Even and Odd Functions

If is even then If then both and .

If satisfies the differential inequality then is

Assuming that is even, i.e. for all , simplifies the construction of quadratic majorizers. If an even quadratic satisfies and , then it also satisfies and . If in addition majorizes at either or , then it majorizes at both and , and Theorem \ref{ruitenburg_theorem} implies that it is the best possible majorization at both points. This means we only need an extra condition to guarantee that majorizes . The next theorem, essentially proved in the references [Groenen et al., 2003; Jaakkola and Jordan, 2000; Hunter and Li, 2005] by other techniques, highlights an important sufficient condition.


Theorem: Suppose is an even, differentiable function on such that the ratio is decreasing on . Then the even quadratic is the best majorizer of at the point .

Proof: It is obvious that is even and satisfies the tangency conditions and . For the case , we have where the inequality comes from the assumption that is decreasing. It follows that . The case is proved in similar fashion, and all other cases reduce to these two cases given that and are even. QED


There is an condition equivalent to the sufficient condition of Theorem \ref{declining_ratio} that is sometimes easier to check.


Theorem: The ratio is decreasing on if and only is concave. The set of functions satisfying this condition is a closed under the formation of (a) positive multiples, (b) convex combinations, (c) limits, and (d) composition with a concave increasing function .

Proof: Suppose is concave in and . Then the two inequalities are valid. Adding these, subtracting the common sum from both sides, and rearranging give Dividing by yields the desired result Conversely, suppose the ratio is decreasing and . Then the mean value expansion for leads to the concavity inequality. The asserted closure properties are all easy to check. QED


As examples of property (d) of Theorem \ref{convex_sqrt}, note that the functions and are concave and increasing. Hence, if is concave, then and are concave as well.