<s>[INST] better upper bound of Cauchy - Schwarz inequality for 4 variables Given $a,b,c,d,x,y,z,w \geq 0$. 
By the Cauchy - Schwarz inequality we have that:
\begin{equation}
\label{1}\tag{1}
\left( ax + by + cz + dw \right) ^{2} \leq \left( a^{2} + b^{2} + c^{2} + d^{2} \right) \left( x^{2} + y^{2} + z^{2} + w^{2} \right) .
\end{equation}
Also, by the Cauchy - Schwarz inequality we also obtain:
\begin{equation}
\label{2}\tag{2}
\left( ax + by + cz + dw \right) ^{2} \leq 4 a^{2} x^{2} + 4 b^{2} y^{2} + 4 c^{2} z^{2} + 4 d^{2} w^{2} .
\end{equation}
In general it is better to use \eqref{1} or \eqref{2} as I am looking for the tightest estimate as possible. That is, generally we have
\begin{align}
\label{3}\tag{3}
\left( ax + by + cz + dw \right) ^{2} & \leq \left( a^{2} + b^{2} + c^{2} + d^{2} \right) \left( x^{2} + y^{2} + z^{2} + w^{2} \right) \\
& \leq 4 a^{2} x^{2} + 4 b^{2} y^{2} + 4 c^{2} z^{2} + 4 d^{2} w^{2}
\end{align}
or
\begin{align}
\label{4}\tag{4}
\left( ax + by + cz + dw \right) ^{2} & \leq 4 a^{2} x^{2} + 4 b^{2} y^{2} + 4 c^{2} z^{2} + 4 d^{2} w^{2} \\
& \leq \left( a^{2} + b^{2} + c^{2} + d^{2} \right) \left( x^{2} + y^{2} + z^{2} + w^{2} \right) .
\end{align}
Edit: I forgot one assumption that $d = kc$ for some $k > 0$ and also $x,y,z,w > 0$ so that there is no trivial answer. The question is when will \eqref{3} and \eqref{4} corresponding to the choice of $k$.
 [/INST] Both your inequalities are wrong.
$$\left( a^{2} + b^{2} + c^{2} + d^{2} \right) \left( x^{2} + y^{2} + z^{2} + w^{2} \right) 
 \leq 4 a^{2} x^{2} + 4 b^{2} y^{2} + 4 c^{2} z^{2} + 4 d^{2} w^{2}
$$ is wrong for $a=y=c=w\rightarrow0^+$.
$$\left( a^{2} + b^{2} + c^{2} + d^{2} \right) \left( x^{2} + y^{2} + z^{2} + w^{2} \right) 
 \geq 4 a^{2} x^{2} + 4 b^{2} y^{2} + 4 c^{2} z^{2} + 4 d^{2} w^{2}
$$
is wrong for $b=c=d=y=z=w\rightarrow0^+$.
</s>