  For an entire function $f(z) = \sum_{k=0}^\infty a_k z^k,$ $a_k >0,$ we
define the sequence of the second quotients of Taylor coefficients $Q := \left(
\frac{a_k^2}{a_{k-1}a_{k+1}} \right)_{k=1}^\infty$. We find new necessary
conditions for a function with a non-decreasing sequence $Q$ to belong to the
Laguerre--P\'olya class of type I. We also estimate the possible number of
nonreal zeros for a function with a non-decreasing sequence $Q.$
