Wavelet and Linear Algebra

Abstract Abstract: In this paper, we establish extensions of Jensen’s discrete inequality for the class of p-convex functions. Also, we give lower and upper bounds for this inequality. We apply these results in information theory and obtain new and strong bounds for Shannon’s entropy of a probability distribution. Also, We give some applications.

Abstract In this paper, we generalize the Jensen's inequality for $m$-convex functions and we present a correction of Jensen's inequality which is a better than the generalization of this inequality for $m$-convex functions. ّFinally we have found new lower and upper bounds for Jensen's discrete inequality.

Abstract Shannon's entropy plays an important role in information theory, dynamical systems and thermodynamics. In this paper we applying Jensen's inequality in information theory and we obtain some results for the Shannon's entropy of random variables and Shannon's entropy of stochastic process. Also we obtain upper bound and lower bound for Shannon's entropy of information sources.

Abstract Convex sets and convex functions play a fundamental role in the development of various fields
of pure and applied mathematics.  Recently, many new generalizations of inequalities with respect to Hermite-Hadamard  have been proposed in the literature. In this paper,  some  new  inequalities of the Hermite-Hadamard type for differentiable convex functions are given. These new inequalities are based on the second derivative functions.

Abstract It is well known that if a real valued convex function on a compact convex domain
contained in the real numbers attains its maximum,
then it does so at least at one extreme point of its domain.
In this paper,
we consider a matrix convex function on a compact and $C^*$-convex set generated by self--adjoint matrices.
An important issue is so that this function on a compact and $C^*$-convex domain attains its maximum at a $C^*$-extreme point.