Automatic Improvements of Wallis´ Inequality

The mathematical backbone of this article is formed by three classical formulas of Wallis: his product formula for π, an inequality implying the product formula in the limit, and a related definite integral involving powers of the sine function. For the latter we present various evaluations to illustrate recent algorithmic developments. In the main part of the article we automatically refine the bounds in Wallis´ inequality by using methods based on Collins´ cylindrical algebraic decomposition.