After that I've tried understand the problem 6416 [1983, 60] A Second-Order Differential Inequality proposed by Sandford S. Miller in the American Mathematical Monthly (myself proposal is significantly different for other function and domain, inequality...but Mocanu and previous author have literature respect more general context; I am not able to read this high mathematics, but this kind of differential equation seem very interesting), I've asked to me questions likes as the following, where $\eta(s)$ the Dirichlet eta function, and $\zeta(s)$ the Riemann Zeta function
Question. Are there complex numbers $a_2,a_1,a_0$ such that if $$\Im(a_2\eta''(s)+a_1\eta'(s)+a_0\eta(s))<0$$ holds for $0< \Re s<\frac{1}{2} $ implies $$\Re\zeta(s)>\frac{1}{2}?$$ Thanks in advance.
Excuse me, previous question was modified to fix a typo.
I believe that my problem is well-possed. Why it and no other? My only purpose is learn if such question is easy to deduce what are the key computations, and if it is absurd then know one few more about these special functions.
You may check two preprints by Hisashi Kobayashi (2016) :
"Some results on the $ξ(s)$ and $Ξ(t)$ functions associated with Riemann's $ζ(s)$ function" arXiv:1603.02954
"Local Extrema of the $Ξ(t)$ Function and The Riemann Hypothesis" arXiv:1603.02911
And one paper by Sondow, J. and C. Dumitrescu (2010):
“A monotonicity property of Riemann’s xi function and a reformulation of the Riemann hypothesis,” Periodic math. Hung. 60 I (2010), 37-40. Also available at arXiv:1005.1104
Sondow, J. and C. Dumitrescu prove that Riemann's $\xi(\sigma+it)$ function is strictly increasing (respectively, strictly decreasing) in modulus $|\xi(\sigma+it)|$ along every horizontal (in $\sigma$ direction) half-line in any zero-free, open right (respectively, left) half-plane. A corollary is a reformulation of the Riemann Hypothesis.
Kobayashi's preprints showed that the derivatives of the modulus of a function relates to various real or imaginary parts of that function.