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## Über dieses Buch

This monograph presents recent and original work of the author on inequalities in real, functional and fractional analysis. The chapters are self-contained and can be read independently, they include an extensive list of references per chapter.

The book’s results are expected to find applications in many areas of applied and pure mathematics, especially in ordinary and partial differential equations and fractional differential equations. As such this monograph is suitable for researchers, graduate students, and seminars of the above subjects, as well as Science and Engineering University libraries.

## Inhaltsverzeichnis

### Chapter 1. Fractional Polya Integral Inequality

Abstract
Here we establish a fractional Polya type integral inequality with the help of generalised right and left fractional derivatives.
George A. Anastassiou

### Chapter 2. Univariate Fractional Polya Integral Inequalities

Abstract
Here we establish a series of various fractional Polya type integral inequalities with the help of generalised right and left fractional derivatives.
George A. Anastassiou

### Chapter 3. About Multivariate General Fractional Polya Integral Inequalities

Abstract
Here we present a set of multivariate general fractional Polya type integral inequalities on the ball and shell.
George A. Anastassiou

### Chapter 4. Balanced Canavati Fractional Opial Inequalities

Abstract
Here we present $$L_{p}$$, $$p>1$$, fractional Opial type inequalities subject to high order boundary conditions.
George A. Anastassiou

### Chapter 5. Fractional Representation Formulae Using Initial Conditions and Fractional Ostrowski Inequalities

Abstract
Here we present very general fractional representation formulae for a function in terms of the fractional Riemann-Liouville integrals of different orders of the function and its ordinary derivatives under initial conditions.
George A. Anastassiou

### Chapter 6. Basic Fractional Integral Inequalities

Abstract
Here we present basic fractional integral inequalities for left and right Riemann-Liouville, generalized Riemann-Liouville, Hadamard, Erdelyi-Kober and multivariate Riemann-Liouville fractional integrals.
George A. Anastassiou

### Chapter 7. Harmonic Multivariate Ostrowski and Grüss Inequalities Using Several Functions

Abstract
Here we derive very general multivariate Ostrowski and Grüss type inequalities for several functions by involving harmonic polynomials.
George A. Anastassiou

### Chapter 8. Fractional Ostrowski and Grüss Inequalities Using Several Functions

Abstract
Using Caputo fractional left and right Taylor formulae we establish mixed fractional Ostrowski and Grüss type inequalities involving several functions.
George A. Anastassiou

### Chapter 9. Further Interpretation of Some Fractional Ostrowski and Grüss Type Inequalities

Abstract
We further interpret and simplify earlier produced fractional Ostrowski and Grüss type inequalities involving several functions.
George A. Anastassiou

### Chapter 10. Multivariate Fractional Representation Formula and Ostrowski Inequality

Abstract
Here we derive a multivariate fractional representation formula involving ordinary partial derivatives of first order. Then we prove a related multivariate fractional Ostrowski type inequality with respect to uniform norm.
George A. Anastassiou

### Chapter 11. Multivariate Weighted Fractional Representation Formulae and Ostrowski Inequalities

Abstract
Here we derive multivariate weighted fractional representation formulae involving ordinary partial derivatives of first order. Then we present related multivariate weighted fractional Ostrowski type inequalities with respect to uniform norm.
George A. Anastassiou

### Chapter 12. About Multivariate Lyapunov Inequalities

Abstract
We transfer here basic univariate Lyapunov inequalities to the multivariate setting of a shell by using the polar method.
George A. Anastassiou

### Chapter 13. Ostrowski Type Inequalities for Semigroups

Abstract
Here we present Ostrowski type inequalities on Semigroups for various norms. We apply our results to the classical diffusion equation and its solution, the Gauss-Weierstrass singular integral. It follows [3].
George A. Anastassiou

### Chapter 14. About Ostrowski Inequalities for Cosine and Sine Operator Functions

Abstract
Here we present Ostrowski type inequalities on Cosine and Sine Operator Functions for various norms.
George A. Anastassiou

### Chapter 15. About Hilbert-Pachpatte Inequalities for Semigroups, Cosine and Sine Operator Functions

Abstract
Here we present Hilbert-Pachpatte type general Lp inequalities regarding Semigroups, Cosine and Sine Operator functions.
George A. Anastassiou

### Chapter 16. About Ostrowski and Landau Type Inequalities for Banach Space Valued Functions

Abstract
Very general univariate Ostrowski type inequalities are presented regarding Banach space valued functions.
George A. Anastassiou

### Chapter 17. Multidimensional Ostrowski Type Inequalities for Banach Space Valued Functions

Abstract
Here we are dealing with smooth functions from a real box to a Banach space. For these we establish vector multivariate sharp Ostrowski type inequalities to all possible directions.
George A. Anastassiou

### Chapter 18. About Fractional Representation Formulae and Right Fractional Inequalities

Abstract
Here we prove fractional representation formulae involving generalized fractional derivatives, Caputo fractional derivatives and Riemann-Liouville fractional derivatives.
George A. Anastassiou

### Chapter 19. About Canavati Fractional Ostrowski Inequalities

Abstract
Here we present Ostrowski type inequalities involving left and right Canavati type generalised fractional derivatives. Combining these we obtain fractional Ostrowski type inequalities of mixed form. Then we establish Ostrowski type inequalities for ordinary and fractional derivatives involving complex valued functions defined on the unit circle.
George A. Anastassiou

### Chapter 20. The Most General Fractional Representation Formula for Functions and Consequences

Abstract
Here we present the most general fractional representation formulae for a function in terms of the most general fractional integral operators due to Kalla [46]. The last include most of the well-known fractional integrals such as of Riemann-Liouville, Erdé lyi-Kober and Saigo, etc. Based on these we derive very general fractional Ostrowski type inequalities. It follows [2].
George A. Anastassiou

### Chapter 21. Rational Inequalities for Integral Operators Using Convexity

Abstract
Here we present integral inequalities for convex and increasing functions applied to products of ratios of functions and other important mixtures. As applications we derive a wide range of fractional inequalities of Hardy type. They involve the left and right Riemann-Liouville fractional integrals and their generalizations, in particular the Hadamard fractional integrals.
George A. Anastassiou

### Chapter 22. Fractional Integral Inequalities with Convexity

Abstract
Here we present general integral inequalities involving convex and increasing functions applied to products of functions.
George A. Anastassiou

### Chapter 23. Vectorial Inequalities for Integral Operators Involving Ratios of Functions Using Convexity

Abstract
Here we present vectorial integral inequalities for products of multivariate convex and increasing functions applied to vectors of ratios of functions. As applications we derive a wide range of vectorial fractional inequalities of Hardy type.
George A. Anastassiou

### Chapter 24. About Vectorial Splitting Rational $$L_{p}$$ L p Inequalities for Integral Operators

Abstract
Here we present $$L_{p}$$, $$p>1$$, vectorial integral inequalities for products of multivariate convex and increasing functions applied to vectors of ratios of functions.
George A. Anastassiou

### Chapter 25. About Separating Rational $$L_{\!p}$$ L p Inequalities for Integral Operators

Abstract
Here we present $$L_{p}$$, $$p>1$$, integral inequalities for convex and increasing functions applied to products of ratios of functions and other important mixtures. As applications we derive a wide range of fractional inequalities of Hardy type.
George A. Anastassiou

### Chapter 26. About Vectorial Hardy Type Fractional Inequalities

Abstract
Here we present vectorial integral inequalities for products of multivariate convex and increasing functions applied to vectors of functions.
George A. Anastassiou

### Chapter 27. About Vectorial Fractional Integral Inequalities Using Convexity

Abstract
Here we present vectorial general integral inequalities involving products of multivariate convex and increasing functions applied to vectors of functions. As specific applications we derive a wide range of vectorial fractional inequalities of Hardy type.
George A. Anastassiou

### Backmatter

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