  Continuous DR-submodular functions are a class of functions that satisfy the
Diminishing Returns (DR) property, which implies that they are concave along
non-negative directions. Existing works have studied monotone continuous
DR-submodular maximization subject to a convex constraint and have proposed
efficient algorithms with approximation guarantees. However, in many
applications, e.g., computing the stability number of a graph and mean-field
inference for probabilistic log-submodular models, the DR-submodular function
has the additional property of being \emph{strongly} concave along non-negative
directions that could be utilized for obtaining faster convergence rates. In
this paper, we first introduce and characterize the class of \emph{strongly
DR-submodular} functions and show how such a property implies strong concavity
along non-negative directions. Then, we study $L$-smooth monotone strongly
DR-submodular functions that have bounded curvature, and we show how to exploit
such additional structure to obtain algorithms with improved approximation
guarantees and faster convergence rates for the maximization problem. In
particular, we propose the SDRFW algorithm that matches the provably optimal
$1-\frac{c}{e}$ approximation ratio after only $\lceil\frac{L}{\mu}\rceil$
iterations, where $c\in[0,1]$ and $\mu\geq 0$ are the curvature and the strong
DR-submodularity parameter. Furthermore, we study the Projected Gradient Ascent
(PGA) method for this problem and provide a refined analysis of the algorithm
with an improved $\frac{1}{1+c}$ approximation ratio and a linear convergence
rate. Given that both algorithms require knowledge of the smoothness parameter
$L$, we provide a \emph{novel} characterization of $L$ for DR-submodular
functions showing that in many cases, computing $L$ could be formulated as a
convex problem, i.e., a geometric program, that could be solved efficiently.
