A new proof of the monotonicity property of power means. (English) Zbl 1051.26023

Witkowski, Alfred

JIPAM, J. Inequal. Pure Appl. Math. 5, No. 3, Paper No. 73, 2 p. (2004).

Summary: If \(M_r\) is the weighted power mean of the numbers \(x_j\in[a,b]\) then \(Q_r(a,b,x)= (a^r+b^r-M_r^r)^{1/r}\) is increasing in \(r\). A new proof of this fact is given.

MSC:

26D15	Inequalities for sums, series and integrals
26E60	Means
26A51	Convexity of real functions in one variable, generalizations

Keywords:

convexity; monotonicity; weighted power means; Jensen inequality