  In 1907 M.Petrovitch initiated the study of a class of entire functions all
whose finite sections are real-rooted polynomials. An explicit description of
this class in terms of the coefficients of a series is impossible since it is
determined by an infinite number of discriminantal inequalities one for each
degree. However, interesting necessary or sufficient conditions can be
formulated. In particular, J.I.Hutchinson has shown that an entire function
p(x)=a_0+a_1x+...+a_nx^n+... with strictly positive coefficients has the
property that any its finite segment a_ix^i+...+a_jx^j has all real roots if
and only if for all i=1,2,... one has a_i^2/a_{i-1}a_{i+1} is greater than or
equal to 4. In the present paper we give sharp lower bounds on the ratios
a_i^2/a_{i-1}a_{i+1} for the class considered by M.Petrovitch. In particular,
we show that the limit of these minima when i tends to infinity equals the
inverse of the maximal positive value of the parameter for which the classical
partial theta function belongs to the Laguerre-Polya class.
