Some discrete inequalities such as the Sobolev inequality give useful a priori estimates for numerical schemes. Although they had been known for the simplest forward difference operator, those for central difference type opereators had been left open until quite recently in Kojima-Matsuo-Furihata (2016) a unified way to discuss them was found. Still, due to some technical reasons, the result was limited to a narrow range of central difference operators. In this paper, we provide a new proof that gives a complete answer regarding the discrete Sobolev inequality and the discrete Gagliardo-Nirenberg inequality with the nonlinear Schrödinger equation index.