Nonextensive Statistical Mechanics

6.2 Further Generalizing 213∫ W{∫ +∞−1S {F} =F(κ) u dκ} κ du . (6.13)1 −∞The generalized logarithm of Eq. (6.11) appears to be isomorphic to the generalizedlogarithm introduced recently by Naudts [401] who started from a differentperspective.We can straightforwardly prove that, assuming that F(κ) is normalized, the followingproperties hold:henceln {F} 1 = 0 , (6.14)exp {F} 0 = 1 . (6.15)Alsoddx ln {F} x∣ =x=1ddx exp {F} x ∣∣∣∣x=0= 1 , (6.16)as well as monotonicity, more preciselyandddx ln {F} x > 0, ∀x ∈ (0, +∞) , (6.17)ddx exp {F} x > 0, ∀x ∈ A exp {F}, (6.18)where ∈ A exp{F} is the set of admissible values of x for the nonnegative exp {F} xfunction. When no negative q contributes (i.e., if F(κ) = 0, ∀κ < 0), then thefollowing properties hold also:andd 2dx 2 ln {F} x < 0 (concavity) , (6.19)d 2dx exp 2 {F} x < 0 (convexity) . (6.20)Analogously, when no positive q contributes (i.e., if F(κ) = 0, ∀κ >0), thend 2dx 2 ln {F} x > 0 (convexity) , (6.21)