  It is shown that $\sum_{j=-m}^m (-1)^j \frac{f(x-j)(f(x+j)}{(m-j)! (m+j)!}
\ge 0,$ $m=0,1,...,$ where $f(x)$ is either a real polynomial with only real
zeros or an allied entire function of a special type, provided the distance
between two consecutive zeros of $f(x)$ is at least $\sqrt{4-\frac{6}{m+2}}.$
These inequalities are a surprisingly similar discrete analogue of higher
degree generalizations of the Laguerre and Turan inequalities. Being applied to
the classical discrete orthogonal polynomials, they yield sharp, explicit
bounds uniform in all parameters involved, on the polynomials and their extreme
zeros. We will illustrate it for the case of Krawtchouk polynomials
