Skip to main content Accessibility help
×
Home

Inequalities for Entire Functions of Exponential Type

  • Clément Frappier (a1)

Abstract

Bernstein's inequality says that if f is an entire function of exponential type τ which is bounded on the real axis then

Genchev has proved that if, in addition, hf (π/2) ≤0, where hf is the indicator function of f, then

Using a method of approximation due to Lewitan, in a form given by Hörmander, we obtain, to begin, a generalization and a refinement of Genchev's result. Also, we extend to entire functions of exponential type two results first proved for polynomials by Rahman. Finally, we generalize a theorem of Boas concerning trigonometric polynomials vanishing at the origin.

    • Send article to Kindle

      To send this article to your Kindle, first ensure no-reply@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about sending to your Kindle. Find out more about sending to your Kindle.

      Note you can select to send to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be sent to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

      Find out more about the Kindle Personal Document Service.

      Inequalities for Entire Functions of Exponential Type
      Available formats
      ×

      Send article to Dropbox

      To send this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Dropbox.

      Inequalities for Entire Functions of Exponential Type
      Available formats
      ×

      Send article to Google Drive

      To send this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Google Drive.

      Inequalities for Entire Functions of Exponential Type
      Available formats
      ×

Copyright

References

Hide All
1. Ankeny, N. C. and Rivlin, T. J., On a theorem of S. Bernstein, Pacific J. Math. 5, (1955), 849-852.
2. Bernstein, S. N., Sur Vordre de la meilleure approximation des fonctions continues par les polynömes de degré donné, Mémoire de l'académie Royale de Belgique, vol. 4 (1912), 1-103.
3. Bernstein, S. N., Sur une propriété des fonctions entières, Comptes rendus de l'académie des sciences, Paris, vol. 176 (1923), 1602-1605.
4. Boas, R. P., Entire functions, Acad. Press, New York (1954).
5. Boas, R. P., Inequalities for asymmetric entire functions, Illinois J. Math. 1, (1957), 94-97.
6. Boas, R. P., Inequalities for polynomials with a prescribed zero, Studies in mathematical analysis and related topics (Essays in Honour of George Pölya), Stanford Univ. Press, Stanford, Calif. (1962), 42-47.
7. De Bruijn, N. G., Inequalities concerning polynomials in the complex domain, Nederl. Akad. Wetensch. Proc. 50, (1947), 1265-1272.
8. Duffin, R. J. and Schaeffer, A. C., Some inequalities concerning functions of exponential type, Bull. Amer. Math. Soc. 43, (1937), 554-556.
9. Genchev, T. G., Inequalities for asymmetric entire functions of exponential type, Soviet Math. Dokl., vol. 19 (1978), No. 4, 981-985.
10. Giroux, A. and Rahman, Q. I., Inequalities for polynomials with a prescribed zero, Trans. Amer. Math. Soc. 193, (1974), 67-98.
11. Hörmander, L., Some inequalities for functions of exponential type, Math. Scand. 3, (1955), 21-27.
12. Lax, P. D., Proof of a conjecture of P. Erdos on the derivative of a polynomial, Bull. Amer. Math. Soc. 50, (1944), 509-513.
13. Lewitan, B. M. Über eine Verallgemeinerung der Ungleichungen von S. Bernstein und H. Bohr. Doklady Akad. Nauk SSSR 15 (1937), 169-172.
14. Rahman, Q. I., Functions of exponential type, Trans. Amer. Math. Soc. 135, (1969), 295-309.
15. Szegö, G., Űber einen Satz des Herrn Serge Bernstein, Schr. Königsb. gelehrt. Ges., Vol, 22 (1928), 59-70.
MathJax
MathJax is a JavaScript display engine for mathematics. For more information see http://www.mathjax.org.

Keywords

Inequalities for Entire Functions of Exponential Type

  • Clément Frappier (a1)

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed.