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Lemma 2 If f ∈ C 1 , f ( 0, 0 ) = 0 and Q ∈ C 1 ( [ 0, 1 ] × [ 0, 1 ] ), then F α β ∈ C 1 [ 0, 1 ] and F α β ( 0 ) = 0. Proof Using the explicit form of the Green's function given in (10), it is not complicated to prove that function F α β, defined by formulas (16) and (19), belongs to the class of functions C 1 [ 0, 1 ] and F α β ( 0 ) = 0. Lemma 2 is proved.

It is easy to prove that the function (f x,y)= sqrt[p]{x}sqrt[q]{y}) is a concave function on (D={(x,y) mid xgeq0, ygeq0}).

It is easy to prove that the function (f x,y)= (sqrt[p]{x}+sqrt[p]{y})^{p}) is a concave function on (D={(x,y) mid xgeq0, ygeq0}).

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).

Hence it remains to prove that the function D 2 is also positive and increasing.

It is easy to prove that this function satisfies conditions (A1 - A3).

It is easy to prove that the function s ↦ [ y 0, y 1, y 2 ; f s ] is exponentially convex.

To prove that the function C(p) is convex, we show that C"(p) is non-negative for 0 ≤ p ≤ 1.

In order to prove that the function is convex over, we derive its Hessian and then show that it is positive semidefinite.

From the above discussion, we only need to prove that the function (f(theta)) obtained by (1.9) belongs to (L^{2}[-pi, pi]).

It is easy to prove that the function s ↦ [ y 0, y 1, y 2 ; k s ] is also exponentially convex.