Inequalities for α-fractional differentiable functions

In this article, we present an identity and several Hermite-Hadamard type inequalities for conformable fractional integrals. As applications, we establish some inequalities for certain special means of two positive real numbers and give the error estimations for the trapezoidal formula.


Introduction
A real-valued function ψ : I ⊆ R → R is said to be convex on I if the inequality ψ θξ + (θ )ζ ≤ θψ(ξ ) + (θ )ψ(ζ ) (.) holds for all ξ , ζ ∈ I and θ ∈ [, ]. ψ is said to be concave on I if inequality (.) is reversed. Let ψ : I ⊆ R → R be a convex function on the interval I, and c  , c  ∈ I with c  < c  . Then the double inequality is known in the literature as the Hermite-Hadamard inequality for convex functions [-]. Both inequalities hold in the reversed direction if ψ is concave on the interval I. In particular, many classical inequalities for means can be derived from (.) for appropriate particular selections of the function ψ.
Recently, the improvements, generalizations, refinements and applications for the Hermite-Hadamard inequality have attracted the attention of many researchers [-].
Dragomir and Agarwal [] proved the following results connected with the right hand part of (.).
Making use of Theorem ., Pearce and Pečarić [] established Theorem . as follows.
Next, we recall several elementary definitions and important results in the theory of conformable fractional calculus, which will be used throughout the article, we refer the interested reader to [-].
The conformable fractional derivative of order  < α ≤  for a function ψ : (, ∞) → R at ξ >  is defined by , and the fractional derivative at  is defined as D α (ψ)() = lim ξ → + D α (ψ)(ξ ). The (left) fractional derivative starting from c  of a function ψ : [c  , ∞) → R of order  < α ≤  is defined by For more details see []. Let α ∈ (, ] and ψ, φ be α-differentiable at ξ > . Then we have where ψ is differentiable at φ(ξ ) in equation (.). In particular, Remark . We clearly see that inequalities (.) and (.) reduce to inequality (.) if The main purpose of the article is to present an identity and several Hermite-Hadamard type inequalities for conformable fractional integrals, establish some inequalities for certain special means of two positive real numbers and give the error estimations for the trapezoidal formula.

Main results
In order to prove our main results we need a lemma, which we present in this section.
Then making use of integration by parts, we get Therefore, Lemma . follows easily from (.).

Remark .
We clearly see that the identity given in Lemma . reduces to the identity given in Theorem . if α = .
Proof It follows from Lemma . and the convexities of the functions ξ → ξ α- and ξ → Remark . Let α = . Then inequality (.) becomes Proof From Lemma . and the well-known Hölder mean inequality together with the convexity of |ψ | q on the interval [c  , c  ] we clearly see that Therefore, inequality (.) follows easily from (.)-(.).

Remark . Let α = . Then inequality (.) becomes
where A  (α) and B  (α) are defined as in Theorem ., and C  (α) and C  (α) are defined by Proof It follows from the concavity of |ψ | q and the Hölder mean inequality that which implies that |ψ | is also concave. Making use of Lemma . and the Jensen integral inequality, we have Therefore, inequality (.) follows easily from (.)-(.).

Applications to special means of real numbers
Let α ∈ (, ], r ∈ R, r = , -α and a, b >  with a = b. Then the arithmetic mean A(a, b), logarithmic mean L(a, b) and (α, r)th generalized logarithmic mean L (α,r) (a, b) of a and b are defined by respectively. Then from Theorems . and . together with the convexities of the functions ξ → ξ r and ξ → /ξ on the interval (, ∞) we get several new inequalities for the arithmetic, logarithmic and generalized logarithmic means as follows.

Applications to the trapezoidal formula
Let be a division c  = ξ  < ξ  < · · · < ξ n- < ξ n = c  of the interval [c  , c  ] and consider the quadrature formula is the trapezoidal version and E α (ψ, ) denotes the associated approximation error. In this section, we are going to derive several new error estimations for the trapezoidal formula.
Proof Applying Theorem . on the subinterval [ξ i , ξ i+ ] (i = , , . . . , n -) of the division , we have It follows from (.) and the convexity of |ψ (ξ )| on the interval [c  , c  ] that Making use of arguments analogous to the proof of Theorem ., we get Theorem . immediately.

Conclusion
In this work, we find an identity and several Hermite-Hadamard type inequalities for conformable fractional integrals, present some new inequalities for the arithmetic, logarithmic and generalized logarithmic means of two positive real numbers and provide the error estimations for the trapezoidal formula.