Consider the following polynomials
$f(x)=a_0+a_1x+\cdots a_mx^m$
$g(x)=b_0+b_1x+\cdots+b_nx^n$,
such that the roots of $f$ are all real, and the roots of $g$ are all real and of the same sign, then the Hadamard product
$$f\circ g(x)=a_0b_0+a_1b_1x+a_2b_2x^2+\cdots$$
has all roots real.
This is proved in
E. Malo, Note sur les équations algébriques dont toutes les racines
sont réelles, Journal de Mathématiques Spéciales, (4), vol. 4 (1895)Question : Can this result be extended to Analytic functions (represented by infinite series of course) with the same conditions?
Best Answer
Partial answer: If $f,g$ are of the Laguerre-Polya class and $g$ is type I (compact approximation by polynomials with only negative real roots say) then the result holds by Malo's theorem since $f$ can be compactly approximated by polynomials $P_k$ with real zeroes (equivalent definition of L-P class), $g$ can be compactly approximated by polynomials $Q_k$ with negative real zeroes. Then if we denote the convolution (Hadamard product) by $*$ as usual, the integral representation:
$2\pi if*g(z)=\int_{|w|=r}{f(\frac{z}{w})g(w)}\frac{dw}{w}$, shows that $P_k*Q_k \to f*g$ compactly and then Malo's theorem applies so $P_k*Q_k$ has only real roots, hence $f*g$ does (trivially by Hurwitz) and it also follows that $\sum{a_nb_nz^n}=f*g(z)$ is in the Laguerre-Polya class by definition.