Lindelöf's theorem

Lindelöf's theorem Not to be confused with Lindelöf's lemma in topology.

In mathematics, Lindelöf's theorem is a result in complex analysis named after the Finnish mathematician Ernst Leonard Lindelöf. It states that a holomorphic function on a half-strip in the complex plane that is bounded on the boundary of the strip and does not grow "too fast" in the unbounded direction of the strip must remain bounded on the whole strip. The result is useful in the study of the Riemann zeta function, and is a special case of the Phragmén–Lindelöf principle. Also, see Hadamard three-lines theorem.

Statement of the theorem Let Ω be a half-strip in the complex plane: {displaystyle Omega ={zin mathbb {C} |x_{1}leq mathrm {Re} (z)leq x_{2}{text{ and }}mathrm {Im} (z)geq y_{0}}subsetneq mathbb {C} .,} Suppose that ƒ is holomorphic (i.e. analytic) on Ω and that there are constants M, A and B such that {displaystyle |f(z)|leq M{text{ for all }}zin partial Omega ,} and {displaystyle {frac {|f(x+iy)|}{y^{A}}}leq B{text{ for all }}x+iyin Omega .,} Then f is bounded by M on all of Ω: {displaystyle |f(z)|leq M{text{ for all }}zin Omega .,} Proof Fix a point {displaystyle xi =sigma +itau } inside {displaystyle Omega } . Choose {displaystyle lambda >-y_{0}} , an integer {displaystyle N>A} and {displaystyle y_{1}>tau } large enough such that {displaystyle {frac {By_{1}^{A}}{(y_{1}+lambda )^{N}}}leq {frac {M}{(y_{0}+lambda )^{N}}}} . Applying maximum modulus principle to the function {displaystyle g(z)={frac {f(z)}{(z+ilambda )^{N}}}} and the rectangular area {displaystyle {zin mathbb {C} |x_{1}leq mathrm {Re} (z)leq x_{2}{text{ and }}y_{0}leq mathrm {Im} (z)leq y_{1}}} we obtain {displaystyle |g(xi )|leq {frac {M}{(y_{0}+lambda )^{N}}}} , that is, {displaystyle |f(xi )|leq Mleft({frac {|xi +lambda |}{y_{0}+lambda }}right)^{N}} . Letting {displaystyle lambda rightarrow +infty } yields {displaystyle |f(xi )|leq M} as required.

References Edwards, H.M. (2001). Riemann's Zeta Function. New York, NY: Dover. ISBN 0-486-41740-9. Categories: Theorems in complex analysis

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