Multivariate fractional Ostrowski type inequalities

Optimal upper bounds are given for the deviation of a value of a multivariate function of a fractional space from its average, over convex and compact subsets of RN,N≥2RN,N≥2. In particular we work over rectangles, balls and spherical shells. These bounds involve the supremum and L∞L∞ norms of related multivariate fractional derivatives of the function involved. The inequalities produced are sharp, namely they are attained. This work has been motivated by the works of Ostrowski [A. Ostrowski, Über die Absolutabweichung einer differentiebaren Funcktion von ihrem Integralmittelwert, Commentarii Mathematici Helvetici 10 (1938) 226–227], 1938, and of the author [G.A. Anastassiou, Fractional Ostrowski type inequalities, Communications in Applied Analysis 7 (2) (2003) 203–208], 2003.