From some my previous questions here and here and well-known rearrangement inequality. I pose an inequality as follows and I am looking for the proof or a reference.

Inequality: Let $y=f(x,y)$ is positive continuous function in interval $[a, b] \times [c, d]$. Such that: $f'_x(x,y)>0$ and $f'_y(x,y)>0$ in $[a, b] \times [c, d]$. If $a \le x_1 \le x_2 \le \cdots \le x_n \le b$ and $c \le y_1 \le y_2 \le \cdots \le y_n \le d$ then

$$n\sum_{i=1}^{n}f(x_i, y_i) \ge \sum_{i=1}^{n}\sum_{j=1}^{n}f(x_i, y_j)$$

if If $a \le x_1 \le x_2 \le \cdots \le x_n \le b$ and $d \ge y_1 \ge y_2 \ge \cdots \ge y_n \ge c$ then

$$n\sum_{i=1}^{n}f(x_i, y_i) \le \sum_{i=1}^{n}\sum_{j=1}^{n}f(x_i, y_j)$$

EDIT: Allow me adding condition $f''_{xy}>0$ $f''_{yx}>0$

Remarks: The result be can general to $n$ variant.

  • 1
    $\begingroup$ This is more like Chebyshev's sum inequality than the rearrangement one. $\endgroup$ – Max Alekseyev Jul 21 '18 at 8:15
  • $\begingroup$ @MaxAlekseyev Thank You very much. Yes, I think so. Should change tilte? $\endgroup$ – Đào Thanh Oai Jul 21 '18 at 8:21
  • 4
    $\begingroup$ I'm voting to close this question because the OP should put more thought into these questions before asking them. The editing of the question seems to present a moving target $\endgroup$ – Yemon Choi Jul 21 '18 at 14:41
  • $\begingroup$ @YemonChoi Don't close this because I check this true with many function. The form of inequality is nice so we should find condition for it true. $\endgroup$ – Đào Thanh Oai Jul 21 '18 at 15:06

A counterexample: $f(x,y)=\log(x+y)$ with $n=2$, $a=c=x_1=y_1=1$ and $b=d=x_2=y_2=2$. Then the l.h.s. of the first inequality equals $2\log(2\cdot 4)=\log(64)$, while its r.h.s. equals $\log(2\cdot 3\cdot 3\cdot 4)=\log(72)$.

  • $\begingroup$ Which condition is the inequality abovetrue? $\endgroup$ – Đào Thanh Oai Jul 21 '18 at 9:41
  • $\begingroup$ $f(x,y)=xy$ or $f(x,y)=(x+y)^2$ the inequality above is also true. Dear Sir @MaxAlekseyev $\endgroup$ – Đào Thanh Oai Jul 21 '18 at 9:46
  • $\begingroup$ Allow me adding condition $f''_{xy}>0$ $f''_{yx}>0$ ?? $\endgroup$ – Đào Thanh Oai Jul 21 '18 at 10:04

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.