Bisymmetric difference means Patrick Flandrin Ecole Normale Sup´ erieure de Lyon Laboratoire de Physique (Umr 5672 Cnrs) 46 all´ ee d’Italie, 69364 Lyon Cedex 07, FRANCE email : flandrin@physique.ens-lyon.fr February 1, 1999 Abstract — We consider the difference means {D a,b (x,y); a,b ∈ IR,x,y ∈ IR + } introduced by Stolarsky and we characterize all pairs (a,b) guaranteeing that the property of bisymme- try : D a,b (D a,b (x 1 ,x 2 ),D a,b (x 3 ,x 4 )) = D a,b (D a,b (x 1 ,x 3 ),D a,b (x 2 ,x 4 )) is satisfied. We show that the power means with exponent k ∈ IR are the only bisymmetric difference means. Difference means have been introduced by Stolarsky [7] as a way of generalizing the logarithmic mean. For any pair of distinct positive numbers x and y, the basic form of their difference mean is given by D a,b (x,y)=  b a x a − y a x b − y b  1 a-b (1) if ab(a − b)(x − y) = 0, with a continuous extension to the whole domain defined by {(a,b,x,y) |a,b ∈ IR ; x,y ∈ IR + } (see [6] for detailed expressions). Apart from the log- arithmic mean D 0,1 , numerous standard means happen to be special cases of difference means : for instance, D 2,1 , D 0,0 and D -2,-1 correspond to the arithmetic, geometric and harmonic means, respectively, whereas power means with exponent k = 0 can be obtained as D 2k,k . Difference means offer therefore a nice and versatile framework for most of the useful means, as well as a unified setting for evaluating their properties. By construction, difference means are symmetric in the sense that, for any pair of numbers x,y ∈ IR + , we have D a,b (x,y)= D a,b (y,x). However, it is not guaranteed that, 1