# Can we prove $\sum_{i=1}^m \frac{1}{t_i +1} \ge \sum_{i=1}^m \frac{1}{\lambda_i +1}$ or not?

Let $$t_1,\dots,t_m \ge 0$$, and $$\lambda_1, \dots, \lambda_m \ge 0$$. Suppose \begin{aligned} & t_1 + \dots + t_m = m(\lambda_1 + \dots + \lambda_m), \\ & t_1 \times \dots \times t_m \leq \lambda_1 \times \dots \times \lambda_m. \end{aligned} Can we prove the following inequality: $$$$\sum_{i=1}^m \frac{1}{t_i +1} \ge \sum_{i=1}^m \frac{1}{\lambda_i +1}.$$$$ I am stuck for a long time and have tried many methods, but have no idea at all. Can anyone prove it or provide a counterexample? Any help will be appreciated.

• How is this linear algebra? – Arturo Magidin Mar 18 at 3:17
• fine, I edited it. – Ze-Nan Li Mar 18 at 3:23
• the thing is close to majorization theory related to linear algebra in a way. – Toni Mhax Mar 18 at 3:50
• @ToniMhax Thanks for the hint ! – Ze-Nan Li Mar 18 at 3:51
• maybe $t_1+t_2=\lambda_1+\lambda_2$ yields the inequality when $m=2$. etc – Toni Mhax Mar 18 at 4:16

A preliminary calculation gives the following counterexample, $$m=2,t_1=\frac{3+\sqrt 7}{2}, t_2=\frac{3-\sqrt 7}{2}, \lambda_1=1,\lambda_2=\frac{1}{2}.$$ You can double check if it is indeed a counterexample or there is a mistake in my calculation.
• Actually, it should not be hard to check that under your conditions, the inequality is true if and only if $\lambda_1 \lambda_2\ge 1$. From this, it should be easy to get a counterexample. – QZ0 Mar 18 at 3:50
The sum constraint may be the wrong one if $$t_1=4$$ and $$t_2=0$$, $$\lambda_1=1.9$$, $$\lambda_2=0.1$$, $$$$\sum_{i=1}^2 \frac{1}{t_i +1}=1.2 \le \sum_{i=1}^2 \frac{1}{\lambda_i +1}\approx 1.25$$$$