# Nonnegativity of values of holomorphic functions #

We show that if `f`

is holomorphic on an open disk `B(c,r)`

and all iterated derivatives of `f`

at `c`

are nonnegative real, then `f z ≥ 0`

for all `z ≥ c`

in the disk; see
`DifferentiableOn.nonneg_of_iteratedDeriv_nonneg`

. We also provide a
variant `Differentiable.nonneg_of_iteratedDeriv_nonneg`

for entire functions and versions
showing `f z ≥ f c`

when all iterated derivatives except `f`

itseld are nonnegative.

A function that is holomorphic on the open disk around `c`

with radius `r`

and whose iterated
derivatives at `c`

are all nonnegative real has nonnegative real values on `c + [0,r)`

.

An entire function whose iterated derivatives at `c`

are all nonnegative real has nonnegative
real values on `c + ℝ≥0`

.

An entire function whose iterated derivatives at `c`

are all nonnegative real (except
possibly the value itself) has values of the form `f c + nonneg. real`

on the set `c + ℝ≥0`

.

An entire function whose iterated derivatives at `c`

are all real with alternating signs
(except possibly the value itself) has values of the form `f c + nonneg. real`

along the
set `c - ℝ≥0`

.