# A generalization of Chebyshev's sum inequality

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.

• This is more like Chebyshev's sum inequality than the rearrangement one. – Max Alekseyev Jul 21 '18 at 8:15
• @MaxAlekseyev Thank You very much. Yes, I think so. Should change tilte? – Đào Thanh Oai Jul 21 '18 at 8:21
• 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 – Yemon Choi Jul 21 '18 at 14:41
• @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. – 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 $\log(2\cdot 3\cdot 3\cdot 4)=\log(72)$.
• $f(x,y)=xy$ or $f(x,y)=(x+y)^2$ the inequality above is also true. Dear Sir @MaxAlekseyev – Đào Thanh Oai Jul 21 '18 at 9:46
• Allow me adding condition $f''_{xy}>0$ $f''_{yx}>0$ ?? – Đào Thanh Oai Jul 21 '18 at 10:04