MathDB

A smooth function

f:I\to \mathbb{R}

is said to be totally convex if

(-1)^k f^{(k)}(t) > 0

for all

t\in I

and every integer

k>0

(here

I

is an open interval).
Prove that every totally convex function

f:(0,+\infty)\to \mathbb{R}

is real analytic.
Note: A function

f:I\to \mathbb{R}

is said to be smooth if for every positive integer

k

the derivative of order

k

f

is well defined and continuous over

\mathbb{R}

. A smooth function

f:I\to \mathbb{R}

is said to be real analytic if for every

t\in I

there exists

\epsilon> 0

such that for all real numbers

h

with

|h|<\epsilon

the Taylor series

\sum_{k\geq 0}\frac{f^{(k)}(t)}{k!}h^k

converges and is equal to

f(t+h)

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