For the productions $f(k,l) $ that are continuously differentiable, is the proposition that "$ f(k,l) \: is \: decreasing\:return\:to\:scale \Leftrightarrow f_{ll}f_{kk}-f_{kl}^2>0$" always true, I have checked for Cobb-Douglas functions and believe it will also hold for all Constant-Elastisity fuction, can someone give me some hints or show some counter-examples?

Here we define $f(k,l)$ to be decreasing return to scale if $ f(tk,tl)<tf(k,l)\: for\: t>1\: for\: \forall (k,l)\:in \:the\:domain$

For a single output production function, decreasing returns to scale does imply concavity of $f$. I believe you can find this result in Ch5 of MWG, either as a proposition or an exercise. The condition you have comes from considering the Hessian of $f$, and showing that the determinant is positive. For more on the Hessian Conditions, this is a handy reference: https://mjo.osborne.economics.utoronto.ca/index.php/tutorial/index/1/cvn/t

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