Ufa Mathematical Journal, vol.13, no.1, pp.119-130, 2021 (Journal Indexed in ESCI)
The role of convexity theory in applied problems, especially in optimization problems, is well known.The integral Hermite-Hadamard inequality has a special place in this theory since it provides an upper bound for the mean value of a function. In solving applied problems from different fields of science and technology, along with the classical integro-differential calculus, fractional calculus plays an imp ortant role. Alot of research is devoted to obtaining an upper bound in the Hermite-Hadamard inequality using operatörs off ractional calculus.
The article formulates and proves the identity with the participation of the fractional integration operator. Based on this identity, new generalized Hadamard-type integral inequalities are obtained for functions for which the second derivatives are convex and take values at intermediate points of the integration interval. It is shown that the upper limit of the absolute error of inequality decreases by approximately $n^{2}$ times ($n$ is the number of intermediate points). In a particular case, the obtained estimates are consistent wich are those available in the literature. The results obtained in the article can be used in further research of integro-differential fractional calculus.