Extensions of Schwarz Lemma in the Half-Plane

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Schwarz Lemma in the Half-Plane</h5> <div class="card-body"> <p class="card-text"><strong>Authors:</strong> <a href="https://publications.waset.org/abstracts/search?q=Nicolae%20Pascu">Nicolae Pascu</a> </p> <p class="card-text"><strong>Abstract:</strong></p> Aside from being a fundamental tool in Complex analysis, Schwarz Lemma-which was finalized in its most complete form at the beginning of the last century-generated an important area of research in various fields of mathematics, which continues to advance even today. We present some properties of analytic functions in the half-plane which satisfy the conditions of the classical Schwarz Lemma (Carath茅odory functions) and obtain a generalization of the well-known Aleksandrov-Sobolev Lemma for analytic functions in the half-plane (the correspondent of Schwarz-Pick Lemma from the unit disk). Using this Schwarz-type lemma, we obtain a characterization for the entire class of Carath茅odory functions, which might be of independent interest. We prove two monotonicity properties for Carath茅odory functions that do not depend upon their normalization at infinity (the hydrodynamic normalization). The method is based on conformal mapping arguments for analytic functions in the half-plane satisfying appropriate conditions, in the spirit of Schwarz lemma. According to the research findings in this paper, our main results give estimates for the modulus and the argument for the entire class of Carath茅odory functions. As applications, we give several extensions of Julia-Wolf-Carath茅odory Lemma in a half-strip and show that our results are sharp. <iframe src="https://publications.waset.org/abstracts/105458.pdf" style="width:100%; height:400px;" frameborder="0"></iframe> <p class="card-text"><strong>Keywords:</strong> <a href="https://publications.waset.org/abstracts/search?q=schwarz%20lemma" title="schwarz lemma">schwarz lemma</a>, <a href="https://publications.waset.org/abstracts/search?q=Julia-wolf-carat%C3%A9odory%20lemma" title=" Julia-wolf-carat茅odory lemma"> Julia-wolf-carat茅odory lemma</a>, <a href="https://publications.waset.org/abstracts/search?q=analytic%20function" title=" analytic function"> analytic function</a>, <a href="https://publications.waset.org/abstracts/search?q=normalization%20condition" title=" normalization condition"> normalization condition</a>, <a href="https://publications.waset.org/abstracts/search?q=carat%C3%A9odory%20function" title=" carat茅odory function"> carat茅odory function</a> </p> <a href="https://publications.waset.org/abstracts/105458/extensions-of-schwarz-lemma-in-the-half-plane" class="btn btn-primary btn-sm">Procedia</a> <a href="https://publications.waset.org/abstracts/105458.pdf" target="_blank" class="btn btn-primary btn-sm">PDF</a> <span class="bg-info text-light px-1 py-1 float-right rounded"> Downloads <span class="badge badge-light">227</span> </span> </div> </div> </div> </main> <footer> <div id="infolinks" class="pt-3 pb-2"> <div class="container"> <div style="background-color:#f5f5f5;" class="p-3"> <div class="row"> <div class="col-md-2"> <ul class="list-unstyled"> About <li><a href="https://waset.org/page/support">About Us</a></li> <li><a href="https://waset.org/page/support#legal-information">Legal</a></li> <li><a target="_blank" rel="nofollow" href="https://publications.waset.org/static/files/WASET-16th-foundational-anniversary.pdf">WASET celebrates its 16th foundational anniversary</a></li> </ul> </div> <div class="col-md-2"> 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Extensions of Schwarz Lemma in the Half-Plane