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Suppose $f(0)=0$, $f^{\prime }(0)=c$, $h=(f^{\prime }{)}^{-1}$ is differentiable, and $k_{-}<h^{\prime }/2<k_{+}<0$. Then $h(c)=g(c)=0$ for the concave Legendre dual $g$, andIntegrate the derivative bounds from $c$ to $z$, then integrate $g^{\prime }=h$. Multiplying the first inequality by a negative $z-c$ reverses both comparisons. A bound valid without cases is $|h(z)|\le -2k_{-}|z-c|$.

Let $f$ be smooth and strictly concave, with $f(0)=0$ and $f^{\prime }$ bijective, and let $u_0$ be smooth with compact support. For $0<t<t_{\ast }$ in the smooth concave-flux characteristic lifespan, the primitive $U_x=u$ solves the Hamilton-Jacobi equation $U_t+f(U_x)=0$ and obeyswhere $g$ is the concave Legendre dual. The tangent inequality for a concave function bounds every candidate by $U(t,x)$, and the backward characteristic curve attains equality. Its foot $x_0$ is the unique maximizer, and $u(t,x)=h((x-x_0)/t)$ for $h=(f^{\prime }{)}^{-1}$. This statement concerns the smooth solution, without asserting a post-crossing continuation.

Because $f^{\prime }$ is a decreasing bijection, its inverse $h$ exists and is continuous. For any $y$, choose $s=h((x-y)/t)$ in the preceding inequality. The line from $(0,y)$ then reaches $(t,x)$, giving $U(t,x)\ge U(0,y)+tg((x-y)/t)$.

Now let $x_0=X_t^{-1}(x)$ and $s=u_0(x_0)$. On this characteristic curve, $u=s$, so the tangent-line inequality is an equality throughout. This proves attainment andIn fact the maximizing foot is unique. The concave Legendre dual can be written $g(z)={\inf }_s(zs-f(s))$ and satisfies $g^{\prime }(z)=h(z)$: compare the minimizing values at $z$ and $z+\epsilon$ and use continuity of $h$. This avoids assuming differentiability of $h$, which strict concavity by itself does not guarantee. Differentiating the maximizing expression with respect to $y$ gives $u_0(y)-h((x-y)/t)=0$, hence $X_t(y)=x$ and $y=x_0$. The maximum representation for a concave conservation law therefore reproduces the characteristic solution of a scalar conservation law throughout the smooth lifespan.

For increasing strictly concave wealth value $v$, the dual $J(z)={\sup }_x[v(x)-xz]$ is convex. At an interior optimum $z=v^{\prime }(x)$, $J^{\prime }=-x$, $J^{\prime \prime }=-1/v^{\prime \prime }>0$, and $v=J-z{J}^{\prime }$. This sign convention is the negative of the concave Legendre dual. It can turn the optimized portfolio term of a Hamilton-Jacobi-Bellman equation into a linear second derivative.