If a real valued, positive, increasing, differentiable function defined on a convex subset of the real line is geometrically convex, is it also convex?
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what is your definition of geometrically convex? do you mean that the graph is convex? that clearly is not true. However, the epigraph of a convex function I believe is convex (as a set) – bazookabear Sep 18 '16 at 01:43
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1If you google for "geometrically convex" you will find that it has been used for a notion of convexity analogous to the geometric mean. E.g., see http://arxiv.org/pdf/1312.7725v1.pdf. I think the OP has given no evidence of work on this hence my close vote. – Rob Arthan Sep 18 '16 at 01:56
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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:
bazookabear
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