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## Refinements of mean-square stochastic integral inequalities on convex stochastic processes

### Aequationes mathematicae (2016-08-01) 90: 765-772 , August 01, 2016

Recently, in the class of convex stochastic processes, Kotrys (Aequat Math 83:143–151, 2012; Aequat Math 86:91–98, 2013) proposed upper and lower bounds of mean-square stochastic integrals by using Hermite–Hadamard inequality. This paper shows that these bounds can be refined. Our results extend and refine the corresponding ones in the literature. Finally, an open problem for further investigations is given.

## Topological loops with three-dimensional solvable left translation group

### Aequationes mathematicae (2010-03-01) 79: 83-97 , March 01, 2010

We classify all connected topological loops having a three-dimensional solvable Lie group *G* as the group topologically generated by their left translations. It is surprising that to the non-nilpotent Lie group *G* having precisely one one-dimensional normal subgroup there are topological but no differentiable strongly left alternative loops.

## Integral functional equations on locally compact groups with involution

### Aequationes mathematicae (2016-10-01) 90: 967-982 , October 01, 2016

Our main goal is to introduce some integral-type generalizations of the cosine and sine equations for complex-valued functions defined on a group *G* that need not be abelian. These equations provide a joint generalization of many trigonometric type functional equations such as d’Alembert’s, Cauchy’s, Gajda’s, Kannappan’s and Van Vleck’s equations. We prove that the continuous solutions for the first type and the central continuous solutions for the second one of these equations can be expressed in terms of characters, additive maps and matrix elements of irreducible, 2-dimensional representations of the group *G*. So the theory is part of harmonic analysis on groups.

## Convexity with respect to families of means

### Aequationes mathematicae (2015-02-01) 89: 161-167 , February 01, 2015

In this paper we investigate continuity properties of functions
$${f : \mathbb {R}_+ \to \mathbb {R}_+}$$
that satisfy the (*p*, *q*)-Jensen convexity inequality
$$f\big(H_p(x, y)\big) \leq H_q(f(x), f(y)) \qquad(x, y > 0),$$
where *H*_{p} stands for the *p*th power (or Hölder) mean. One of the main results shows that there exist discontinuous multiplicative functions that are (*p*, *p*)-Jensen convex for all positive rational numbers *p*. A counterpart of this result states that if *f* is (*p*, *p*)-Jensen convex for all
$${p \in P \subseteq \mathbb {R}_+}$$
, where *P* is a set of positive Lebesgue measure, then *f* must be continuous.

## On continuous on rays solutions of a composite-type equation

### Aequationes mathematicae (2015-06-01) 89: 583-590 , June 01, 2015

Let *X* be a real linear space. We characterize solutions
$${f, g : X \rightarrow \mathbb{R}}$$
of the equation *f*(*x* + *g*(*x*)*y*) = *f*(*x*)*f*(*y*), where *f* is continuous on rays. Our result refers to papers by Brzdȩk (Acta Math Hungar 101:281–291, 2003), Chudziak (Aequat Math, doi:
10.1007/s00010-013-0228-4
, 2013) and Jabłońska (J Math Anal Appl 375:223–229, 2011).

## Posets having a unique decomposition into the minimum number of antichains

### Aequationes mathematicae (1978-02-01) 17: 41-43 , February 01, 1978

## Simultaneous Abel equations

### Aequationes mathematicae (2012-07-01) 83: 283-294 , July 01, 2012

Let $${\mathcal{S}}$$ be a set of homeomorphisms of an open interval such that the group generated by $${\mathcal{S}}$$ is disjoint, i.e., the graphs of any two distinct functions in it do not intersect. We give necessary and sufficient conditions for the system of Abel equations $$\phi(f(x))=\phi(x)+\lambda(f),\quad f \in \mathcal{S}$$ to have a continuous solution, where $${{\lambda}:\mathcal{S}\to{\mathbb {R}}}$$ is a given map. We describe this solution and show that there exists a specific map λ for which the above system always has a continuous solution. As an application we give a criterion for the embeddability of a noncyclic disjoint group of continuous functions in a continuous iteration group.

## Cauchy—Kovalevskaya theory for equations with deviating variables

### Aequationes mathematicae (1999-08-01) 58: 143-156 , August 01, 1999

### Summary.

We prove existence theorems of the Cauchy—Kovalevskaya type for linear partial differential equations with deviating variables. Our results generalize to strongly coupled systems and equations with deviations dependent on the unknown function. We give also sufficient conditions for a nontrivial case where the deviations depend on the unknown function.

## Some questions concerning superderivations on $${\mathbb {Z}}_{2}$$ Z 2 -graded rings

### Aequationes mathematicae (2017-08-01) 91: 725-738 , August 01, 2017

In this paper we pose some questions about superderivations on $${\mathbb {Z}}_{2}$$ -graded rings. Then we consider the quaternion rings and upper triangular matrix rings with special $${\mathbb {Z}}_{2}$$ -gradings and we check the answer to these questions about them.

## An alternative Cauchy functional equation on a semigroup

### Aequationes mathematicae (2013-03-01) 85: 131-163 , March 01, 2013

More than 33 years ago M. Kuczma and R. Ger posed the problem of solving the alternative Cauchy functional equation
$${f(xy) - f(x) - f(y) \in \{ 0, 1\}}$$
where
$${f : S \to \mathbb{R}, S}$$
is a group or a semigroup. In the case when the Cauchy functional equation is stable on *S*, a method for the construction of the solutions is known (see Forti in Abh Math Sem Univ Hamburg 57:215–226, 1987). It is well known that the Cauchy functional equation is not stable on the free semigroup generated by two elements. At the 44th ISFE in Louisville, USA, Professor G. L. Forti and R. Ger asked to solve this functional equation on a semigroup where the Cauchy functional equation is not stable. In this paper, we present the first result in this direction providing an answer to the problem of G. L. Forti and R. Ger. In particular, we determine the solutions
$${f : H \to \mathbb{R}}$$
of the alternative functional equation on a semigroup
$${H = \langle a, b| a^2 = a, b^2 = b \rangle }$$
where the Cauchy equation is not stable.