Thread: complex analysis, inequality contradiction

I realise this is not calculus, yet still I'd appreciate if someone could help me

I have an entire function, f
and I need to show that the following is NOT true for ALL n=1,2,....

|f⁽ⁿ⁾(0)| >= nⁿ *n!

(ie, the modulus of the nth derivative of f at zero is greater or equal to n to the power of n, times n factorial)

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so, my first attempt was by counter example, I picked a simple polynomial and showed the above expression is not true for all n.
however, I'm not sure if the question just wants a counter example, or a generalised proof for all such entire functions. If so, I'm slightly stumped... the only thought I have in this case is to look at Taylor series, but what will that yield?

You are in the correct path. The negation of a "for all" is "there exists" so to disprove this you only need one counter example. A polynomial would work very well as after taking derivatives of an nth degree polynomial they are all zero! The right hand side is always non negative.

Originally Posted by TheEmptySet

The negation of a "for all" is "there exists" so to disprove this you only need one counter example.

ah, ah, here is where I became unsure about the way to go about this.. it doesn't say "FOR ALL entire functions f .....", it just says "let f be entire, and then for all n we have ....".
so if we pick a different function, we could assume the statement to still be true (even though we know it's not)

do you see a difference? maybe i need to prove it more generally
or am i just making things more complicated..

ah Okay then. Do a google search for Liouville's theorm. The jist of this is that the constants in a taylor expansion are bounded by a polynomial.

Liouville's theorem (complex analysis) - Wikipedia, the free encyclopedia