Contributors here take the task on, with papers on such aspects as the maximal characterization of Hardy-Sobolev spaces on manifolds, approximately Gaussian marginals and the hyperplane conjecture, on the existence of sub-Gaussian directions for log-concave measures, isoperimetric bounds on convex manifolds, and the log-convex
We note that 0-convex function is simply log-convex function and 1-convex function is ordinary convex function.
If r = 0 and p [not equal to] 0, then f is log-convex, it follows from (1) and (2) that
i[phi]] (v - u) = v - u, the [phi]-convex set K becomes the convex set and consequently, [phi]-convex, [phi]-invex, and log-[phi]-invex functions reduce to convex and log-convex functions.
Dragomir and Mond  proved the following Hermite-Hadamard type inequalities for the log-convex functions:
Pachpatte  has also obtained some other refinement of the Hermite-Hadamard inequality for differentiable log-convex functions.
Mond: Integral inequalities of Hadamard type for log-convex functions, Demonst.
Pachpatte: A note on integral inequalities involving two log-convex functions, Math.
Keywords and Phrases: Ostrowski inequality, Gruss inequality, Cebysev inequality, Convex functions, Log-convex functions.
Recall the following definitions of a convex function and a log-convex function:
The functions f and g are called convex on [a, b] and log-convex on [a, b], respectively, if
2]) If |f'| and |g'| are log-convex on [a, b], then