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?