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1

(C-K)-strongly n-convex functions in linear spaces

Kominek Z.

Commentationes Mathematicae

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2013

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Vol. 53, No. 2

89--96

EN

The notion of strongly n-convex functions with a control symmetric n-linear function φ in the class of functions acting from one real linear space to another one are introduced. Some connections between such functions and n-convex functions are also given.

2

Continuity of real valued subquadratic functions

Kominek Z., Troczka-Pawelec K.

Commentationes Mathematicae

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2011

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Vol. 51, [Z] 1

71-75

EN

The aim of this paper is to prove a regularity theorem for real valued subquadratic mappings that are solutions of the inequality [formula], where X = (X, +) is a topological group.

3

Remarks on the stability of some quadratic functional equations

Kominek Z.

Opuscula Mathematica

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2008

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Vol. 28, no. 4

517-527

EN

Stability problems concerning the functional equations of the form f(2x + y) = 4ƒ(x) + ƒ(y) + ƒ(x + y) - ƒ(x - y), and ƒ(2x + y) + ƒ(2x -y) = 8ƒ(x) + 2ƒ(y) are investigated. We prove that if the norm of the difference between the LHS and the RHS of one of equations (1) or (2), calculated for a function g is say, dominated by a function φ in two variables having some standard properties then there exists a unique solution ƒ of this equation and the norm of the difference between g and ƒ is controlled by a function depending on φ.

4

Some remarks on subquadratic functions

Kominek Z., Troczka K.

Demonstratio Mathematica

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2006

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Vol. 39, nr 4

751-758

EN

Some basic properties of subquadratic functions, i.e. functions fulfilling the inequality phi (x + y) + phi(x - y) is less than or equal to 2 phi(x) + 2 phi(y) are proved. In this note X be always a real linear space and R be denotes the set of all reals. Every function phi : X approaches R satisfying the following inequality (1) phi(x + y) + phi(x - y) is less than or equal 2phi(x) + 2phi(y), x, y is an element of X, is called subquadratic. If the sign "is less than or equal to" is replaced by "is more than or equal to" then phi is called superquadratic and if we have "=" instead of " is less than or equal to" in (1) then we say that phi is quadratic function. There are plenty papers devoted to quadratic functions [1], [2], [3] (and references there). In this note some properties of the solutions of (1) will be proved, particularly we will investigate nonpositive solutions of (1). Also interesting question of finding sucient conditions on subquadratic function to be quadratic one will be considered.

5

On Hyers-Ulam stability of the Pexider equation

Kominek Z.

Demonstratio Mathematica

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2004

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Vol. 37, nr 2

373--376

6

On functionals with the Cauchy difference bounded by a homogeneous functional

Baron K., Kominek Z.

Bulletin of the Polish Academy of Sciences. Mathematics

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2003

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Vol. 51, no 3

301-307

EN

Assume V is a real vector space. We consider the functional inequality f(x+y)-f(x)-f(y) is more than or equal to Phi(x,y) (x,y is an element of V) for f : V --> R and Phi : V x V --> R such that Phi(x, .) is homogeneous for every x is an element of V and we provide conditions on f which force the representation f(x)=L(x)+B(x,x) (x is an element of V) with a linear L : V --> R and a bilinear and symmetric B : V x V --> R.