caa a22 4500 465825338 CHVBK 20180323112147.0 cr unu---uuuuu 170327e19901201xx s 000 0 eng 10.1007/BF02112299 doi (NATIONALLICENCE)springer-10.1007/BF02112299 Sándor J. Forteni Nr. 79, R-4136, Jud. Harghita, Romania aut On the identric and logarithmic means [Elektronische Daten] [J. Sándor] Summary: Leta, b > 0 be positive real numbers. The identric meanI(a, b) of a andb is defined byI = I(a, b) = (1/e)(b b /a a ) 1/(b−a) , fora ≠ b, I(a, a) = a; while the logarithmic meanL(a, b) ofa andb isL = L(a, b) = (b − a)/(logb − loga), fora ≠ b, L(a, a) = a. Let us denote the arithmetic mean ofa andb byA = A(a, b) = (a + b)/2 and the geometric mean byG =G(a, b) = $$\sqrt {ab}$$ . In this paper we obtain some improvements of known results and new inequalities containing the identric and logarithmic means. The material is divided into six parts. Section 1 contains a review of the most important results which are known for the above means. In Section 2 we prove an inequality which leads to some improvements of known inequalities. Section 3 gives an application of monotonic functions having a logarithmically convex (or concave) inverse function. Section 4 works with the logarithm ofI(a, b), while Section 5 is based on the integral representation of means and related integral inequalities. Finally, Section 6 suggests a new mean and certain generalizations of the identric and logarithmic means. Birkhäuser Verlag, 1990 Primary 26D99 nationallicence Secondary 26D15, 26D20 nationallicence aequationes mathematicae Birkhäuser-Verlag 40/1(1990-12-01), 261-270 0001-9054 40:1<261 1990 40 10 https://doi.org/10.1007/BF02112299 text/html Onlinezugriff via DOI 1 research-article jats NATIONALLICENCE 856 40 https://doi.org/10.1007/BF02112299 text/html Onlinezugriff via DOI NATIONALLICENCE 100 1- Sándor J. Forteni Nr. 79, R-4136, Jud. Harghita, Romania aut NATIONALLICENCE 773 0- aequationes mathematicae Birkhäuser-Verlag 40/1(1990-12-01), 261-270 0001-9054 40:1<261 1990 40 10 Metadata rights reserved Springer special CC-BY-NC licence nationallicence BK010053 XK010053 XK010000 NATIONALLICENCE NATIONALLICENCE NL-springer