Consider a projectile moving in a plane. One of many different models for this problem is the following ordinary differential equation
\begin{align} x''(t) &= -Ex'(t), \\ y''(t) &= -Ey'(t)- g, \end{align}
together with the initial condition
\begin{alignat}{2} x(0) &= 0, \qquad x'(0) &= v_0 \cos(\theta), \\ y(0) &= 0, \qquad y'(0) &= v_0 \sin(\theta). \end{alignat}
Here $E \ge 0$ is a non-negative function which specifies the friction, $g > 0$ is the acceleration due to gravity, $v_0$ is the muzzle velocity of the gun and $\theta$ is the elevation of the gun. The case of $\theta=\frac{\pi}{2}$ corresponds to shooting straight into the air. Cases of special interest include \begin{equation} E = f(v)v, \end{equation} where $f$ reflects the geometry of the projectile or the more general \begin{equation} E = E(v,y), \end{equation} which also allows the atmosphere to vary with the height of the projectile above the ground level $y=0$.
The time of flight $\tau = \tau(\theta)$ from the muzzle of the gun to the point of impact is the unique positive solution of the equation
\begin{equation} y(\theta,\tau(\theta)) = 0. \end{equation}
The range $r$ of the gun is defined as
\begin{equation} r(\theta) = x(\theta,\tau(\theta)), \end{equation}
i.e. the $x$ coordinate of the shell at the point of impact.
I have enough numerical evidence to suggest that the range is a concave function of the elevation, but no proof. I imagine that this is a result which is at about 90 years old, but I have not found a proof in the literature and I have not had any luck myself.
The result is significant in the context of computing a firing solution for a target with coordinates $(d,0)$, i.e. solving the non-linear equation
\begin{equation} r(\theta) = d \end{equation}
with respect to elevation $\theta$. While a precomputed firing table should always be used to generate a good initial guess $\theta \approx \theta_0$, I have observed that the secant method as well as Newton's method for this equation are both globally convergent. These observations would be explained if the range function could be shown to be concave.
I welcome ideas and suggestions about how to attack this problem, but since I suspect that this is an old result, I am primarily looking for a reference to a specific paper.