Szymon Wąsowicz

Support-type properties of convex functions of higher order and Hadamard-type inequalities

Abstract It is well known that every convex function f : I → R (where I ⊂ R is an interval) admits an affine support at every interior point of I (i.e. for any x 0 ∈ Int I there exists an affine function a : I → R such that a ( x 0 ) = f ( x 0 ) and a ⩽ f on I). Convex functions of higher order (precisely of an odd order) have a similar property: they are supported by the polynomials of degree no greater than the order of convexity. In this paper the attaching method is developed. It is applied to obtain the general result—Theorem 2, from which the mentioned above support theorem and some related properties of convex functions of higher (both odd and even) order are derived. They are applied to obtain some known and new Hadamard-type inequalities between the quadrature operators and the integral approximated by them. It is also shown that the error bounds of quadrature rules follow by inequalities of this kind.

Functions Generating Strongly Schur-Convex Sums

The notion of strongly Schur-convex functions is introduced and functions generating strongly Schur-convex sums are investigated. The results presented are counterparts of the classical Hardy–Littlewood–Polya majorization theorem and the theorem of Ng characterizing functions generating Schur-convex sums. It is proved, among others, that for some (for every) n≥2, the function F(x 1,…,x n )=f(x 1)+⋯+f(x n ) is strongly Schur-convex with modulus c if and only if f is of the form f(x)=g(x)+a(x)+c∥x∥2, where g is convex and a is additive.

On Functional Equations Connected with Quadrature Rules

Abstract The functional equations of the form are considered. They are connected with quadrature rules of the approximate integration. We show that such equations characterize polynomials in the class of continuous functions. It is also shown that if the number of components is sufficiently small, then the continuity is forced by the equation itself. Unique solvability of the considered problem are established.

Some functional equations characterizing polynomials

Abstract We present a method of solving functional equations of the type where f, F: P → P are unknown functions acting on an integral domain P and parameteres are given. We prove that under some assumptions on the parameters involved, all solutions to such kind of equations are polynomials. We use this method to solve some concrete equations of this type. For example, the equation (1) for f, F: ℝ → ℝ is solved without any regularity assumptions. It is worth noting that (1) stems from a well-known quadrature rule used in numerical analysis.

Hermite-Hadamard inequalities for convex set-valued functions

Abstract The following version of the weighted Hermite–Hadamard inequalities for set-valued functions is presented: Let Y be a Banach space and F : [a, b] → cl(Y) be a continuous set-valued function. If F is convex, then F(xμ)⊃1μ([a,b])∫abF(x) dμ(x)⊃b−xμb−aF(a)+xμ−ab−aF(b),

On the stability of the equation stemming from Lagrange MVT

F(x_\mu ) \supset {1 \over {\mu ([a,b])}}\int\limits_a^b {F(x)\;d\mu (x) \supset {{b - x_\mu } \over {b - a}}} F(a) + {{x_\mu - a} \over {b - a}}F(b),

On some inequality of Hermite-Hadamard type

where μ is a Borel measure on [a, b] and xμ is the barycenter of μ on [a, b]. The converse result is also given.

Probabilistic characterization of strong convexity

A new family of fuzzy implications, motivated by classic Sheffer stroke operator, is introduced. Sheffer stroke, which is a negation of a conjunction and is called NAND as well, is one of the two operators that can be used by itself, without any other logical operators, to constitute a logical formal system. Classical implication can be presented just by Sheffer stroke operator in two ways which leads to two new families of fuzzy implication functions. It turns out that one of them is mainly a subclass of QL-operations, while the other one, called in our paper as SS\(_{qq}\)-implications, is independent of other well-known families of fuzzy implications. Basic properties of Sheffer stroke implications are also analysed.