## Monotonicity of ratio between the generalized logarithmic means.(English)Zbl 1127.26021

The generalized logarithmic means are defined as follows
$L_r(a,b)= \begin{cases} \left(\dfrac{b^{r+1}-a^{r+1}}{(r+1)(b-a)}\right)^{1/r}, & r \neq -1, 0,\\ \dfrac{b-a}{\ln{b}-\ln{a}}, & r=-1,\\ \dfrac{1}{e}\left(\dfrac{b^b}{a^a}\right)^{1/(b-a)}, & r=0, \end{cases}$
for $$a \neq b$$. $$L_r(a,b)=a$$, for $$a = b$$. In this short note, for $$c>b>a>0$$ being real numbers, the authors prove that the function $$f(r)={L_r(a,b)}/{L_r(a,c)}$$ is strictly decreasing for $$-\infty <r<\infty$$; this result answers a conjecture of Sampaio in integral form.

### MSC:

 26D15 Inequalities for sums, series and integrals 26E60 Means
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