TITLE: Generalization of log-convexity (log-concavity): log-log-convexity (log-log-concavity)?
QUESTION [6 upvotes]: $\underline{\mathrm{Background\; on\; function\; Convexity}}$
A function, $f$, is convex if:
$$f( x\theta+y(1-\theta) ) \leq \theta f(x) + (1-\theta)f(y).$$
$f$ is concave if $-f$ is convex,  [1]. If we can demonstrate the convexity (or concavity) of a function, then function optimization can be performed using the well understood theory of convex optimization. Thus, if we can, it is beneficial to be able to do so.
In some situations, a function may not be convex (or concave). However, its logarithm may still be. This is termed log-convexity (log-concavity). Determining log-convexity (log-concavity) allows us to exploit the same convex optimization procedures.
$\underline{\mathrm{Question}}$
I want to know if the idea of log-convexity can be extended further. In particular, can we describe a function as being log-log-convex (log-log-concave)? By log-log-convex (log-log-concave) I mean that if the logarithm of a function is plotted on a set of axes in which the abscissa has been expressed using a logarithmic scale, convexity (concavity) is observed. Said in a different way, if we plot a function on a set of axes where both the ordinate and abscissa have been expressed using a logarithmic scale, convexity (concavity) is observed.

REPLY [1 votes]: The answer is yes, at least for log concavity. It also depends what you are after. If you are trying to approximate a linear function you do not generally want log convexity, but you may want to strengthen log concavity.
You may be interested in this line of research see 
Saumard, Adrien; Wellner, Jon A., Log-concavity and strong log-concavity: a review,  ZBL06386273.

They mention many different stronger forms its a detailed study of strong forms of log concavity. There are stronger forms of then Strong log concavity/log convexity as well (strong log concavity, Central Convex uni modal log concavity and whole bunch of properties. these even have midpoint like forms as jensen's convexity/concavity does in the discrete/non continuous case. 
See Marshall, Albert W.; Olkin, Ingram, Inequalities via majorization - an introduction, General inequalities 3, 3rd int. Conf., Oberwolfach 1981, ISNM 64, 165-187 (1983). ZBL0572.26013. for this
. There is also a term called log log concavity mentioned in the paper 
These forms of stronger log concavity or convexity put  tighter bounds on the functions. These forms or at least their stronger variants can often be stronger then concavity unlike log concavity ( and like log convexity in that implies convexity) . 
See the following paper. Its quite detailed. It also mentions ULTRA LOG CONCAVITY and many variants and types. There are presumably infinitely many types that one could produce.
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