A simple bound for the Poisson kernel for the upper half plane.












0












$begingroup$


Let $P_z(t)=frac{1}{pi}Im(frac{1}{t-z})=frac{1}{pi} frac{y}{(x-t)^2+y^2}$



where $0<y=Im(z)$ and $x=Re(z)$



$P$ is the Poisson kernel for the upper half plane.



If $t in Bbb{R}$ how can we derive that $P_z(t) leq frac{c_z}{1+t^2}$ where $c_z$ is a constant depending on $z$?



Thank you in advance for your help.










share|cite|improve this question









$endgroup$

















    0












    $begingroup$


    Let $P_z(t)=frac{1}{pi}Im(frac{1}{t-z})=frac{1}{pi} frac{y}{(x-t)^2+y^2}$



    where $0<y=Im(z)$ and $x=Re(z)$



    $P$ is the Poisson kernel for the upper half plane.



    If $t in Bbb{R}$ how can we derive that $P_z(t) leq frac{c_z}{1+t^2}$ where $c_z$ is a constant depending on $z$?



    Thank you in advance for your help.










    share|cite|improve this question









    $endgroup$















      0












      0








      0





      $begingroup$


      Let $P_z(t)=frac{1}{pi}Im(frac{1}{t-z})=frac{1}{pi} frac{y}{(x-t)^2+y^2}$



      where $0<y=Im(z)$ and $x=Re(z)$



      $P$ is the Poisson kernel for the upper half plane.



      If $t in Bbb{R}$ how can we derive that $P_z(t) leq frac{c_z}{1+t^2}$ where $c_z$ is a constant depending on $z$?



      Thank you in advance for your help.










      share|cite|improve this question









      $endgroup$




      Let $P_z(t)=frac{1}{pi}Im(frac{1}{t-z})=frac{1}{pi} frac{y}{(x-t)^2+y^2}$



      where $0<y=Im(z)$ and $x=Re(z)$



      $P$ is the Poisson kernel for the upper half plane.



      If $t in Bbb{R}$ how can we derive that $P_z(t) leq frac{c_z}{1+t^2}$ where $c_z$ is a constant depending on $z$?



      Thank you in advance for your help.







      real-analysis complex-analysis complex-numbers






      share|cite|improve this question













      share|cite|improve this question











      share|cite|improve this question




      share|cite|improve this question










      asked Jan 9 at 11:36









      Marios GretsasMarios Gretsas

      8,48011437




      8,48011437






















          1 Answer
          1






          active

          oldest

          votes


















          1












          $begingroup$

          Certainly. Since your bound can depend on $z$ all that you are asking is if $frac {1+t^{2}} {(x-t)^{2}+y^{2}}$ is bounded in $t$. This is a continuous function which tends to $1$ as $|t| to infty$ so it is bounded.






          share|cite|improve this answer









          $endgroup$













            Your Answer





            StackExchange.ifUsing("editor", function () {
            return StackExchange.using("mathjaxEditing", function () {
            StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix) {
            StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
            });
            });
            }, "mathjax-editing");

            StackExchange.ready(function() {
            var channelOptions = {
            tags: "".split(" "),
            id: "69"
            };
            initTagRenderer("".split(" "), "".split(" "), channelOptions);

            StackExchange.using("externalEditor", function() {
            // Have to fire editor after snippets, if snippets enabled
            if (StackExchange.settings.snippets.snippetsEnabled) {
            StackExchange.using("snippets", function() {
            createEditor();
            });
            }
            else {
            createEditor();
            }
            });

            function createEditor() {
            StackExchange.prepareEditor({
            heartbeatType: 'answer',
            autoActivateHeartbeat: false,
            convertImagesToLinks: true,
            noModals: true,
            showLowRepImageUploadWarning: true,
            reputationToPostImages: 10,
            bindNavPrevention: true,
            postfix: "",
            imageUploader: {
            brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
            contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
            allowUrls: true
            },
            noCode: true, onDemand: true,
            discardSelector: ".discard-answer"
            ,immediatelyShowMarkdownHelp:true
            });


            }
            });














            draft saved

            draft discarded


















            StackExchange.ready(
            function () {
            StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3067348%2fa-simple-bound-for-the-poisson-kernel-for-the-upper-half-plane%23new-answer', 'question_page');
            }
            );

            Post as a guest















            Required, but never shown

























            1 Answer
            1






            active

            oldest

            votes








            1 Answer
            1






            active

            oldest

            votes









            active

            oldest

            votes






            active

            oldest

            votes









            1












            $begingroup$

            Certainly. Since your bound can depend on $z$ all that you are asking is if $frac {1+t^{2}} {(x-t)^{2}+y^{2}}$ is bounded in $t$. This is a continuous function which tends to $1$ as $|t| to infty$ so it is bounded.






            share|cite|improve this answer









            $endgroup$


















              1












              $begingroup$

              Certainly. Since your bound can depend on $z$ all that you are asking is if $frac {1+t^{2}} {(x-t)^{2}+y^{2}}$ is bounded in $t$. This is a continuous function which tends to $1$ as $|t| to infty$ so it is bounded.






              share|cite|improve this answer









              $endgroup$
















                1












                1








                1





                $begingroup$

                Certainly. Since your bound can depend on $z$ all that you are asking is if $frac {1+t^{2}} {(x-t)^{2}+y^{2}}$ is bounded in $t$. This is a continuous function which tends to $1$ as $|t| to infty$ so it is bounded.






                share|cite|improve this answer









                $endgroup$



                Certainly. Since your bound can depend on $z$ all that you are asking is if $frac {1+t^{2}} {(x-t)^{2}+y^{2}}$ is bounded in $t$. This is a continuous function which tends to $1$ as $|t| to infty$ so it is bounded.







                share|cite|improve this answer












                share|cite|improve this answer



                share|cite|improve this answer










                answered Jan 9 at 11:54









                Kavi Rama MurthyKavi Rama Murthy

                62.7k42262




                62.7k42262






























                    draft saved

                    draft discarded




















































                    Thanks for contributing an answer to Mathematics Stack Exchange!


                    • Please be sure to answer the question. Provide details and share your research!

                    But avoid



                    • Asking for help, clarification, or responding to other answers.

                    • Making statements based on opinion; back them up with references or personal experience.


                    Use MathJax to format equations. MathJax reference.


                    To learn more, see our tips on writing great answers.




                    draft saved


                    draft discarded














                    StackExchange.ready(
                    function () {
                    StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3067348%2fa-simple-bound-for-the-poisson-kernel-for-the-upper-half-plane%23new-answer', 'question_page');
                    }
                    );

                    Post as a guest















                    Required, but never shown





















































                    Required, but never shown














                    Required, but never shown












                    Required, but never shown







                    Required, but never shown

































                    Required, but never shown














                    Required, but never shown












                    Required, but never shown







                    Required, but never shown







                    Popular posts from this blog

                    Human spaceflight

                    Can not write log (Is /dev/pts mounted?) - openpty in Ubuntu-on-Windows?

                    張江高科駅