Let $f(x,y,z)=ax^2+by^2+cz^2+dxy+exz+fyx$ a quadratic form. So my question is if it's possible that for three sequences of natural numbers $(m_{1,k})$, $(m_{2,k})$ and $(m_{3,k})$ we have the following behavior $$ f(m_{1,k},m_{2,k},m_{3,k})\leq (m_{1,k}+m_{2,k}+m_{3,k})+\sqrt{k} $$
and the sequence has the property that there is $a_1,a_2,a_3$ positive real number such that $$ \lim_{k\rightarrow +\infty}\frac{(a_1m_{1,k}+a_2m_{2,k}+a_3m_{3,k})^2}{k}=a \in (0,\infty) $$
or we can conclude that $f$ needs to be zero?