f-0-0-is-convex-function-for-n-2-an-integer-prove-f-1-f-1-f-2-f-2-f-n-f-n-1-f-1-f-2-f-n-f-1-f-2-f-n-1-n-f-1-f-n-

Question Number 161162 by metamorfose last updated on 13/Dec/21

f :] 0, + \infty [\to] 0, + \infty [i s c o n v e x f u n c t i o n

f o r n ⩾ 2 a n i n t e g e r, p r o v e :

{(f {(1)}^{f (1)} f {(2)}^{f (2)} \dots f {(n)}^{f (n)})}^{\frac{1}{f (1) + f (2) + \dots + f (n)}} + {(f (1) f (2) \dots f (n))}^{\frac{1}{n}} ⩽ f (1) + f (n)