I have objective function f(D,h), and I want to minimize it, but first I want to check if the function is convex and I don't know how to prove it, I'm thinking about the sum of convex functions on the same interval is convex. and I can see that some of them are probably a convex (e.g., log function), but I'm not sure about others. any help?
$$f(D, h) = \frac 3{D^2} + \frac 1{h^2} \left(\frac{\pi^2}6 + \frac{1.3003}{\left(\frac Dh\right)^{1.067}}\right) + \frac 2{D^3h}\left(0.2658 + \frac hD + 1.0799\log\left(\frac Dh\right)\right)^2$$
where $1.5 < h \le 86$, $28 \le D \le 125$
I used MAPLE to check the eigenvalues of the Hessian of f(D,h) w.r.t. D and h for a variety of values of D and h within the prescribed bounds, and all were positive. So I think, but have not proven, that f(D,h) is jointly convex in D and h within the region specified by the bounds.
Nevertheless, I used BARON to find the global minimum of f(D,h) within the prescribed bounds and obtained the global optimum at the boundary h = 86, D = 125. Even if f(D,h) is not convex, this is the globally optimal solution. I also obtained this same solution using several local optimizers (for which any local minimum found would be a global optimum if the objective function is indeed convex).
The above conclusions hold whether your log is base e or base 10.
Edit: The region in which joint convexity (apparently) holds is limited. Small values of D (somewhat below the specified lower bound of 28) can result in an indefinite Hessian, thereby showing non-convexiity. For example, at D=5, h=86, the Hessian has one negative eigenvalue, whether the log is base e or base 10. This does not affect my conclusions above about the global minimum within the specified region.