The main motivation for preparation this presentation is to show some aspects of generalizations, refinements, and variants of famous Hardy, Turan, Sidon, Bernstein and related inequalities, involving different type of analytical kernels in certain fractional calculus operators. Namely, integral operators with general non-negative kernel on measure spaces with positive finite measure are considered and some new Hardy type inequalities for convex functions are obtained. The Turan type inequalities are illustrated for two parameter Mittag-Leffler functions. Particularly we have considered Riemann-Liouville fractional integral, Hilfer fractional derivative, Hilfer-Prabhakar derivative, Hadamard fractional integrals, and Weyl fractional derivative. These operators are fundamental part of fractional calculus and have been of great importance during the last few decades. Recently, Sandev and Tomovski (2019) added remarkable contribution in the theory of fractional differential equations and models in statistical mechanics, stochastic processes, anomalous diffusion, relaxation and wave models, etc. More details on analytical inequalities in fractional calculus, can be found in the new book, published in 2020 by Iqbal, Pecaric and Tomovski.  The presentation is finished by fractional generalization of the Sidon type inequality by using Weyl fractional derivative operator and Bernstein type inequality for trigonometric polynomials in L^p -space. Furthermore new necessary and sufficient conditions for L^1- convergence of fractional derivatives of trigonometric series are presented.