The purpose of this paper is to show that functions that derivate the two-variable product function and one of the exponential, trigonometric or hyperbolic functions are also standard derivations. The more general problem considered is to describe finite sets of differentiable functions such that derivations with respect to this set are automatically standard derivations. A n := {f : Ω → R | Ω ⊂ R n is open, nonempty and f is differentiable}
The purpose of this paper is to investigate the equality problem of generalized Bajraktarević means, i.e., to solve the functional equation Date: April 16, 2019. 2010 Mathematics Subject Classification. 39B30, 39B40, 26E60.
The purpose of this paper is to investigate the following invariance equation involving two 2-variable generalized Bajraktarević means, i.e., we aim to solve the functional equationwhere I is a nonempty open real interval and f, g : I → R are continuous, strictly monotone and p 1 , p 2 , q 1 , q 2 : I → R + are unknown functions. The main result of the paper shows that, assuming four times continuous differentiability of f , g, twice continuous differentiability of p 1 and p 2 and assuming that p 1 differs from p 2 on a dense subset of I, a necessary and sufficient condition for the equality above is that the unknown functions are of the formwhere u, v, w, z : I → R are arbitrary solutions of the second-order linear differential equation F ′′ = γF (γ ∈ R is arbitrarily fixed) such that v > 0 and z > 0 holds on I and {u, v} and {w, z} are linearly independent.
The purpose of this paper is to introduce the notion of a generalized derivation which derivates a prescribed family of smooth vector-valued functions of several variables. The basic calculus rules are established and then a result derived which shows that if a function f satisfies an addition theorem whose determining operation is derivable with respect to an additive function d, then the function f is itself derivable with respect to d. As an application of this approach, new proof of a generalization of a recent result of Maksa is obtained. We also extend the result of Nishiyama and Horinouchi and formulate two open problems.
The purpose of this paper is to introduce the notion of a generalized derivation which derivates a prescribed family of smooth vector-valued functions of several variables. The basic calculus rules are established and then a result derived which shows that if a function f satisfies an addition theorem whose determining operation is derivable with respect to an additive function d, then the function f is itself derivable with respect to d. As an application of this approach, new proof of a generalization of a recent result of Maksa is obtained. We also extend the result of Nishiyama and Horinouchi and formulate two open problems.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.