SpringerBriefs in Mathematics For further volumes: http://www.springer.com/series/10030
George A. Anastassiou Advances on Fractional Inequalities 123
George A. Anastassiou Department of Mathematical Sciences University of Memphis Memphis, TN 38152 USA ganastass@memphis.edu ISSN 2191-8198 e-issn 2191-8201 ISBN 978-1-4614-0702-7 e-isbn 978-1-4614-0703-4 DOI 10.1007/978-1-4614-0703-4 Springer New York Dordrecht Heidelberg London Library of Congress Control Number: 2011932688 Mathematics Subject Classification (2010): 26A33, 26D10, 26D15 George A. Anastassiou 2011 All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer Science+Business Media, LLC, 233 Spring Street, New York, NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks, and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights. Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com)
To my wife Koula and my daughters Angela and Peggy The measure of success for a person is the magnitude of his/her ability to convert negative conditions to positive ones and achieve goals The author
Preface This short monograph is a spin-off of the author s Fractional Differentiation Inequalities, a research monograph published by Springer, New York, 2009. It continues and complements the earlier book to various interesting and important directions. In this short monograph we use primarily the Caputo fractional derivative, as the most important in applications, and we present first fractional differentiation inequalities of Opial type where we involve the so-called balanced fractional derivatives. We continue with right and mixed fractional differentiation Ostrowski inequalities in the univariate and multivariate cases. Then we present right and left, as well as mixed, Landau fractional differentiation inequalities in the univariate and multivariate cases. The inequalities are given for various norms. Fractional differentiation inequalities are by themselves an important and great mathematical topic for research. Furthermore they have many applications, the most important ones are in establishing uniqueness of solution in fractional differential equations and systems and in fractional partial differential equations. Also they provide upper bounds to the solutions of the above equations. In this brief monograph we give several applications. Each chapter is self-contained and can be read independently of the others and several graduate courses and seminars can be taught out of this monograph. The final preparation of this book took place in Memphis, USA, during 2010 2011. Fractional calculus has become very useful over the last 40 years due to its many applications in almost all applied sciences. We see now applications in acoustic wave propagation in inhomogeneous porous material, diffusive transport, fluid flow, dynamical processes in self-similar structures, dynamics of earthquakes, optics, geology, viscoelastic materials, biosciences, bioengineering, medicine, economics, probability and statistics, astrophysics, chemical engineering, physics, splines, tomography, fluid mechanics, electromagnetic waves, nonlinear control, signal processing, control of power electronics, converters, chaotic dynamics, polymer vii
viii Preface science, proteins, polymer physics, electrochemistry, statistical physics, rheology, thermodynamics, neural networks, etc. By now almost all fields of research in science and engineering use fractional calculus as better describing them. So as expected this book being a part of fractional calculus is useful for researchers and graduate students for research, seminars and advanced graduate courses, in pure and applied mathematics, engineering, and all other applied sciences. I would like to thank my family members for their support and for their tolerance to accept my continuous mathematics habit. Also I am greatly indebted and thankful to my Ph.D. student Razvan Mezei for the heroic and fantastic technical preparation of the manuscript in a very short time. Memphis, TN, USA George A. Anastassiou
Contents 1 Opial-Type Inequalities for Balanced Fractional Derivatives... 1 1.1 Introduction... 1 1.2 Background... 2 1.3 Main Results... 3 References... 18 2 Univariate Right Caputo Fractional Ostrowski Inequalities... 21 2.1 Introduction... 21 2.2 Main Results... 22 References... 27 3 Multivariate Right Caputo Fractional Ostrowski Inequalities... 29 3.1 Introduction... 29 3.2 Main Results... 30 References... 39 4 Univariate Mixed Fractional Ostrowski Inequalities... 41 4.1 Introduction... 41 4.2 Main Results... 42 References... 48 5 Multivariate Radial Mixed Fractional Ostrowski Inequalities... 51 5.1 Introduction... 51 5.2 Main Results... 52 References... 60 6 Shell Mixed Caputo Fractional Ostrowski Inequalities... 63 6.1 Introduction... 63 6.2 Main Results... 64 References... 75 7 Left Caputo Fractional Uniform Landau Inequalities... 77 7.1 Introduction... 77 7.2 Main Results... 78 ix
x Contents 7.3 Addendum... 83 References... 84 8 Left Caputo Fractional L p -Landau-Type Inequalities... 85 8.1 Introduction... 85 8.2 Main Results... 86 References... 91 9 Right Caputo Fractional L p -Landau-Type Inequalities... 93 9.1 Introduction... 93 9.2 Main Results... 94 References... 99 10 Mixed Caputo Fractional L p -Landau-Type Inequalities... 101 10.1 Introduction... 101 10.2 Main Results... 102 References... 110 11 Multivariate Caputo Fractional Landau Inequalities... 113 11.1 Introduction... 113 11.2 Main Results... 114 References... 121