Cascante, Ma. Carme (Maria Carme)Palacios Torrell, Roger2023-06-022023-06-022023-01-24https://hdl.handle.net/2445/198841Treballs Finals de Grau de Matemàtiques, Facultat de Matemàtiques, Universitat de Barcelona, Any: 2023, Director: Ma. Carme Cascante[en] The Maximum Modulus Principle, which is one of the most important results in complex analysis, states that a holomorphic function defined on a bounded domain of $\mathbb{C}$, takes its maximum value at some point from the domain's boundary. Hence, the objective of this work is to introduce and apply the Phragmén-Lindelöf method in order to extend the conclusions given by the Maximum Modulus Principle to unbounded domains. Furthermore, this method will be used to see some applications such as: the Hadamard Three Lines Theorem, which provides good enough bounds for holomorphic functions on vertical strips; the Riesz-Thorin Interpolation Theorem, which establishes that a linear operator between measurable function spaces is bound in certain Lebesgue spaces $L^p$; and the Hardy's Uncertainty Principle, which claims that a measurable function and its Fourier transform cannot simultaneously have compact support, unless they both are identically zero.43 p.application/pdfcatcc-by-nc-nd (c) Roger Palacios Torrell, 2023http://creativecommons.org/licenses/by-nc-nd/3.0/es/Teoria geomètrica de funcionsTreballs de fi de grauOperadors linealsAnàlisi harmònicaGeometric function theoryBachelor's thesesLinear operatorsHarmonic analysisinfo:eu-repo/semantics/bachelorThesisinfo:eu-repo/semantics/openAccess