Convex Function

A function whose value at the Midpoint of every Interval in its Domain does not exceed the Average of its values at the ends of the Interval. In other words, a function $f(x)$ is convex on an Interval $[a,b]$ if for any two points $x_1$ and $x_2$ in $[a,b]$,

$$f[{\textstyle{1\over 2}}(x_1+x_2)]\leq {\textstyle{1\over 2}}[f(x_1)+f(x_2)].$$

If $f(x)$ has a second Derivative in $[a,b]$, then a Necessary and Sufficient condition for it to be convex on that Interval is that the second Derivative $f''(x)>0$ for all $x$ in $[a,b]$.

See also Concave Function, Logarithmically Convex Function

References

Eggleton, R. B. and Guy, R. K. ``Catalan Strikes Again! How Likely is a Function to be Convex?'' Math. Mag. 61, 211-219, 1988.

Gradshteyn, I. S. and Ryzhik, I. M. Tables of Integrals, Series, and Products, 5th ed. San Diego, CA: Academic Press, p. 1100, 1980.