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On the identric and logarithmic means. (English) Zbl 0717.26014
After a survey of existing results, several new ones are offered for the identric mean $$I(a,b)=e^{-1}(a^{-a}b^ b)^{1/(b-a)}\quad (a\neq b),\quad I(a,a)=a,$$ the logarithmic mean $$L(a,b)=(b-a)\ln^{- 1}(b/a)\quad (a\neq b),\quad L(a,a)=a\quad (a>0,\quad b>0)$$ and the arithmetic and geometric mean; for instance $L(a,b)I(a,b)^{t- 1}<L(a,b)(b^ t-a^ t)/(t(b-a))<(a^ t+b^ t)/2\quad (a\neq b,\quad t\neq 0).$ Logarithmic convexity and integral representations of the above means are used.
The definition of a “new mean” is unfortunately misprinted: it should be $J(a,b):=1/I(1/a,1/b)\quad (=\quad e(b^ aa^{-b})^{1/(a- b)}\text{ for } b\neq a,\quad J(a,a)=a).$
Reviewer: J.Aczél

MSC:
 26D15 Inequalities for sums, series and integrals 26A51 Convexity of real functions in one variable, generalizations 26A48 Monotonic functions, generalizations
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