Structure of the Nuttall partition for some class of four-sheeted Riemann surfaces

The structure of a Nuttall partition into sheets of some class of four-sheeted Riemann surfaces is studied. The corresponding class of multivalued analytic functions is a special class of algebraic functions of fourth order generated by the function inverse to the Zhukovskii function. We show that in this class of four-sheeted Riemann surfaces, the boundary between the second and third sheets of the Nuttall partition of the Riemann surface is completely characterized in terms of an extremal problem posed on the two-sheeted Riemann surface of the function <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="w"> <mml:semantics> <mml:mi>w</mml:mi> <mml:annotation encoding="application/x-tex">w</mml:annotation> </mml:semantics> </mml:math> </inline-formula> defined by the equation <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="w squared equals z squared minus 1"> <mml:semantics> <mml:mrow> <mml:msup> <mml:mi>w</mml:mi> <mml:mn>2</mml:mn> </mml:msup> <mml:mo>=</mml:mo> <mml:msup> <mml:mi>z</mml:mi> <mml:mn>2</mml:mn> </mml:msup> <mml:mo>−</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">w^2=z^2-1</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. In particular, we show that in this class of functions the boundary between the second and third sheets intersects neither the boundary between the first and second sheets nor that between the third and fourth sheets.