On a functional equation related to competition

The functional equation $$f \left(\frac{x + y}{1 - xy}\right) = \frac{f\left(x\right) + f\left(y\right)} {1 + f\left(x\right) f\left(y\right)}, \quad xy < 1,$$ (introduced by the first author in a competition model) is considered. The main result says that a function \({f : \mathbb{R} \rightarrow \mathbb{R}}\) satisfies this equation if, and only if, \({f = {\rm tanh} \circ \, \alpha \circ {\rm tan}^{-1}}\) , where \({\alpha : \mathbb{R} \rightarrow \mathbb{R}}\) is an additive function.     

On an alternative functional equation related to the Cauchy equation

We consider the following problem: Let (G, +) be an abelian group,B a complex Banach space,a, bB,b0,M a positive integer; find all functionsf:G B such that for every (x, y) G ×G the Cauchy differencef(x+y)–f(x)–f(y) belongs to the set {a, a+b, a+2b, ...,a+Mb}. We prove that all solutions of the above problem can be obtained by means of the injective homomorphisms fromG/H intoR/Z, whereH is a suitable proper subgroup ofG.     

On the functional equation of certain Dirichlet series

Inventiones mathematicae -     

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 functional equation related to derivations in prime rings

In this paper we prove the following result. Let m ≥ 1, n ≥ 1 be fixed integers and let R be a prime ring with m + n + 1 ≤ char(R) or char(R) = 0. Suppose there exists an additive nonzero mapping D : R → R satisfying the relation 2D(x ^n+m+1) = (m + n + 1)(x ^m D(x)x ⁿ + x ⁿ D(x)x ^m ) for all \({x\in R}\). In this case R is commutative and D is a derivation.     

On finite groups with a certain number of centralizers

LetG be a finite group and #Cent(G) denote the number of centralizers of its elements.G is calledn-centralizer if #Cent(G)=n, and primitiven-centralizer if #Cent(G)=#Cent(G/Z(G))=n. In this paper we investigate the structure of finite groups with at most 21 element centralizers. We prove that such a group is solvable and ifG is a finite group such thatG/Z(G)?A₅, then #Cent(G)=22 or 32. Moroever, we prove that A₅ is the only finite simple group with 22 centralizers. Therefore we obtain a characterization of A₅ in terms of the number of centralizers     

On a functional equation related to the Matsumoto—Yor property

Summary. A functional equation arising from the independence properties of some transformations of independent generalized inverse Gaussian and gamma variables is completely solved under the local integrability assumption.