Related papers: Characterizations of monotone right continuous fun…

Associativity of a two-place function $T: [0,1]^2\rightarrow [0,1]$ defined by $T(x,y)=f^{(-1)}(F(f(x),f(y)))$ where $F:[0,\infty]^2\rightarrow[0,\infty]$ is an associative function, $f: [0,1]\rightarrow [0,\infty]$ is a monotone function…

This article introduces a weak pseudo-inverse of a monotone function, which is applied to characterize the associativity of a two-place function $T: [0,1]^2\rightarrow [0,1]$ defined by $T(x,y)=t^{[-1]}(F(t(x),t(y)))$ where…

This article characterizes the associativity of two-place functions $T: [0,1]^2\rightarrow [0,1]$ defined by $T(x,y)=f^{(-1)}(F(f(x),f(y)))$ where $F:[0,1]^2\rightarrow[0,1]$ is a triangular norm (even a triangular subnorm), $f:…

Let $T^*:[0,1]^2\rightarrow[0,1]$ be a continuous, non-decreasing and associative function with neutral element, $f: [0,1]\rightarrow [0,1]$ be a strictly monotone function and $f^{(-1)}:[0,1]\rightarrow[0,1]$ be the pseudo-inverse of $f$.…

Let a function $F: [0,1]^2\rightarrow [0,1]$ be given by $F(x,y)= f^{(-1)}(T(f(x), f(y)))$ where $f :[0,1]\rightarrow [0,1]$ is a monotone function, $f^{(-1)}$ is the pseudo-inverse of $f$ and $T$ is a triangular norm. This article…

For topological spaces $X$ and $Y$, a (not necessarily continuous) function $f:X \rightarrow Y$ naturally induces a functor from the category of closed subsets of $X$ (with morphisms given by inclusions) to the category of closed subsets of…

We prove that a continuous function $f:(0,\infty) \to (0,\infty)$ is operator monotone increasing if and only if $f(A \: !_t \: B) \leqs f(A) \: !_t \: f(B)$ for any positive operators $A,B$ and scalar $t \in [0,1]$. Here, $!_t$ denotes the…

The matrix convexity and the matrix monotony of a real $C^1$ function $f$ on $(0,\infty)$ are characterized in terms of the conditional negative or positive definiteness of the Loewner matrices associated with $f$, $tf(t)$, and $t^2f(t)$.…

We characterize real functions $f$ on an interval $(-\alpha,\alpha)$ for which the entrywise matrix function $[a_{ij}] \mapsto [f(a_{ij})]$ is positive, monotone and convex, respectively, in the positive semidefiniteness order. Fractional…

A real arithmetic function f is multiplicatively monotonous if f (mn) -- f (m) has constant sign for m, n positive integers. Properties and examples of such functions are discussed, with applications to positive hermitian…

We mainly establish a monotonicity property between some special Riemann sums of a convex function $f$ on $[a,b]$, which in particular yields that $\frac{b-a}{n+1}\sum_{i=0}^n f\left(a+i\frac{b-a}{n}\right)$ is decreasing while…

If $\sigma$ is a symmetric mean and $f$ is an operator monotone function on $[0, \infty)$, then $$f(2(A^{-1}+B^{-1})^{-1})\le f(A\sigma B)\le f((A+B)/2).$$ Conversely, Ando and Hiai showed that if $f$ is a function that satisfies either one…

Two classical results characterizing regularity of a convergence space in terms of continuous extensions of maps on one hand, and in terms of continuity of limits for the continuous convergence on the other, are extended to…

For an operator monotone function $f(t)$ on the positive real line, we show the operator monotonicity of the type of the functions $(t-a)(t-b)/(f(t)-f(a))(f^\sharp(t)-f^\sharp(b))$.

If $f(x,y)$ is a real function satisfying $y>0$ and $\sum_{r=0}^{n-1}f(x+ry,ny)=f(x,y)$ for $n=1,2,3,\ldots$, we say that $f(x,y)$ is an invariant function. Many special functions including Bernoulli polynomials, Gamma function and Hurwitz…

We give necessary and sufficient conditions on a function $f:[0,1]\to {0,1,2,...,\omega,\continuum}$ under which there exists a continuous function $F:[0,1]\to [0,1]$ such that for every $y\in[0,1]$ we have $|F^{-1}(y)|=f(y)$.

A monotone function interval is the set of monotone functions that lie pointwise between two fixed monotone functions. We characterize the set of extreme points of monotone function intervals and apply this to a number of economic settings.…

A metric space (X,d) is monotone if there is a linear order < on X and a constant c>0 such that d(x,y) < c d(x,z) for all x<y<z in X. Properties of continuous functions with monotone graph (considered as a planar set) are investigated. It…

Let ${\mathbb T}=({\bf T},\leq)$ and ${\mathbb T}_{1}=({\bf T}_{1},\leq_{1})$ be linearly ordered sets and $\mathscr{X}$ be a topological space. The main result of the paper is the following: If function $\boldsymbol{f}(t,x):{\bf…

In this note we prove a condition of monotonicity for the integral functional $ F(g) = \int_a^b h(x)\, d[-g(x)] $ with respect to $g$, a function of bounded variation. This condition is applied to analyze the behavior of a generalized…