On the Equality Problem of Conjugate Means

Let ${I\subset\mathbb{R}}$ be a nonvoid open interval and let L : I ²→ I be a fixed strict mean. A function M : I ²→ I is said to be an L-conjugate mean on I if there exist ${p,q\in\,]0,1]}$ and ${\varphi\in CM(I)}$ such that $$M(x,y):=\varphi^{-1}(p\varphi(x)+q\varphi(y)+(1-p-q) \varphi(L(x,y)))=:L_\varphi^{(p,q)}(x,y),$$ for all ${x,y\in I}$ . Here L(x, y) : = A _χ(x, y) ${(x,y\in I)}$ is a fixed quasi-arithmetic mean with the fixed generating function ${\chi\in CM(I)}$ . We examine the following question: which L-conjugate means are weighted quasi-arithmetic means with weight ${r\in\, ]0,1[}$ at the same time? This question is a functional equation problem: Characterize the functions ${\varphi,\psi\in CM(I)}$ and the parameters ${p,q\in\,]0,1]}$ , ${r\in\,]0,1[}$ for which the equation $$L_\varphi^{(p,q)}(x,y)=L_\psi^{(r,1-r)}(x,y)$$ holds for all ${x,y\in I}$ .     

Zur Definition der Nörlundschen Hauptlösung von Differenzengleichungen

φ: R→R. Nörlund [4] defined the principal solution f_N of the difference equation $$V (x, y) \varepsilon R \times R_ + : \frac{1}{y}\left[ {g(x + y, y) - g(x, y)} \right] = \phi (x)$$ by V (x, y) ? [b, ∞) ×R₊: $$f_N (x, y) : = \mathop {\lim }\limits_{s \to 0 + } ( \int\limits_a^\infty {\phi (t) e^{ - st} dt} - y \sum\limits_{\nu = 0}^\infty { \phi (x + \nu y) e^{ - s(x + \nu y )} } )$$ with suitable a,bεR and proved the existence of f_N under certain restrictions onφ. In this paper, another way of defining a principal solution of the difference equation above, which includes Nörlund's, is gone. As an application, we construct in an easy manner a class of limitation methods for getting a principal solution, generalizing results from Nörlund [5].¹)     

On a problem of Nicole Brillou?t-Belluot

We solve the problem posed by Nicole Brillou?t-Belluot (During the Forty-ninth International Symposium on Functional Equations, 2011) of determining all continuous bijections f : I → I satisfying $$f(x)f^{-1}(x) = x^2 \quad{\rm for\, every}\, x \in I,$$ where I is an arbitrary subinterval of the real line.     

The equivalence of two functional equations involving the arithmetic mean, the geometric mean and their Gauss composition

In this paper, the equivalence of the two functional equations $$f\left(\frac{x+y}{2} \right)+f\left(\sqrt{xy} \right)=f(x)+f(y)$$ and $$2f\left(\mathcal{G}(x,y)\right)=f(x)+f(y)$$ will be proved by showing that the solutions of either of these equations are constant functions. Here I is a nonvoid open interval of the positive real half-line and ${\mathcal{G}}$ is the Gauss composition of the arithmetic and geometric means.     

On certain generalizations of the Gagliardo–Nirenberg inequality and their applications to capacitary estimates and isoperimetric inequalities

We derive the inequality $$\int_\mathbb{R}M(|f'(x)|h(f(x))) dx\leq C(M,h)\int_\mathbb{R}M\left({\sqrt{|f''(x)\tau_h(f(x))|}\cdot h(f(x))}\right)dx$$ with a constant C(M, h) independent of f, where f belongs locally to the Sobolev space ${W^{2,1}(\mathbb{R})}$ and f′ has compact support. Here M is an arbitrary N-function satisfying certain assumptions, h is a given function and ${\tau_h(\cdot)}$ is its given transform independent of M. When M(λ) =  λ ^p and ${h \equiv 1}$ we retrieve the well-known inequality ${\int_\mathbb{R}|f'(x)|^{p}dx \leq (\sqrt{p - 1})^{p}\int_\mathbb{R}(\sqrt{|f''(x) f(x)|})^{p}dx}$ . We apply our inequality to obtain some generalizations of capacitary estimates and isoperimetric inequalities due to Maz’ya (1985).     

A Tauberian theorem for ℓ-adic sheaves on \mathbb{A} 1

Let K ∈ L ¹(?) and let f ∈ L ^∞(?) be two functions on ?. The convolution $$ \left( {K*F} \right)\left( x \right) = \int_\mathbb{R} {K\left( {x - y} \right)f\left( y \right)dy} $$ can be considered as an average of f with weight defined by K. Wiener’s Tauberian theorem says that under suitable conditions, if $$ \mathop {\lim }\limits_{x \to \infty } \left( {K*F} \right)\left( x \right) = \mathop {\lim }\limits_{x \to \infty } \int_\mathbb{R} {\left( {K*A} \right)\left( x \right)} $$ for some constant A, then $$ \mathop {\lim }\limits_{x \to \infty } f\left( x \right) = A $$ We prove the following ?-adic analogue of this theorem: Suppose K, F, G are perverse ?-adic sheaves on the affine line $ \mathbb{A} $ over an algebraically closed field of characteristic p (p ≠ ?). Under suitable conditions, if $ \left( {K*F} \right)|_{\eta _\infty } \cong \left( {K*G} \right)|_{\eta _\infty } $ , then $ F|_{\eta _\infty } \cong G|_{\eta _\infty } $ , where η _∞ is the spectrum of the local field of $ \mathbb{A} $ at ∞.     

Weighted approximation of entire functions of exponential type of several variables by trigonometric polynomials

Letf be an entire function (in Cⁿ) of exponential type for whichf(x)=0(?(x)) on the real subspace \(\mathbb{R}^w (\phi \geqslant 1,{\mathbf{ }}\mathop {\lim }\limits_{\left| x \right| \to \infty } \phi (x) = \infty )\) and ?δ>0?C_δ>0 $$\left| {f(z)} \right| \leqslant C_\delta \exp \left\{ {h_s (y) + S\left| z \right|} \right\},z = x + iy$$ where h, (x)=sup〈3, x〉, S being a convex set in ?ⁿ. Then for any ?, ?>0, the functionf can be approximated with any degree of accuracy in the form p→ \(\mathop {\sup }\limits_{x \in \mathbb{R}^w } \frac{{\left| {P(x)} \right|}}{{\varphi (x)}}\) by linear combinations of functions x→expi〈λx〉 with frequenciesX belonging to an ?-neighborhood of the set S.     

On a functional equation of Bruce Ebanks

In the paper Brillouët-Belluot and Ebanks (Aequationes Math 60:233–242, 2000), the authors found all continuous functions f: [0, 1] → [0, + ∞) which verify f(0) = f(1) = 0 and the functional equation $$f(xy +c f(x) f(y)) = x f(y) + y f(x) +d \, f(x) f(y)$$ where c and d are given real numbers with c ≠ 0. In the present paper we obtain all continuous solutions ${f: \mathbb{R} \rightarrow \mathbb{R}}$ of the functional equation (1).     

The equality problem in the class of conjugate means

Let ${I\subset\mathbb{R}}$ be a nonempty open interval and let ${L:I^2\to I}$ be a fixed strict mean. A function ${M:I^2\to I}$ is said to be an L-conjugate mean on I if there exist ${p,q\in{]}0,1]}$ and a strictly monotone and continuous function φ such that $$M(x,y):=\varphi^{-1}(p\varphi(x)+q\varphi(y)+(1-p-q)\varphi(L(x,y)))=:L_\varphi^{(p,q)}(x,y),$$ for all ${x,y\in I}$ . Here L(x, y) is a fixed quasi-arithmetic mean. We will solve the equality problem in this class of means.     

On mappings preserving orthogonality of non-singular vectors

Letq be a regular quadratic form on a vector space (V,F) and letf be the bilinear form associated withq. Then, \(\dot V: = \{ z \in V|q(z) \ne 0\} \) is the set of non-singular vectors ofV, and forx, y ∈ \(\dot V\) , ?(x, y) ?f(x, y) ²/(q(x) · q(y)) is theq-measure of (x, y), where ?(x,y)=0 means thatx, y are orthogonal. For an arbitrary mapping \(\sigma :\dot V \to \dot V\) we consider the functional equations $$\begin{gathered} (I)\sphericalangle (x,y) = 0 \Leftrightarrow \sphericalangle (x^\sigma ,y^\sigma ) = 0\forall x,y \in \dot V, \hfill \\ (II)\sphericalangle (x,y) = \sphericalangle (x^\sigma ,y^\sigma )\forall x,y \in \dot V, \hfill \\ (III)f(x,y)^2 = f(x^\sigma ,y^\sigma )^2 \forall x,y \in \dot V, \hfill \\ \end{gathered} $$ and we state conditions on (V,F,q) such thatσ is induced by a mapping of a well-known type. In case of dimV ∈N?{0, 1, 2} ∧ ∣F∣ > 3, each of the assumptions (I), (II), (III) implies that there exist aρ-linear injectionξ :V →V and a fixed λ ∈F?{0} such thatF x ^σ =F x ^ξ ?x ∈ \(\dot V\) andf(x ^ξ,y ^ξ)=λ · (f(x, y))^ρ ?x, y ∈V. Moreover, (II) implies ρ =id _F ∧q(x ^ξ) = λ ·q(x) ?x ∈ \(\dot V\) , and (III) implies ρ=id _F ∧ λ ∈ {1,?1} ∧x ^σ ∈ {x ^ξ, ?x ^ξ} ?x ∈ \(\dot V\) . Other results obtained in this paper include the cases dimV = 2 resp. dimV ?N resp. ∣F∣ = 3.     

Some functional inequalities

LetM, N, O be open subsets of ? ⁿ and letF:M×N→O,f:O→?,g: M→?,h: N→? be functions, satisfying the functional inequality $$\forall (x,y) \in M \times N:f[F(x,y)] \leqslant g(x) + h(y).$$ IfF belongs to a certain extensive class of functions, we prove in this note, thatf is bounded above on every compact subset of ? ⁿ , wheneverh is bounded above on a Lebesgue-measurable set of positive Lebesgue-measure, contained inN (no assumptions aboutg are needed). Moreover we give applications of this theorem to generalized convex and subadditive functions.     

Invariant Translative Mappings and a Functional Equation

Let \({K,M,N : \mathbb{R}^{2} \rightarrow \mathbb{R}}\) be translative functions. Then K is invariant with respect to the mapping \({(M,N) : \mathbb{R}^{2} \rightarrow \mathbb{R}^{2}}\) if and only if the functions \({h = K(\cdot, 0), f = M(\cdot, 0), g = N(\cdot, 0)}\) satisfy the functional equation $$h(x) = h(f(x) - g(x)) + g(x),\,\, x\in \mathbb{R}.$$ If K, M, N are means, then h(0) =  f(0) =  g(0) = 0. The formal power solutions and analytic solutions of this functional equation, satisfying these initial conditions, are considered.     

Remarques sur un article de Israel Aharoni sur les prolongements lipschitziens dansc 0

We give a purely metric proof of the following result: let (X,d) be a separable metric space; for all ?>0 there is an injectionf ofX inC ₀ ⁺ such that: $$\forall x,y \in X,d(x, y) \leqq \parallel f(x) - f(y)\parallel _\infty \leqq (3 + \varepsilon )d(x, y).$$ It is a more precise version of a result of I. Aharoni. We extend it to metric space of cardinal α⁺ (for infinite α).     

Additivity and exponentiality are alien to each other

Let (S, +) be a (semi)group and let (R,+, ·) be an integral domain. We study the solutions of a Pexider type functional equation $$f(x+y) + g(x+y) = f(x) + f(y) + g(x)g(y)$$ for functions f and g mapping S into R. Our chief concern is to examine whether or not this functional equation is equivalent to the system of two Cauchy equations $$\left\{\begin{array}{@{}ll} f(x+y) = f(x) + f(y)\\ g(x+y) = g(x)g(y)\end{array}\right.$$ for every ${x,y \in S}$ .     

On an equation involving weighted quasi-arithmetic means

Let I ? ? be an interval and κ, λ ∈ ? / {0, 1}, µ, ν ∈ (0, 1). We find all pairs (φ, ψ) of continuous and strictly monotonic functions mapping I into ? and satisfying the functional equation $$ \kappa x + (1 - \kappa )y = \lambda \phi ^{ - 1} (\mu \phi (x) + (1 - \mu )\phi (y)) + (1 - \lambda )\psi ^{ - 1} (\nu \psi (x) + (1 - \nu )\psi (y)) $$ which generalizes the Matkowski-Sutô equation. The paper completes a research stemming in the theory of invariant means.     

Fractional integrals on spaces of homogeneous type

We study fractional integrals on spaces of homogeneous type defined byI _α f(x)=∫_Xf(y)|B(x,d(x,y))|^ga?1dμ(y), 0<α<1. If \(1< p\frac{1}{\alpha },\frac{1}{q} = \frac{1}{p} - \alpha \) , we show that I_αf is of strong type (p,q) and is of weak type ( \(\left( {1,\frac{1}{{1 - \alpha }}} \right)\) ). We also consider the necessary and sufficient conditions on two weights for which I_αf is of weak type (p,q) with respect to (w,v).     

A method of constructing integrable linear equations and its application to Hill's equation

Starting with a given equation of the form $$\ddot x + [\lambda + \varepsilon f(t)] x = 0$$ , where λ > 0 and ? ? l is a small parameter [heref(t) may be periodic, and so Hill's equation is included], we construct an equation of the form y + [λ + ?f (t) + ?² _g (t)]y = 0, integrable by quadratures, close in a certain sense to the original equation. For x₀ = y₀ and x ₀ ^′ = y ₀ ^′ , an upper bound is obtained for ¦y—x¦ on an interval of length Δt.     

A generalized ring homomorphisms equation

We deal with the functional equation $$af(xy) + bf(x)f(y) + cf(x+y) + df(x) + kf(y) = 0\quad\quad\quad\quad\quad\quad\quad(\ast)$$ yielding a joint generalization of equations that has been studied by Dhombres (Aequationes Math 35:186–212, 1988), H. Alzer (private communication) and Ger (Publ Math Debrecen 52:397–417, 1998; Rocznik Nauk-Dydakt Prace Mat 17:101–115, 2000). We are looking for solutions f of equation ${(\ast)}$ mapping a given unitary ring into an integral domain. We continue Dhombres’ studies with the emphasis given upon the dropping of the 2-divisibility assumption in the domain. Among others, our aim is to find suitable conditions under which a function f satisfying ${(\ast)}$ yields a homomorphism between the rings in question.     

Characterization of the Bernoulli-polynomials,exp andΨ by Nörlund's multiplication formula

The system of functional equations $$\forall p\varepsilon N_ + \forall (x,y)\varepsilon D:f(x,y) = \frac{1}{p}\sum\limits_{k = 0}^{p - 1} {f(x + ky,py)}$$ is suited to characterize the functions $$(x,y) \mapsto y^m B_m \left( {\frac{x}{y}} \right),m\varepsilon N,$$ B _m means them-th Bernoulli-polynomial, $$(x,y) \mapsto \exp (x)y(\exp (y) - 1)^{ - 1}$$ (for these functionsD =R ×R ₊) and $$(x,y) \mapsto \log y + \Psi \left( {\frac{x}{y}} \right)(D = R_ + \times R_ + )$$ as those continuous solutions of this system which allow a certain separation of variables and take on some prescribed function values.