Let for $\Re s>0$ $$\eta(s)=\sum_{n=1}^\infty\frac{(-1)^{n-1}}{n^{s}}$$ the Dirichlet eta function. After I read Example 5 and Corollary 9 from [1], I've asked myself an analogue
Question. If I define for $t\in \left( 0,\infty \right) $ (our $x$ inside the integral is thus a positive real) $$f(t):=-\int_0^t\Re\frac{\eta' \left( \frac{1}{2}+ix \right) }{\eta \left( \frac{1}{2}+ix \right) }dx,$$ a) is it well-defined?, and b) what's about the convexity of this function for $t\in \left( 0,\infty \right) $?
Since I know the calculations of professors in [1] to get such the analogue in which I was inspired, then as an answer for b) if you want only is required the key hint to study/deduce what's about the convexity ($f''(t)\geq 0$, $\forall t>0$) of our $f(t)$. Many thanks.
Was fixed a typo concerning the condition to be convex.
References:
[1] Arias de Reyna and Van de Lune, On the exact location of the non-trivial zeros of Riemann's Zeta function, Acta Arithmetica 163.3 (2014).