| log p$_{Y}$ ( y ; µ$_{1}$ ) | ≤ 1 $_{2}$y 2 + ay + b . Therefore, due to the assumption that µ$_{2}$ ∈ Λ( q ) . Therefore, the dominated convergence theorem provides that Lemma 3 is thus proved. Then µ$_{0}$ ∈ Λ( γ, q ) achieves the capacity if and only if there exists λ ≥ 0 such that λ E$_{µ}$$_{0}$ $^{{}$X 2 − γ } = 0 and E$_{µ}$ { f$_{λ}$ ( X ; µ$_{0}$ ) } ≤ 0 for all µ ∈ Λ( q ) . Proof: Define the Lagrangian where λ is the Lagrange multiplier. Since Λ( q ) is a convex set and I ( µ ) < ∞ on Λ( q ) , µ$_{0}$ is capacity- achieving if and only if there exists λ ≥ 0 such that the following conditions hold [19]: Due to concavity of I ( µ ) , J ( µ ) is also concave. Condition (ii) is then equivalent to that the weak derivative J ′ $_{µ$_{0}$}$( µ ) ≤ 0 for all µ ∈ Λ( q ) . We establish the following sufficient and necessary condition for the optimal input distribution. Lemma 5: Let f$_{λ}$ ( x ; µ ) = d ( x ; µ ) − I ( µ ) − λ ( x 2 − γ ) . (54) lim θ → 0 + 1 θ ∫ ∞ −∞ p$_{Y}$ ( y ; µ$_{θ}$ ) log p$_{Y}$ ( y ; µ$_{θ}$ ) p$_{Y}$ ( y ; µ$_{0}$ ) d y = ∫ ∞ −∞ lim θ → 0 + 1 θ p$_{Y}$ ( y ; µ$_{θ}$ ) log p$_{Y}$ ( y ; µ$_{θ}$ ) p$_{Y}$ ( y ; µ$_{0}$ ) d y (51) = ∫ ∞ −∞ ( p$_{Y}$ ( y ; µ ) − p$_{Y}$ ( y ; µ$_{0}$ )) d y (52) = 0 . (53) J ( µ ) = I ( µ ) − λ E$_{µ}$ $^{{}$X 2 − γ } (55) (i) λ E$_{µ}$$_{0}$ $^{{}$X 2 − γ } = 0 ; (ii) for all µ ∈ Λ( q ) , J ( µ$_{0}$ ) ≥ J ( µ ) . 14 ∫ ∞ −∞ | p$_{Y}$ ( y ; µ$_{2}$ ) log p$_{Y}$ ( y ; µ$_{1}$ ) | d y ≤ ∫ ∞ −∞ p$_{Y}$ ( y ; µ$_{2}$ )$^{(}$ 1 2 y 2 + ay + b$^{)}$ d y (48) = 1 2 E$_{µ}$$_{2}$ $^{{}$X 2 } + a E$_{µ}$$_{2}$ { X } + b + 1 2 (49) < ∞ (50)