In this book the author presents the Opial, Poincaré, Sobolev, Hilbert, and Ostrowski fractional differentiation inequalities. Results for the above are derived using three different types of fractional derivatives, namely by Canavati, Riemann-Liouville and Caputo. The univariate and multivariate cases are both examined. Each chapter is self-contained. The theory is presented systematically along with the applications. The application to information theory is also examined.

This monograph is suitable for researchers and graduate students in pure mathematics. Applied mathematicians, engineers, and other applied scientists will also find this book useful.

From the review:

"Professor Anastassiou considers three definitions of fractional derivatives. … The list of references runs to four hundred and twelve items. … for a specialist in fractional derivative inequalities it would be indispensible." (Underwood Dudley, The Mathematical Association of America, September, 2009)

"In this book, Anastassiou chooses to concentrate on three special cases: operators of Riemann-Liouville type that have been used very intensively by the pure mathematics community, operators of Caputo’s type that have proven to be very important in many applications … and the relatively little-known Canavati operators. For these types of operators, he provides generalizations of the classical differentiation inequalities … . all the chapters are self-contained. … a very useful and easy-to-read reference for readers who are looking for that." (Kai Diethelm, ACM Computing Reviews, November, 2009)

“This book is the first edition of the work on a subject which is not dealt with in a text form, as this is, before. … each chapter has almost identical format with detailed proof of theorems, which will be proved fruitful for both young and matured researchers to understand the subject. … References at the end is exhaustive and fruitful. … The present monograph centers its attention mainly in the aspect of the fractional inequalities and contains a wealth of interesting material … .” (P. K. Banerji, Zentralblatt MATH, 2010)

“The text will be very useful to researchers in fractional calculus and its applications to the existence and uniqueness problems of fractional differential and partial differential equations.” (R. N. Kalia, Mathematical Reviews, Issue 2010 g)