In $1977$, G. Bennett proved in a non-deterministic way inequalities that play a fundamental role in a series of optimization problems. More precisely, Bennett’s inequality for $p_{1},p_{2} \in\lbrack1,\infty]$ and all positive integers $n_{1},n_{2}$ It shows that the bilinear form $ exists. A_{n_{1},n_{2}}\colon\left( \mathbb{R}^{n_{1}},\left\Vert \cdot\right\Vert _{p_{1}}\right) \times\left( \mathbb{R}^{n_{2}},\left\Vert \cdot\right\Vert _{p_{2}}\right) \longrightarrow\mathbb{R}$ coefficient $\ pm1 $satisfied\[ \left\Vert A_{n_{1},n_{2}}\right\Vert \leq
C_{p_{1},p_{2}}\max\left\{ n_{1}^{1-\frac{1}{p_{1}}}n_{2}^{\max\left\{
\frac{1}{2}-\frac{1}{p_{2} },0\right\}
},n_{2}^{1-\frac{1}{p_{2}}}n_{1}^{\max\left\{ \frac{1}{2}
-\frac{1}{p_{1}},0\right\} }\right\} \] For certain constants $C_{p_{1},p_{2}}$, it depends only on $p_{1},p_{2}$. Furthermore, the exponents of $n_{1},n_{2}$ cannot be improved. In this paper, using a constructive approach, $C_{p_{1},p_{2}}\leq\sqrt{8/5}$ is always $p_{1},p_{2}\in \left[
2,\infty\right] $ or $p_{1}=p_{2}=p\in\left[ 1,\infty\right] $. Our technique is applied to provide a constant new upper bound for the combinatorial game known as the Gale-Bergkamp switching game or the unbalanced light problem. As a result, it improves the estimates obtained by Brown and Spencer for $1971 and by Carlson and Stolarski for $2004.
