The function \(\Phi(t)\) and some inequalities involving it (Q2713533)

Let NEWLINE\[NEWLINE \xi (x/2)=8\int^{\infty}_{0}\Phi (t)\cos (xt) dt, NEWLINE\]NEWLINE where NEWLINE\[NEWLINE \Phi (t)=\pi e^{5t}\sum^{\infty}_{n=1}n^2(2n^2 \pi e^{4t}-3)e^{-\pi n^2 e^{4t}}. NEWLINE\]NEWLINE In the proof of Turán's inequalities associated with the moment problem of Pólya, \textit{G. Csordas, T. S. Norfolk} and \textit{R. S. Varga} [Trans. Am. Math. Soc. 296, 521-541 (1986; Zbl 0602.30030)] showed that the function NEWLINE\[NEWLINEK(t)=\int^{\infty}_{t}\Phi (\sqrt{u}) du\quad (t\geq 0) NEWLINE\]NEWLINE is strictly logarithmically concave.NEWLINENEWLINENEWLINEHere the authors show that the first four derivatives of \(\log\Phi(t)\) are negative for all \(t>0\).