caa a22 4500 60620086X CHVBK 20210128100945.0 cr unu---uuuuu 210128e20151101xx s 000 0 eng 10.1007/s00025-015-0446-2 doi (NATIONALLICENCE)springer-10.1007/s00025-015-0446-2 Berrone Lucio Consejo Nacional de Investigaciones Científicas y Técnicas, Lab. de Acústica y Electroacústica, Universidad Nacional de Rosario, Riobamba 45 bis, 2000, Rosario, Argentina aut P-means and the Solution of a Functional Equation Involving Cauchy Differences [Elektronische Daten] [Lucio Berrone] Solutions to the functional equation $$f(x + y) - f(x) - f(y) = 2f(\Phi (x, y)), x, y > 0, \qquad\qquad (1)$$ f ( x + y ) - f ( x ) - f ( y ) = 2 f ( Φ ( x , y ) ) , x , y > 0 , ( 1 ) are sought for the admissible pairs $${(f, \Phi)}$$ ( f , Φ ) constituted by a strictly monotonic function f and a strictly increasing in both variables mean $${\Phi}$$ Φ . A related class of means, P-means, is introduced, studied and then employed in solving (1) under additional hypotheses on $${\Phi}$$ Φ . For instance, Ger has proved that the unique P-mean which is also quasiarithmetic is the geometric mean $${G(x,y)=\sqrt{xy}}$$ G ( x , y ) = x y . An elementary proof to this result is given in this paper. Moreover, as a consequence of a fundamental result on the uniqueness of representation of P-means it is proved that the geometric mean G is the unique homogeneous P-mean. Springer Basel, 2015 Composite functional equations in several variables nationallicence means nationallicence uniqueness of representation nationallicence Results in Mathematics Springer Basel 68/3-4(2015-11-01), 375-393 1422-6383 68:3-4<375 2015 68 25 https://doi.org/10.1007/s00025-015-0446-2 text/html Onlinezugriff via DOI BK010053 XK010053 XK010000 Metadata rights reserved Springer special CC-BY-NC licence nationallicence 1 research-article jats NATIONALLICENCE NATIONALLICENCE NL-springer NATIONALLICENCE 856 40 https://doi.org/10.1007/s00025-015-0446-2 text/html Onlinezugriff via DOI NATIONALLICENCE 100 1- Berrone Lucio Consejo Nacional de Investigaciones Científicas y Técnicas, Lab. de Acústica y Electroacústica, Universidad Nacional de Rosario, Riobamba 45 bis, 2000, Rosario, Argentina aut NATIONALLICENCE 773 0- Results in Mathematics Springer Basel 68/3-4(2015-11-01), 375-393 1422-6383 68:3-4<375 2015 68 25