Convex Functions Of Several Variables Research Articles

Multivariate Opial-type Inequalities on Time Scales

Opial inequality was developed to provide bounds for integral of functions and their derivatives. It has become an indispensable tool in the theory of mathematical analysis due to its usefulness. A refined Jensen inequality for multivariate functions is employed to establish new Opial-type inequalities for convex functions of several variables on time scale.

Notes on judgment criteria of convex functions of several variables

By transferring the judgment of convex functions of several variables into the judgment of convex functionsof one variable, the authors discuss the convexity of some convex functions of several variables.

Trapezium-Type Inequalities for an Extension of Riemann–Liouville Fractional Integrals Using Raina’s Special Function and Generalized Coordinate Convex Functions

In this paper, the authors analyse and study some recent publications about integral inequalities related to generalized convex functions of several variables and the use of extended fractional integrals. In particular, they establish a new Hermite–Hadamard inequality for generalized coordinate ϕ-convex functions via an extension of the Riemann–Liouville fractional integral. Furthermore, an interesting identity for functions with two variables is obtained, and with the use of it, some new extensions of trapezium-type inequalities using Raina’s special function via generalized coordinate ϕ-convex functions are developed. Various special cases have been studied. At the end, a brief conclusion is given as well.

Another refinement of the right-hand side of the Hermite\u2013Hadamard inequality for simplices

We establish a new refinement of the right-hand side of the Hermite–Hadamard inequality for convex functions of several variables defined on simplices.

More on operator monotone and operator convex functions of several variables

Let C1,C2,…,Ck be positive matrices in Mn and f be a continuous real-valued function on [0,∞). In addition, consider Φ as a positive linear functional on Mn and defineϕ(t1,t2,t3,…,tk)=Φ(f(t1C1+t2C2+t3C3+…+tkCk)), as a k variables continuous function on [0,∞)×…×[0,∞). In this paper, we show that if f is an operator convex function of order mn, then ϕ is a k variables operator convex function of order (n1,…,nk) such that m=n1n2…nk. Also, if f is an operator monotone function of order nk+1, then ϕ is a k variables operator monotone function of order n. In particular, if f is a non-negative operator decreasing function on [0,∞), then the function t→Φ(f(A+tB)) is an operator decreasing and can be written as a Laplace transform of a positive measure.

Dynamic Inequalities for Convex Functions Harmonized on Time Scales

We present here some general fractional Schl?milch’s type and Rogers-H?lder’s type dynamic inequalities for convex functions harmonized on time scales. First we present general fractional Schl?milch’s type dynamic inequalities and generalize it for convex functions of several variables by using Bernoulli’s inequality, generalized Jensen’s inequality and Fubini’s theorem on diamond-α calculus. To conclude our main results, we present general fractional Rogers-H?lder’s type dynamic inequalities for convex functions by using general fractional Schl?milch’s type dynamic inequality on diamond-α calculus for pi>1 with .

A refinement of the left-hand side of Hermite-Hadamard inequality for simplices

In this paper, we establish a new refinement of the left-hand side of Hermite-Hadamard inequality for convex functions of several variables defined on simplices.

Certain Inequalities for Convex Functions

This is a review paper on some new inequalities for convex functions of one and several variables. The most important result presented for convex functions of one variable is the extension of Jensen’s inequality to affine combinations. The most interesting results presented for convex functions of several variables refer to inequalities concerning simplexes and its cones. Mathematics subject classification (2010): 26A51, 26B25, 52A20, 52B11.

Some operator convex functions of several variables

We obtain operator concavity (convexity) of some functions of two or three variables by using perspectives of regular operator mappings of one or several variables. As an application, we obtain, for 0<p<1, concavity, respectively convexity, of the Fréchet differential mapping associated with the functions t→t1+p and t→t1−p.

Integral Variants of Jensen's Inequality for Convex Functions of Several Variables

The paper focuses on the derivation of the integral variants of Jensen's inequality for convex functions of several variables. The work is based on the integral method, using convex combinations as input, and set barycentres as output.

The Applications of Functional Variants of Jensen's Inequality

The paper is inspired by McShane's results on the functional form of Jensen's inequality for convex functions of several variables. The work is focused on applications and generalizations of this important result. At that, the generalizations of Jensen's inequality are obtained using the positive linear functionals.

Integral, discrete and functional variants of Jensen's inequality

We deal with convex functions on bounded closed convex sets with common barycenter. More precisely, the integral arithmetic means of convex function $f$ are compared on the sets $A$ and $B$ with $A\subseteq B$. The paper shows series of inequalities hold for convex function of one variable, but it doesn't hold generally for convex functions of several variables.

Some majorization inequalities for convex functions of several variables

Some majorization inequalities for convex functions of several variables

An algorithm for linearizing convex extremal problems

This paper suggests a method of approximating the solution of minimization problems for convex functions of several variables under convex constraints is suggested. The main idea of this approach is the approximation of a convex function by a piecewise linear function, which results in replacing the problem of convex programming by a linear programming problem. To carry out such an approximation, the epigraph of a convex function is approximated by the projection of a polytope of greater dimension. In the first part of the paper, the problem is considered for functions of one variable. In this case, an algorithm for approximating the epigraph of a convex function by a polygon is presented, it is shown that this algorithm is optimal with respect to the number of vertices of the polygon, and exact bounds for this number are obtained. After this, using an induction procedure, the algorithm is generalized to certain classes of functions of several variables. Applying the suggested method, polynomial algorithms for an approximate calculation of the -norm of a matrix and of the minimum of the entropy function on a polytope are obtained.Bibliography: 19 titles.

A variant of Jensen's inequality for convex functions of several variables

Two Jensen's type inequalities for convex functions defined on R k which involve elements of convex hulls are given. As their consequences the comparison theorem for weighted L-conjugate means and a Mercer's result are obtained.

Jensen’s operator inequality for functions of several variables

The operator convex functions of several variables are characterized in terms of a non-commutative generalization of Jensen’s inequality, extending previous results of F. Hansen and G.K. Pedersen for functions of one variable and of F. Hansen for functions of two variables.

A Correction to: 'Operator Convex Functions of Several Variables'

A Correction to: 'Operator Convex Functions of Several Variables'

Operator Convex Functions of Several Variables

The functional calculus for functions of several variables associates to each tuple x= (x_1, ⋯, x_k) of selfadjoint operators on Hilbert spaces H_1, ⋯,H_k an operator f(x) in the tensor product B(H_1)⊗ ⋯ ⊗B(H_k) . We introduce the notion of generalized Hessian matrices associated with f . Those matrices are used as the building blocks of a structure theorem for the second Fréchet differential of the map x → f(x) . As an application we derive that functions with positive semi-definite generalized Hessian matrices of arbitrary order are operator convex. The result generalizes a theorem of Kraus [15] for functions of one variable.

Some inequalities for generalized convex functions of several variables

Some inequalities for generalized convex functions of several variables

Lp approximation by subsets of convex functions of several variables

If S is a bounded convex subset of R m , the problem is to find a best approximation to a function in L p ( S), 1 ⩽ p ⩽ ∞, by an arbitrary subset of convex functions. An existence theorem for a best approximation is established under a certain condition on the subset. In particular, a best convex approximation exists. Also investigated are properties of norm-bounded subsets and L p -convergent sequences of convex functions.