345 research outputs found
On a functional equation related to two-variable weighted quasi-arithmetic means
In this paper, we are going to describe the solutions of the functional
equation
concerning the unknown functions and defined on an open interval.
In our main result only the continuity of the function and a
regularity property of the set of zeroes of are assumed. As application, we
determine the solutions of the functional equation under monotonicity and
differentiability conditions on the unknown functions
On derivations with respect to finite sets of smooth functions
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
On the equality problem of generalized Bajraktarevi\'c means
The purpose of this paper is to investigate the equality problem of
generalized Bajraktarevi\'c means, i.e., to solve the functional equation
\begin{equation}\label{E0}\tag{*}
f^{(-1)}\bigg(\frac{p_1(x_1)f(x_1)+\dots+p_n(x_n)f(x_n)}{p_1(x_1)+\dots+p_n(x_n)}\bigg)=g^{(-1)}\bigg(\frac{q_1(x_1)g(x_1)+\dots+q_n(x_n)g(x_n)}{q_1(x_1)+\dots+q_n(x_n)}\bigg),
\end{equation} which holds for all , where ,
is a nonempty open real interval, the unknown functions
are strictly monotone, and denote
their generalized left inverses, respectively, and
and
are also unknown functions. This
equality problem in the symmetric two-variable (i.e., when ) case was
already investigated and solved under sixth-order regularity assumptions by
Losonczi in 1999. In the nonsymmetric two-variable case, assuming three times
differentiability of , and the existence of such that
either is twice continuously differentiable and is continuous
on , or is twice differentiable and is once differentiable
on , we prove that \eqref{E0} holds if and only if there exist four
constants with such that \begin{equation*}
cf+d>0,\qquad
g=\frac{af+b}{cf+d},\qquad\mbox{and}\qquad q_\ell=(cf+d)p_\ell\qquad
(\ell\in\{1,\dots,n\}). \end{equation*} In the case , we obtain the
same conclusion with weaker regularity assumptions. Namely, we suppose that
and are three times differentiable, is continuous and there exist
with such that are
differentiable
Approximate Hermite-Hadamard type inequalities for approximately convex functions
In this paper, approximate lower and upper Hermite--Hadamard type
inequalities are obtained for functions that are approximately convex with
respect to a given Chebyshev system
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