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We establish various fractional convex inequalities of the Hermite-Hadamard type which generalize the previously obtained results in the literature. Various types of such inequalities are obtained and given as corollaries. The main motivation of the paper is to generalize the recently published results in terms of the (α, h − m) − p convexity with k-p Riemann Liouville fractional operator. The application of H¨olders inequality is given in tandem with the k-p fractional operator of the convex type. [ABSTRACT FROM AUTHOR]

We establish various fractional convex inequalities of the Hermite–Hadamard type with addition to many other inequalities. Various types of such inequalities are obtained, such as (p , h) fractional type inequality and many others, as the (p , h) -convexity is the generalization of the other convex inequalities. As a consequence of the (h , m) -convexity, the fractional inequality of the (s , m) -type is obtained. Many consequences of such fractional inequalities and generalizations are obtained. [ABSTRACT FROM AUTHOR]

We establish various fractional convex inequalities of the Hermite–Hadamard type with addition to many other inequalities. Various types of such inequalities are obtained, such as (p,h) fractional type inequality and many others, as the (p,h)-convexity is the generalization of the other convex inequalities. As a consequence of the (h,m)-convexity, the fractional inequality of the (s,m)-type is obtained. Many consequences of such fractional inequalities and generalizations are obtained.

Here we present low order Riemann-Liouville left and right multivariate fractional inequalities without any initial or boundary conditions. These are of Opial and Poincaré type on a spherical shell. [ABSTRACT FROM AUTHOR]

Here we present Lp, p > 1, fractional Opial type inequalities subject to high order boundary conditions. They involve the right and left Canavati type generalised fractional derivatives. These derivatives are mixed together into the balanced Canavati type generalised fractional derivative. This balanced fractional derivative is introduced and activated here for the first time. [ABSTRACT FROM AUTHOR]

Here we present basic Lp fractional integral inequalities for left and right Riemann-Liouville, generalized Riemann-Liouville, Hadamard, Erdelyi- Kober and multivariate Riemann-Liouville fractional integrals. Then we derive basic Lp fractional inequalities regarding the left Riemann-Liouville, the left and right Caputo and the left and right Canavati type fractional derivatives. [ABSTRACT FROM AUTHOR]

Here we present basic Lp fractional integral inequalities for left and right Riemann-Liouville, generalized Riemann-Liouville, Hadamard, Erdelyi-Kober and multivariate Riemann-Liouville fractional integrals. Then we derive basic Lp fractional inequalities regarding the left Riemann-Liouville, the left and right Caputo and the left and right Canavati type fractional derivatives. [ABSTRACT FROM AUTHOR]

Here we present Lp, p > 0, fractional Opial type inequalities subject to high order boundary conditions. They involve the right and left Caputo, Riemann-Liouville fractional derivatives. These derivatives are blended together into the balanced Caputo, Riemann-Liouville, respectively, fractional derivative. We give an application to a balanced fractional boundary value problem by proving uniqueness of the solution. [ABSTRACT FROM AUTHOR]

Here is introduced the concept of Caputo fractional radial derivative for a function defined on a spherical shell. Using polar coordinates we are able to derive multivariate Opial type inequalities over a spherical shell of ℝN, N ≥ 2, by studying the topic in all possibilities. Our results involve one, two or more functions. We produce many univariate Caputo fractional Opial type inequalities several of these used to establish results on the shell. We give application to prove uniqueness of solution of a general partial differential equation on the shell. Also we apply our results for Riemann-Liouville fractional derivatives. [ABSTRACT FROM AUTHOR]

Optimal upper bounds are given to the deviation of a value of a univariate or multivariate function of a Caputo fractional derivative related space from its average, over convex and compact subsets of ℝN, N ⩾ 1. In particular we work over closed intervals, rectangles, balls and spherical shells. These bounds involve the supremum and L∞ norms of related univariate or multivariate Caputo fractional derivatives of the involved functions. The derived inequalities are sharp, namely they are attained by simple functions. This work has been motivated by the works of Ostrowski [1], 1938, and of the author's Anastassiou [2], 2003 and [3], 2007. [ABSTRACT FROM AUTHOR]

Here we present univariate Sobolev-type fractional inequalities involving fractional derivatives of Canavati, Riemann-Liouville and Caputo types. The results are general Lp inequalities forward and converse on a closed interval. We give an application to a fractional ODE. We present also the mean Sobolev-type fractional inequalities. [ABSTRACT FROM AUTHOR]

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 . In particular we work over rectangles, balls and spherical shells. These bounds involve the supremum and 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. [Copyright &y& Elsevier]