In this paper, we present a general version of operator Bell-man inequality. Also, the refinement of inequality due to J. Aujla and F. Silva for the convex functions is given as well.

The aim of inequalities is to develop tools for analyzing the problems in pure and applied mathematics. Our primary objective in this research is to introduce some new type inequalities connected with Caputo fractional derivative. In the light of the operator, we extend and generalize some important inequalities to this fractional calculus dealing with synchronous functions.

Hardy–Sobolev–type inequalities associated with the operator L := x .Ñ are established, using an improvement to the Sobolev embedding theorem obtained by M. Ledoux. The analysis involves the determination of the operator semigroup {e_tL*L} t>0.

In this paper, we establish new cebysev type integral inequalities involving functions whose derivatives belong to Lp spaces via certain integral identities.

For 0 < r < 1 and 1£ p£q < ¥ we find necessary and sufficient conditions for the validity of the following inequality: (òba u(x) (òxa | g(x)-g (t)| r w(t)dt)q/r dx)1/q £ c(òba v(x)| g’ (x)|p dx)1/pwhere u(·), v(·), and w(·) are weight functions.

In this paper, we present some re nements of the famous Young type inequality. As application of our result, we obtain some matrix inequalities for the Hilbert-Schmidt norm and the trace norm. The results obtained in this paper can be viewed as re nement of the derived results by H. Kai [Young type inequalities for matrices, J. East China Norm. Univ. 4 (2012) 12{17].

Some new refined multidimensional Hardy-type inequalities for p³2 and their duals are derived and discussed. Moreover, these inequalities hold in the reversed direction when 1 < p £ 2. The results obtained are based mainly on some new results for superquadratic and subquadratic functions. In particular, our results further extend the recent results in [J.A. Oguntuase and L.-E. Persson, Refinement of Hardy’s inequalities via superquadratic and subquadratic functions, J. Math. Anal. Appl., 339 (2008), no. 2, 1305– 1312] to a multidimensional setting.

In this paper, we establish some Bernstein type inequalities for the complex polynomial. Our results constitute generalizations and re nements of some well-known polynomial inequalities.