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

Qi, Feng; Chen, Shouxin; Chen, Chaoping

Math. Inequal. Appl. 10, No. 3, 559-564 (2007).

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.

Reviewer: Qiu-Ming Luo (Shanghai)

MSC:

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

Keywords:

monotonicity; inequality; ratio; generalized logarithmic mean; extended logarithmic mean; identric mean; exponential mean