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

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
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