Let g be a strictly increasing function on a , b , having a continuous derivative g′ on a , b . For the Lebesgue integrable function f : a , b → C , we define the k-g-left-sided fractional integral of f by S k , g , a + f x = ∫ a x k g x - g t g ′ t f t d t , x ∈ a , b and the k-g-right-sided fractional integral of f by S k , g , b - f x = ∫ x b k g t - g x g ′ t f t d t , x ∈ [ a , b ) , where the kernel k is defined either on 0 , ∞ or on 0 , ∞ with complex values and integrable on any finite subinterval. In this paper we establish some Ostrowski and trapezoid type inequalities for the k-g-fractional integrals of functions of bounded variation. Applications for mid-point and trapezoid inequalities are provided as well. Some examples for a general exponential fractional integral are also given.
Bivariate functions of bounded variation: Fractal dimension and fractional integral