb) dt + t) f t)a + t ) dt 4 [ t h t) f ) + ht) f b) ) dt + t) h t) f a) + ht) f ) ) dt = 4 [ f ) t h t)dt This completes the reuired proof. + f a) + f b) ) t ht)dt. Corollary. In Theorem, if f is convex, then we get the following ineualities ) Γ + ) ) [Ι a+b ) ) 4 + ) [ f ) + f a) + f b) + ) Γ + ) ) [I a + ) + I + b b a ) 4 + ) [ + f ) + [ f a) + f b). Corollary. In Theorem, if f is s-convex, then we get the following ineualities ) Γ + ) ) [Ι a+b ) ) 6

7 4 [ f ) + s + + Γ + )Γs + ) Γ + s + ) + f a) + f b) ) 4 [ f Γ + ) ) [I a + ) + I + b b a ) ) + Γ + )Γs + ) Γ + s + ) + f a) + f b) + s + Theorem.. Let f: I R R be differentiable on I, a, b I, with a < b, f L[a, b). If f is h convex on [a, b; p, > ; + =, then following p ineualities hold ) Γ + ) ) [Ι a+b 4 p + ) ht)dt) p ) ) [ f a) + f ) Γ + ) ) [I a + ) + I b ) ) + f b) + f ) ) 4 p + ) ht)dt) p [ f a) + f ) ) + f b) + f ) ). Proof. From Lemma. using the Hőlder s integrals ineuality, we have 7

8 ) Γ + ) ) [Ι a+b t 4 [ p p + t) dt) dt) p p ) f t)a + t f t) ) ) + t b) dt) dt) 4 p + ) ht)dt) p [ f a) + f ) In the analogous way, we can prove the second ineuality. ) + f b) + f ) ). Theorem.3. Let f: I R R be differentiable on I, a, b I, with a < b, f L[a, b). If f,, is h - convex on [a, b, then the following ineualities hold: ) Γ + ) ) [Ι a+b 4 + ) ) [ f a) t h t)dt ) + f ) t ht)dt) + f b) t h t)dt Γ + ) ) [I a + + f ) t ht)dt) ) + I b ) 8

9 4 + ) [ f a) t ht)dt + f ) t h t)dt) + f b) t ht)dt + f ) t h t)dt). Proof. From Lemma. using the well known power mean ineuality, we have ) Γ + ) ) [Ι a+b 4 [ t 4 t dt) + t) 4 + ) ) ) f t)a + t ) dt + t) f t) [ t f t)a + t ) dt) f t) + t b) dt) [ f a) t h t)dt + f ) t ht)dt) + t b) dt + f b) t h t)dt + f ) t ht)dt) In analogous way we can prove the second ineuality.. 9

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