Certain Chebyshev Type Integral Inequalities Involving Hadamard’s Fractional Operators

and Applied Analysis 3 we obtain 󵄨󵄨󵄨󵄨H (τ, ρ) 󵄨󵄨󵄨󵄨 ≤ 󵄨󵄨󵄨󵄨󵄨󵄨󵄨 ∫∫ ρ τ 󵄨󵄨󵄨󵄨 f 󸀠 (y) 󵄨󵄨󵄨󵄨 r dydz 󵄨󵄨󵄨󵄨󵄨󵄨󵄨 r −1 󵄨󵄨󵄨󵄨󵄨󵄨󵄨 ∫∫ ρ τ 󵄨󵄨󵄨󵄨 g 󸀠 (z) 󵄨󵄨󵄨󵄨 s dydz 󵄨󵄨󵄨󵄨󵄨󵄨󵄨 s −1 . (15) Since 󵄨󵄨󵄨󵄨󵄨󵄨󵄨 ∫∫ ρ τ 󵄨󵄨󵄨󵄨 f 󸀠 (y) 󵄨󵄨󵄨󵄨 r dydz 󵄨󵄨󵄨󵄨󵄨󵄨󵄨 r −1 = 󵄨󵄨󵄨󵄨τ − ρ 󵄨󵄨󵄨󵄨 r −1 󵄨󵄨󵄨󵄨󵄨󵄨󵄨 ∫ ρ τ 󵄨󵄨󵄨󵄨 f 󸀠 (y) 󵄨󵄨󵄨󵄨 r dy 󵄨󵄨󵄨󵄨󵄨󵄨󵄨 r −1

If ,  : [, ] → R + are absolutely continuous functions, whose first derivatives   and where ‖ ⋅ ‖ ∞ denotes the norm in  ∞ [, ].The Chebyshev functional (1) has many applications in numerical quadrature, transform theory, probability, study of existence for solutions of differential equations, and in statistical problems.Moreover, in the theory of approximations, under various assumptions (Chebyshev inequalities, Grüss inequality, etc.), Chebyshev functionals are useful to give lower bound or upper bounds for the functions.
Our work in the present paper is based on a weighted version of the Chebyshev functional (see [1]): where  and  are two integrable functions on [, ] and () is a positive and integrable function on [, ].In 2000, Dragomir [15] derived the following inequality: where  and  are two differentiable functions and   ∈   (, ),   ∈   (, ),  > 1, and Abstract and Applied Analysis Dahmani et al. [16] added one more dimension to this study by introducing generalization of inequality (4), involving Riemann-Liouville fractional integrals.Moreover, Purohit and Raina [17][18][19] and Baleanu et al. [20,21] introduced certain generalized integral inequalities for synchronous functions, involving the various fractional hypergeometric integral operators, while Tariboon et al. [22] studied Riemann-Liouville fractional integral inequalities.
In 1892, Hadamard [23] introduced a fractional derivative, which differs from the Riemann-Liouville and Caputo derivatives in the sense that the kernel of the integral contains logarithmic function of arbitrary exponent.For details and fundamental properties of Hadamard fractional derivative and integral can be found in [24][25][26][27][28]. Recently, some results on fractional integral inequalities have been derived by using Hadamard fractional integrals (see [29][30][31][32]).
In this paper, we establish certain integral inequalities related to the weighted Chebyshev's functional (3) in the case of differentiable functions whose derivatives belong to the space   ([1, ∞)), involving Hadamard fractional integral operators [23].We also develop some integral inequalities for the fractional integrals by suitably choosing the function (), as special cases of our findings.
Firstly, we mention below the basic definitions and notations of some well-known operators of fractional calculus, which shall be used in the sequel.
The Hadamard fractional integral of order  ∈ R + of a function (), for all  > 1, is defined as [28]    { ( Further, the Hadamard fractional derivative of order  ∈ [− 1, ),  ∈ Z + , of a function () is given by Our results in this paper are based on the following preliminary assertions giving composition formula of Hadamard fractional integral and derivatives with a power function ( [4]) where 0 <  < 1.

Main Results
Our results in this section are related to the Chebyshev's functional (3)  ( Proof.We define (, ) = (log (/)) We observe that the function (, ) remains positive, for all  ∈ (1, ) ( > 1).Multiplying both sides of ( 9) by (, )() and integrating with respect to  from 1 to , we get Next, multiplying both sides of (11) by (, )() and integrating with respect to  from 1 to , we can write In view of ( 9), we have Using the following Hölder's inequality for  > 1 and  −1 + It follows from ( 12) that Applying again Hölder's inequality on the right-hand side of (18), we get On making use of ( 23) and ( 24), the left-hand side of inequality (8) follows very easily.Now, to prove the right-hand side of inequality (8), we observe that 1 ≤  ≤ , 1 ≤  ≤ , and therefore, Evidently, from (23), we get which completes the proof of Theorem 1. Now, we establish the following integral inequality, which may be regarded as a generalization of Theorem 1.

Theorem 2.
Let  be a positive function and let  and  be two differentiable functions on [1, ∞).
Proof.To prove the above theorem, we use inequality (11).
Multiplying both sides of ( 11) by which remains positive and integrating with respect to  from 1 to , we get Now making use of ( 17), (29) gives Applying Hölder's inequality on the right-hand side of (30), we get In view of ( 29) and (32) which completes the proof of Theorem 2.

Special Cases
As implications of our main results, we consider some consequent results of Theorems 1 and 2 by suitably choosing the function ().Other classes of no weighted inequalities are also obtained.To this end, let us set () = (log )  ( ∈ [0, ∞),  ∈ (1, ∞)); then Theorems 1 and 2 yield the following results.
We conclude our paper by remarking that we have introduced new general Chebyshev type inequalities involving Hadamard fractional integral operators.By suitably specializing the arbitrary function (), one can further easily obtain additional fractional integral inequalities from our main results (Theorems 1 and 2).