For analytic functions , the so-called principle of modulus maximum is valid, which prescribes a clear arrangement of the modulus maximum for an analytic function in a certain limited region exclusively on the boundary of this region. In the general case, for unbounded domains, this assumption is false. However, when some additional restrictions are imposed on the function, it can be shown that the function will be bounded modulo and in an unbounded region.
The Phragmén-Lindelöf principle for an unlimited sector
Let the function analytical in the sector {\ displaystyle S = \ {z: - {\ frac {\ pi} {4}} <\ arg z <{\ frac {\ pi} {4}} \}} and continuous on its border. Then, if the inequality and there are constant such that inequality holds across the sector then the inequality fair throughout the sector.
The Phragman-Lindelöf principle for the vertical half-strip
Let be - an infinite vertical half-strip, further, let there be constant such that the inequality , and in the strip itself the inequality {\ displaystyle | f (z) | \ leq B {(\ mathop {\ mathrm {Im}} \, f (z))} ^ {A}} . Then performed in the entire strip.