The Riemann Zeta function, like most other complex functions, are much easier to deal with in the real line, since the values are also real, and definitions can be used in a more straightforward manner. I can easily prove monotony properties on the real line (and in some cases for other horizontal half lines) for Zeta(x), Zeta'(x), Zeta'(x)/Zeta(x), Xi(x), Dirichlet's alternating Zeta etc.
I also 'see' from Wolfram Mathematica plottings, that the monotony (and convexity) seem to hold too if I cut off the singular parts (but an increasing function can turn decreasing, e.g. in case of Zeta'(x)/Zeta(x)). The Laurent-series (or similar series) representations aren't seem to be useful, since these are usually alternating series, and I couldn't find a rule for the signs of the Stieltjes-constants (they seem to be totally random, a quotation from Wolfram Alpha: 'lengths of consecutive signs are 1, 2, 3, 4, 3, 4, 5, 4, 5, 5, 5, ... ').
Is there any other way to deal with this? I'm particularly interested in Zeta'(x)/Zeta(x)+1/(x-1) right now. I would be really grateful if someone could give some ideas.
Thanks, Gabor