Related papers: On subadditive quasi-arithmetic means

Let $[a,b]\subset\mathbb{R}$ be a non empty and non singleton closed interval and $P=\{a=x_0<\cdots<x_n=b\}$ is a partition of it. Then $f:I\to\mathbb{R}$ is said to be a function of $r$-bounded variation, if the expression…

We show that there is an additive $F_\sigma$ subgroup $A$ of $\mathbb{R}$ and $x \in \mathbb{R}$ such that $\mathrm{dim_H} (A) = \frac{1}{2}$ and $A + x A =\mathbb{R}$. However, if $A \subseteq \mathbb{R}$ is a subring of $\mathbb{R}$ and…

In this note we study restrictions on the recently introduced super-additive and sub-additive transformations, $A\mapsto A^*$ and $A\mapsto A_*$, of an aggregation function $A$. We prove that if $A^*$ has a slightly stronger property of…

In this article, we present a new subadditivity behavior of convex and concave functions, when applied to Hilbert space operators. For example, under suitable assumptions on the spectrum of the positive operators $A$ and $B$, we prove that…

We introduce a subfamily of additive enlargements of a maximally monotone operator. Our definition is inspired by the early work of Simon Fitzpatrick. These enlargements constitute a subfamily of the family of enlargements introduced by…

A differentiable function is pseudoconvex if and only if its restrictions over straight lines are pseudoconvex. A differentiable function depending on one variable, defined on some closed interval $[a,b]$ is pseudoconvex if and only if…

The function f:X -> Y is called k-monotonically increasing if there is a partition X = X_1 U ... U X_k such that f|X_i : X_i -> Y is monotonically increasing for i=1,...,k. It is proved that a one-to-one function f:N -> N is k-monotonically…

A double sequence $\{x_{k,l}\}$ is quasi-Cauchy if given an $\epsilon > 0$ there exists an $N \in {\bf N}$ such that $$\max_{r,s= 1\mbox{ and/or} 0} \left \{|x_{k,l} - x_{k+r,l+s}|< \epsilon\right \} .$$ We study continuity type properties…

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…

Theorem 1 Let F:N-->R stand for any function which a) $F$ monotonically weakly increases; b) $F$ tends to infinity; and c) such that $q/F(q)$ tends to infinity. Let Z_F(q) equal the number of divisors of q less than sqrt{F(q)} minus the…

Let $f: \mathbb{N}^2 \mapsto \mathbb{C}$ be an arithmetic function of two variables. We study the existence of the limit: \[\displaystyle \lim_{x \to \infty} \frac{1}{x^2 (\log x)^{k-1}} \sum_{n_1 , n_2 \le x} f (n_1, n_2) \] where $k$ is a…

A real valued function $f$ defined on a subset $E$ of $\textbf{R}$, the set of real numbers, is statistically upward continuous if it preserves statistically upward half quasi-Cauchy sequences, is statistically downward continuous if it…

For a sequence of continuous, monotone functions $f_1,\dots,f_n \colon I \to \mathbb{R}$ ($I$ is an interval) we define the mapping $M \colon I^n \to I^n$ as a Cartesian product of quasi-arithmetic means generated by $f_j$-s. It is known…

Here, we investigate the solutions to equation \[f(f(-x)+x)=f(-f(x))+f(x),\qquad x\in\mathbb{R}\] that are prescribed on the non-positive half-line. We will refer to this prescribed function as the generator of the corresponding solution.…

Let $\Omega\subset\mathbb{R}^n$ be an open, connected subset of $\mathbb{R}^n$, and let $F\colon\Omega-\Omega\to\mathbb{C}$, where $\Omega-\Omega=\{x-y\colon x,y\in\Omega\}$, be a continuous positive definite function. We give necessary and…

An arithmetic function $f$ is Leibniz-additive if there is a completely multiplicative function $h_f$, i.e., $h_f(1)=1$ and $h_f(mn)=h_f(m)h_f(n)$ for all positive integers $m$ and $n$, satisfying $$ f(mn)=f(m)h_f(n)+f(n)h_f(m) $$ for all…

A function in a class $\mathcal{F}(X)$ is said to be subdifferentially determined in $\mathcal{F}(X)$ if it is equal up to an additive constant to any function in $\mathcal{F}(X)$ with the same subdifferential. A function is said to be…

The purpose of this paper is to investigate the following invariance equation involving two $2$-variable generalized Bajraktarevi\'c means, i.e., we aim to solve the functional equation $$…

Let $D$ be a convex subset of a real vector space. It is shown that a radially lower semicontinuous function $f: D\to \mathbf{R}\cup \{+\infty\}$ is convex if and only if for all $x,y \in D$ there exists $\alpha=\alpha(x,y) \in (0,1)$ such…

We show that the family of all holomorphic functions $f$ in a domain $D$ satisfying $$\frac{|f^{(k)}|}{1+|f|}(z)\le C \qquad \mbox{ for all } z\in D$$ (where $k$ is a natural number and $C>0$) is quasi-normal. Furthermore, we give a general…