Is a geometrically convex function also convex?

Question

If a real valued, positive, increasing, differentiable function defined on a convex subset of the real line is geometrically convex, is it also convex?

Answer 1

I am not sure what you mean by "geometrically convex".

However, a function is convex if and only if its epigraph is convex (as a set), see for example here: Convex function and its epigraph PROOF.

epigraph means the set of all points above the graph of a function, for the purpose of clarity. The name stems from the use of the prefix "epi-" to mean "above": http://www.dictionary.com/browse/epi-

Note that we have to use the set of points on or above the graph of the function; it clearly does not work for the graph of the function itself, nor for the set of points on or below the graph of the function. A simple counterexample is provided by the convex function $y=x^2$, which can be seen to be convex by the second derivative test, i.e. $y''=2>0$.

See this image from Wikipedia-- note that the epigraph is sometimes also called the supergraph: