Inequalities of Opial and Jensen. Improvements of Opial-type inequalities with applications to fractional calculus (Q2828644)

Opial's inequality, proved in 1960, is an important tool in the analysis of qualitative properties of solutions to differential equations. Over the last decades, a large amount of work was devoted to it and its applications: successive simplifications of the original proof, various extensions, several generalizations and discrete analogues. This monograph presents many relevant Opial-type inequalities, improvements of them, and also new results in this area. The basic definitions and a list of known Opial inequalities are presented in the first chapter. The next chapters are based on articles published in journals by the authors and some of their co-authors. Chapter 2 is devoted to the fractional integrals and fractional derivatives, in particular to the authors' results concerning composition identities for fractional derivatives. In Chapter 3, the Opial-type inequalities due to Willett, Godunova, Levin and Rozanova are extended and generalized by using Jensen's inequality. Applications to mean value theorems, Stolarsky type means, and construction of exponentially convex functions are given. In Chapter 4, some extensions of Opial-type inequalities are investigated in relation with special classes of convex functions. New inequalities are given for such functions, as well as for some fractional integrals and derivatives. Chapters 5 and 6 are devoted to generalizations and extensions of Opial-type inequalities for fractional derivatives. The best possible constants are investigated. Inequalities for integral operators and Opial-type inequalities related to a Green function are presented in Chapter 7. The last chapter is devoted to integral Opial inequalities in one or several variables and their discrete versions. Connecting the theory of Opial and Jensen inequalities with fractional calculus, it is to be expected that this monograph will open the door for further research.