Let X be a continuous random variable and its probability density function ({p}:Irightarrow ( {0,infty} ) ) be a differentiable log-concave function, and let the twice differentiable function (varphi: Irightarrow J ) satisfy the following conditions: I,J subset 0,infty),qquad {varphi^{{prime}}} ( t ) >0,qquad {varphi^{{primeprime}}} ( t ) geqslant0, quadforall tin I, where I and J are intervals.
Let be a differentiable convex.
Let be a differentiable function, and let be its derivative.
According to Corollary 5, we just need to prove that the function (p_{J}^triangleq pbulletmathscr{A} ^{-1} (mathscr{A} ^{-1} )^{prime} ) is a differentiable log-concave function under the hypotheses of assertions (I) and (II).
Since (p:(alpha,beta)rightarrow 0,infty)) is a differentiable log-concave function, by Proposition 2, we know that the function (log{p} )^{prime}(x)=psi^{prime}(x),quad xin{ alpha,beta)}, is monotone decreasing, hence psi^{prime}(x leqslantpsi^{prime}(t),quad forall {t}: alpha< t leqslant x, forall {x}: alpha< x< beta.
If and is a differentiable GA-convex (concave) function, then (1.3).
Moreover, let be a concave continuously differentiable utility function (of average rate ) associated with user.
(2),, are convex and twice continuous differentiable functions in, for all. . as well as the objective vector function (which can be transformed into a single function in applications),, are twice continuous differentiable concave functions.,, are convex and twice continuous differentiable functions in, for all.
In period 2 the joint venture produces a surplus according to a production function F a,b) which is continuously differentiable, concave in the investment levels and satisfies: begin{array}rcl@ F 0,0)=0,quad frac{partial F}{partial x}>0, quad frac{partial^{2}F}{partial{x}^{2}}<0qquadforall: x:in:{a,b} end{array} (1).
