For monotone Hamburger Hamiltonians with regularly varying parameters, the monodromy matrix grows like r times an inverse-function integral, yielding the exact Nevanlinna order 1/(2(β-1)) in the simply critical Jacobi case with 3/2<β<2.
Riesz, Sur le probl` eme des moments
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Growth estimates for Nevanlinna matrices of order larger than one half
For monotone Hamburger Hamiltonians with regularly varying parameters, the monodromy matrix grows like r times an inverse-function integral, yielding the exact Nevanlinna order 1/(2(β-1)) in the simply critical Jacobi case with 3/2<β<2.