A simple proof is given for the convexity of log det(I +KX) in the positive definite matrix variable X ≻ 0 with a given positive semidefinite K 0. Convexity of functions of covariance matrices often plays an important role in the analysis of Gaussian channels. For example, suppose Y and Z are independent complex Gaussian nvectors with Y ∼ N(0,K) and Z ∼ N(0,X). Then, I(Y;Y + Z) = log det(I +KX)...

In this paper, we establish some Hermite-Hadamard type inequalities for function whose n-th derivatives are logarithmically convex by using Riemann-Liouville integral operator.

In this paper we find a characterization type result for (η1,η2)-convex functions. The Fejér integral inequality related to (η1,η2)-convex functions is obtained as a generalization of Fejér inequality related to the preinvex and η-convex functions. Also some Fejér trapezoid and midpoint type inequalities are given in the case that the absolute value of the derivative of considered function is (...

We study the convexity of mutual information along the evolution of the heat equation. We prove that if the initial distribution is log-concave, then mutual information is always a convex function of time. We also prove that if the initial distribution is either bounded, or has finite fourth moment and Fisher information, then mutual information is eventually convex, i.e., convex for all large ...

in the present paper, we prove subordination, superordination and sandwich-type properties of a certain integral operators for univalent functions on open unit disc, moreover the special behavior of this class is investigated.

We develop a parallel theory to that concerning the concept of integral mean value of a function, by replacing the additive framework with a multiplicative one. Particularly, we prove results which are multiplicative analogues of the Jensen and Hermite-Hadamard inequalities. 1. Introduction Many results in Real Analysis exploits the property of convexity of the subintervals I of R; (A) x; y 2 I...

Abstract In this paper, we prove that the sequences of Stirling numbers first and second kind with higher level are both Pólya frequency log-concave. Then, show some polynomials related to above q -log-convex or strongly -log-convex. Furthermore, establish linear transformation 2 preserves log-convexity.

We prove that the convex least squares estimator (LSE) attains a n-1/2 pointwise rate of convergence in any region where the truth is linear. In addition, the asymptotic distribution can be characterized by a modified invelope process. Analogous results hold when one uses the derivative of the convex LSE to perform derivative estimation. These asymptotic results facilitate a new consistent test...