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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 $I(a,b)=e\sp{-1}(a\sp{-a}b\sp b)\sp{1/(b-a)}\quad (a\ne b),\quad I(a,a)=a,$ the logarithmic mean $L(a,b)=(b-a)\ln\sp{- 1}(b/a)\quad (a\ne b),\quad L(a,a)=a\quad (a>0,\quad b>0)$ and the arithmetic and geometric mean; for instance $$ L(a,b)I(a,b)\sp{t- 1}<L(a,b)(b\sp t-a\sp t)/(t(b-a))<(a\sp t+b\sp t)/2\quad (a\ne b,\quad t\ne 0). $$ Logarithmic convexity and integral representations of the above means are used. The definition of a “new mean” is unfortunately misprinted: it should be $$ J(a,b):=1/I(1/a,1/b)\quad (=\quad e(b\sp aa\sp{-b})\sp{1/(a- b)}\text{ for } b\ne a,\quad J(a,a)=a).$$
Reviewer: J.Aczél

26D15Inequalities for sums, series and integrals of real functions
26A51Convexity, generalizations (one real variable)
26A48Monotonic functions, generalizations (one real variable)
Full Text: DOI EuDML
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