A CONJECTURE (could someone help ? )

[zetafunction]

i have the following conjecture about infinite power series

let be a function f(x) analytic so it can be expanded into a power series

$$f(x)= \sum_{n=0}^{\infty} a_{2n}(-1)^{n} x^{2n}$$

with $$a_{2n} = \int_{-c}^{c}dx w(x)x^{2n}$$ $$w(x) \ge 0$$ and $$w(x)=w(-x)$$ on the whole interval (-c,c) in case c is inifinite this W(x) is ALWAYS positive for every real 'x'

here 'c' is a Real constant , also we can have $$c=\infty$$

and $$a(n)=0$$ for n=1,3,5,7,9,.. 2n-1

the $$a(2n)$$ are ALL positive and NONZERO

then f(x) has ONLY real roots or f(x) has NO roots at all (not real or complex)

amazingly it seems true , for example it holds for $$sin(x)/x$$ the Bessel function of zeroth order $$J_0 (x)$$ and so on

 

[Eynstone]

i have the following conjecture about infinite power series

let be a function f(x) analytic so it can be expanded into a power series

$$f(x)= \sum_{n=0}^{\infty} a_{2n}(-1)^{n} x^{2n}$$

with $$w(x) \ge 0$$ and $$w(x)=w(-x)$$ on the whole interval (-c,c) in case c is inifinite this W(x) is ALWAYS positive for every real 'x'

here 'c' is a Real constant , also we can have $$c=\infty$$

and $$a(n)=0$$ for n=1,3,5,7,9,.. 2n-1

the $$a(2n)$$ are ALL positive and NONZERO

then f(x) has ONLY real roots or f(x) has NO roots at all (not real or complex)

amazingly it seems true , for example it holds for $$sin(x)/x$$ the Bessel function of zeroth order $$J_0 (x)$$ and so on

The series simplifies to 2 *integral( 0 to c) {1 / 1+ (tx)^2}w(t) dt , which is posotive for all
real x. I'm not sure about the complex case, though.