I want to show $$2\,\text{cn}(x|k)^2 \,\text{dn}(x|k)^2\, \text{sn}(2 x|k)-4\, \text{cn}(x|k)\, \text{cn}(2 x|k) \,\text{dn}(2 x|k)\, \text{dn}(x|k) \,\text{sn}(x|k)+k\, \text{cn}(x|k)^2 \,\text{sn}(x|k)^2\, \text{sn}(2 x|k)-4 \,k \,\text{cn}(2 x|k)^2 \, \text{sn}(x|k)^2 \,\text{sn}(2 x|k)+\text{dn}(x|k)^2 \,\text{sn}(x|k)^2\, \text{sn}(2 x|k)-4 \,\text{dn}(2 x|k)^2\, \text{sn}(x|k)^2 \,\text{sn}(2 x|k)<0 $$ for $x\in(0,\pi/2)$ and $0<k<1$. Here $\text{sn}$, $\text{cn}$ and $\text{dn}$ are elliptic functions. This inequality has been verified by Mathematica using the Plot commend for several values of k. However, I cannot give a rigorous proof of it. Any suggestion, idea, or comment is welcome, thanks!
2026-03-26 06:33:41.1774506821
An inequality for elliptic functions
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