Here are plots:

ComplexPlot[Log[Abs[h[v]]], {v, 0 + 0*I, 10 - 10*I}] 

ComplexPlot3D[Log[Abs[h[v]]], {v, 0 + 0*I, 10 - 10*I},PlotPoints -> 50] 

It is difficult with the extrema since ComplexExpand[h[s + I*t]] takes 140.4 MB.The following works.

NMaximize[{ComplexExpand[Log[Abs[h[s + I*t]]]], s >= 0 && s <= 10 && t >= -10 && t <= 0}, {s, t}] 

{56.0063, {s -> 1.21431*10^-9, t -> -10.}}

NMinimize[{ComplexExpand[Log[Abs[h[s+I*t]]]],s>=0&&s<=10&&t>=-10&&t<=0},{s,t}] 

{14.8977 + 0. I, {s -> 8.31982, t -> -1.88003}}

Here are plots:

ComplexPlot[Log[Abs[h[v]]], {v, 0 + 0*I, 10 - 10*I}] 

ComplexPlot3D[Log[Abs[h[v]]], {v, 0 + 0*I, 10 - 10*I},PlotPoints -> 50] 

It is difficult with the extrema since ComplexExpand[h[s + I*t]] takes 140.4 MB.The following works.

NMaximize[{ComplexExpand[Log[Abs[h[s + I*t]]]], s >= 0 && s <= 10 && t >= -10 && t <= 0}, {s, t}] 

{56.0063, {s -> 1.21431*10^-9, t -> -10.}}

NMinimize[{ComplexExpand[Log[Abs[h[s+I*t]]]],s>=0&&s<=10&&t>=-10&&t<=0},{s,t},Method->"DifferentialEvolution"] 

{-13.3409 + 0. I, {s -> 0., t -> -1.48265*10^-9}}

Here are plots:

ComplexPlot[Log[Abs[h[v]]], {v, 0 + 0*I, 10 - 10*I}] 

ComplexPlot3D[Log[Abs[h[v]]], {v, 0 + 0*I, 10 - 10*I},PlotPoints -> 50] 

It is difficult with the extrema since ComplexExpand[h[s + I*t]] takes 140.4 MB.

Here are plots:

ComplexPlot[Log[Abs[h[v]]], {v, 0 + 0*I, 10 - 10*I}] 

ComplexPlot3D[Log[Abs[h[v]]], {v, 0 + 0*I, 10 - 10*I},PlotPoints -> 50] 

It is difficult with the extrema since ComplexExpand[h[s + I*t]] takes 140.4 MB.The following works.

NMaximize[{ComplexExpand[Log[Abs[h[s + I*t]]]], s >= 0 && s <= 10 && t >= -10 && t <= 0}, {s, t}] 

{56.0063, {s -> 1.21431*10^-9, t -> -10.}}

NMinimize[{ComplexExpand[Log[Abs[h[s+I*t]]]],s>=0&&s<=10&&t>=-10&&t<=0},{s,t}] 

{14.8977 + 0. I, {s -> 8.31982, t -> -1.88003}}