Suppose, $B(y)$ is some function of $y$, where $B$ must be convex.
Now, for some functions of $x$, $f(x)$ and $g(x)$, both of which are strictly positive, we consider $h(f)$ and $h(g)$ for these functions $f$ and $g$.
Then, we have, $$B(h(g)) - B(h(f)) - (h(g) - h(f)) B^{'}(h(f)),$$ where, $$B^{'}(h(f)) = \Bigg[ \frac{d}{dy} B(y) \Bigg]; \; \; y = h(f).$$
For example, if $h(f) = f$, then it reduces to: $$B(g) - B(f) - (g - f)B^{'}(f).$$
Now, what would be the choice of $B(.)$ and $h(.)$ so that we get: $$A.g^{A + C} - A.f^{A + C} - (A + C)\bigg(g^A.f^C - f^{A + C} \bigg) \; ?$$
Here, $$A + C = 1 + \alpha;\;\;\;\; A = 1 + \lambda (1 - \alpha); \;\;\; B = \alpha - \lambda(1 - \alpha). \;\;\; $$
Here, $$\alpha \in (0,1), \; \; \; \alpha \in R, \; \; R \textrm{ is the real number line.}$$