engleski

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

In 1960, the Polish mathematician Z. Opial proved integral inequality, which is recognized as a fundamental result in the analysis of qualitative properties of a solution of differential equations. Over the last five decades, an enormous amount of work has been done on the Opial inequality: several simplifications of the original proof, various extensions, generalizations and discrete analogues. Motivated with Opial-type inequalities, together with Jensen's inequality, we improve some known results and obtain new, interesting inequalities. For such inequalities we construct functionals and give its mean value theorems. These Cauchy type mean value theorems are used for Stolarsky type means, all defined by the observed inequalities, and also, they are used to prove the n-exponential convexity for the functionals. We study Opial--type inequalities not only for ordinary derivatives, but also for fractional derivatives which leads us to the fractional calculus. It is a theory of differential and integral operators of non-integer order that has become very useful due to its many applications in almost all the applied sciences. We study the Riemann-Liouville fractional integrals and three types of fractional derivatives (the Riemann-Liouville, the Caputo and the Canavati type), in the real domain. Obtaining improvements of composition identities for the above mentioned fractional derivatives, we apply them on the fractional differentiation inequalities that have applications in the fractional differential equations ; the most important ones are in establishing the uniqueness of the solution of initial problems and giving upper bounds to their solutions. We give refinements, generate new extensions and generalizations of some known Opial-type inequalities, investigate the possibility of obtaining the best possible constant, compare results obtained by different methods and present some new inequalities involving fractional integrals and fractional derivatives.

Opial inequality; Jensen inequality; fractional calculus

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