# Turan Inequalities

A real entire function

$$\psi(x)=\sum_{k=0}^{\infty} \gamma_k\frac{x^k}{k!}$$

is said to be in the Laguerre-Polya class, denoted $\psi(x) \in \mathcal{LP}$, if it can be represented in the form \begin{eqnarray*} \psi(x)=c x^m e^{-\alpha x^{2}+\beta x} \prod_{k=1}^{\infty}\left(1+x/x_{k}\right)e^{- x/x_k}, \end{eqnarray*} where $c$, $\beta$, $x_{k}$ are real numbers, $\alpha \geq 0$, $m$ is a nonnegative integer and $\sum x^{-2}_{k}< \infty$.

It is well known that a necessary condition for a real entire function $\psi(x)$ to belong to $\mathcal{LP}$ class is that its Maclaurin coefficients satisfy the Turan inequalities

\begin{eqnarray} \gamma_{k}^2-\gamma_{k-1}\gamma_{k+1}\geq 0, \end{eqnarray}

for $k\geq 1$.

Question: Who proved the necessary condition? When? How?

First notice that $\gamma_k=\psi^{(k)}(0)$ and the inequalities follow from more general inequalities $$(\psi^{(k)})^2(x)-\psi^{(k-1)}(x)\psi^{(k+1)}(x)\geq 0.$$ As the class of polynomials with all zeros real is closed under differentiation, it is enough to prove this for $k=1$. As the class is also closed with respect to real shifts, what remains to prove is your inequality (at $x=0$) with $k=1$ for polynomials with all zeros real. Wlog $\psi(0)=1$, so $$\psi(x)=\prod\left(1+x/x_k\right),$$ $x_k$ are real, so $$\psi'(0)=\sum\frac{1}{x_k},\quad \psi''(0)=\sum\frac{1}{x_ix_j}$$ and our inequality becomes trivial.