Jensen's functional equation on groups

Summary A natural extension of Jensen's functional equation on the real line is the equationf(xy) + f(xy ^–1 ) = 2f(x), wheref maps a groupG into an abelian groupH. We deduce some basic reduction formulas and relations, and use them to obtain the general solution on special groups.     

On a functional equation for Jacobi's elliptic function cn(z; k)

The purpose of this paper is to give a characterization of Jacobi's elliptic function cn(z; k) by use of a functional equation which is a generalization of the cosine functional equation.     

On a general principle in geometry that leads to functional equations

Summary Suppose that an invariant (or an invariant notion) of some geometry is given, like the distance between two points, the cross ratio of four points, the tangential distance between two spheres (or like the notion of orthogonality, of order, of a circle). One may ask what are the functions preserving (or preserving partially) that invariant (invariant notion). Originating from this principle some functional equation problems are formulated, namely the functional equations of distance, of area, of angle preservance.Dedicated to the memory of Alexander M. Ostrowski on the occasion of the 100th anniversary of his birth.     

A conditional dilatation equation

Summary The following theorem holds true. Theorem. Let X be a normed real vector space of dimension 3 and let k > 0 be a fixed real number. Suppose that f: X X and g: X × X are functions satisfying x – y = k f(x) – f(y) = g(x, y)(x – y) for all x, y X. Then there exist elements and t X such that f(x) = x + t for all x X and such that g(x, y) = for all x, y X with x – y = k.     

On Jensen's functional equation

Summary Letf be a map from a groupG into an abelian groupH satisfyingf(xy) + f(xy ^–1) = 2f(x), f(e) = 0, wherex, y G ande is the identity inG. A set of necessary and sufficient conditions forS(G, H) = Hom(G, H) is given whenG is abelian, whereS(G, H) denotes all the solutions of the functional equation. The case whenG is non-abelian is also discussed.     

On Darboux solutions of the Euler's equation

Summary We construct a non-constant Darboux functionf: R R which is a solution of the Euler's functional equationf(x + f(x)) = f(x) for everyx. This function is a counter-example to a statement given in the literature.     

On a functional equation related to income inequality measures

The functional equationg(u, x)+g(v, y)=g(u, y)+g(v, x) for allu, v, x, y>0 withu+v=x+y is initiated by F. A. Cowell and A. F. Shorrocks in their research on the aggregation of inequality indices. We solve the equation by extension theorems.Dedicated to Professor Janos Aczél on his 60th birthday     

On a generalization of the Cauchy functional equation

Summary While looking for solutions of some functional equations and systems of functional equations introduced by S. Midura and their generalizations, we came across the problem of solving the equationg(ax + by) = Ag(x) + Bg(y) + L(x, y) (1) in the class of functions mapping a non-empty subsetP of a linear spaceX over a commutative fieldK, satisfying the conditionaP + bP P, into a linear spaceY over a commutative fieldF, whereL: X × X Y is biadditive,a, b K\{0}, andA, B F\{0}. Theorem.Suppose that K is either R or C, F is of characteristic zero, there exist A ₁,A ₂,B ₁,B ₂, F\ {0}with L(ax, y) = A ₁ L(x, y), L(x, ay) = A ₂ L(x, y), L(bx, y) = B ₁ L(x, y), and L(x, by) = B ₂ L(x, y) for x, y X, and P has a non-empty convex and algebraically open subset. Then the functional equation (1)has a solution in the class of functions g: P Y iff the following two conditions hold: L(x, y) = L(y, x) for x, y X, (2)if L 0, then A ₁ =A ₂,B ₁ =B ₂,A = A ₁ ² ,and B = B ₁ ² . (3) Furthermore, if conditions (2)and (3)are valid, then a function g: P Y satisfies the equation (1)iff there exist a y ₀ Y and an additive function h: X Y such that if A + B 1, then y ₀ = 0;h(ax) = Ah(x), h(bx) =Bh(x) for x X; g(x) = h(x) + y ₀ + 1/2A ₁ ^-1 B ₁ ^-1 L(x, x)for x P.