Minimization inequality problem

We have that : $2^a + 4^b + 16^c = 2^a + 2^{2b} + 2^{4c} = 2^a + \frac{1}{2} \cdot 2^{2b+1} + \frac{1}{4} \cdot 2^{4c+2}.$ In order to use Jensen's inequality (since the function $x\longmapsto 2^x$ is convex), the sum of the coefficients must be equal to 1. Thus we write : $2^a+4^b+16^c = \frac{7}{4} \cdot \left( \frac{4}{7} \cdot 2^a + \frac{2}{7} \cdot 2^{2b+1} + \frac{1}{7} \cdot 2^{4c+2} \right)$ Jensen gives us : $2^a+4^b+16^c \geq \frac{7}{4}\cdot 2^{\frac{4}{7} \cdot a + \frac{2}{7} \cdot (2b+1) + \frac{1}{7} \cdot (4c+2)} = \frac{7}{4} \cdot 2^{\frac{4}{7}(a+b+c+1)} = \frac{7}{4} \cdot 2^\frac{32}{7} = 7 \cdot 2^{\frac{18}{7}}$

Equality holds if and only if $a = 2b+1 = 4c+2$ which is clearly possible knowing that $a+b+c=7$