110
Views
24
CrossRef citations to date
0
Altmetric
Original Articles

Real-part estimates for solutions of the Riesz system in ℝ3

&
Pages 505-522 | Received 15 Dec 2009, Accepted 24 Jun 2010, Published online: 06 Feb 2011

References

  • Hadamard , J . 1892 . Sur les fonctions entières de la forme e G(X) . C.R. Acad. Sci. , 114 : 1053 – 1055 .
  • Landau , E . 1906 . Über den Picardschen Satz . Vierteljahrschr. Naturf. Gesell. Zür. , 51 : 252 – 318 .
  • Wiman , A . 1914 . Über den Zusammenhang zwischen dem Maximalbetrage einer analytischen Funktion und dem grössten Gliede der zugehörigen Taylorschen Reihe . Acta Math. , 37 : 305 – 326 .
  • Jensen , J . 1919-1920 . Investigation of a class of fundamental inequalities in the theory of analytic functions . Ann. Math. , 21 : 1 – 29 .
  • Koebe , P . 1920 . Über das Schwarzsche lemma und einige damit zusammenhängende ungleicheitsbeziehungen der potentialtheorie und funktionentheorie . Math. Zeit. , 6 : 52 – 84 .
  • Borel , E . 1922 . Méthodes et Problèmes de Théorie des Fonctions , Paris : Gauthier-Villars .
  • Riesz , M . 1927 . Sur les fonctions conjugués . Math. Z. , 27 : 218 – 244 .
  • Littlewood , JE . 1947 . Lectures on the Theory of Functions , Oxford : Oxford University Press .
  • Titchmarsh , E . 1949 . The Theory of Functions, , 2nd , Oxford : Oxford University Press .
  • Rajagopal , C . 1953 . On inequalities for analytic functions . Amer. Math. Monthly , 60 : 693 – 695 .
  • Elkins , J . 1971 . A Borel–Carathéodory inequality and approximation of entire functions by polynomials with restricted zeros . J. Approx. Theory. , 4 : 274 – 278 .
  • Holland , A . 1973 . Introduction to the Theory of Entire Functions , New York, London : Academic Press .
  • Hayman , W . 1974 . The local growth of power series: A survey of the Wiman-Valiron method . Can. Math. Bull. , 17 : 317 – 358 .
  • Levin , B . 1996 . Lectures on Entire Functions, Translation of Mathematical Monographs , Vol. 150 , Providence, RI : American Mathematical Society .
  • G. Kresin and V. Maz'ya, Sharp Real-Part Theorems – A Unified Approach, Lecture Notes in Mathematics, Vol. 1903, 2007.
  • Brackx , F , Delanghe , R and Sommen , F . 1982 . Clifford Analysis , Boston, London, Melbourne : Pitman Publishing .
  • Delanghe , R . 1970 . On regular-analytic functions with values in a Clifford-algebra . Math. Ann. , 185 : 91 – 111 .
  • Delanghe , R , Sommen , F and Soucek , V . 1992 . Residues in Clifford Analysis, Partial Differential Equations with Complex Analysis, Pitman Research Notes in Mathematics Series , Vol. 262 , 61 – 92 . Harlow : Longman Scientific Technical .
  • Delanghe , R , Sommen , F and Soucek , V . 1992 . Clifford Agebras and Spinor-Valued Functions , Dordrecht : Kluwer Academic .
  • Delanghe , R . 2001 . Clifford analysis: History and perspective . Comput. Methods Funct. Theory , 1 ( 1 ) : 107 – 153 .
  • K. Gürlebeck and J. Morais, Hadamard's real part theorem for monogenic functions, AIP Conf. Proc. 1048 (2008), pp. 654–657.
  • Gürlebeck , K , Morais , J and Cerejeiras , P . 2009 . Borel–Carathéodory type theorem for monogenic functions . Complex Anal. Oper. Theory , 3 ( 1 ) : 99 – 112 .
  • Gürlebeck , K and Morais , J . 2009 . On mapping properties of monogenic functions . CUBO A Math. J. , 11 ( 1 ) : 73 – 100 .
  • J. Morais, Approximation by homogeneous polynomial solutions of the Riesz system in ℝ3 , Ph.D. diss., Bauhaus-Universität Weimar, 2009.
  • Stein , EM and Weiss , G . 1968 . Generalization of the Cauchy–Riemann equations and representations of the rotation group . Amer. J. Math. , 90 : 163 – 196 .
  • Gürlebeck , K and Malonek , H . 1999 . A hypercomplex derivative of monogenic functions in ℝ n+1 and its applications . Complex Var. Elliptic Eqns. , 39 ( 3 ) : 199 – 228 .
  • Mitelman , I and Shapiro , M . 1995 . Differentiation of the Martinelli-Bochner integrals and the notion of hyperderivability . Math. Nachr. , 172 ( 1 ) : 211 – 238 .
  • Sudbery , A . 1979 . Quaternionic analysis . Math. Proc. Camb. Philos. Soc. , 85 : 199 – 225 .
  • Leutwiler , H . 2001 . “ Quaternionic analysis in ℝ3 versus its hyperbolic modification ” . In NATO Science Series: II. Mathematics, Physics and Chemistry , Edited by: Brackx , F , Chisholm , JSR and Soucek , V . Vol. 25 , 193 – 211 . Dordrecht, Boston, London : Kluwer Academic .
  • Gürlebeck , K , Habetha , K and Sprössig , W . 2008 . Holomorphic Functions in the Plane and n-dimensional Space , Basel, Boston, Berlin : Birkhäuser Verlag .
  • I. Cação, Constructive approximation by monogenic polynomials, Ph.D. diss., Universidade de Aveiro, 2004.
  • Cação , I , Gürlebeck , K and Bock , S . 2006 . On derivatives of spherical monogenics . Complex Var. Elliptic Equ. , 51 ( 8–11 ) : 847 – 869 .
  • Sansone , G . 1959 . Orthogonal Functions, Pure and Applied Mathematics , Vol. IX , New York : Interscience Publishers .
  • Lohöfer , G . 1998 . Inequalities for the associated legendre functions . J. Approx. Theor. , 95 : 178 – 193 .
  • Andrews , L . 1998 . Special Functions of Mathematics for Engineers, SPIE Optical Engineering Press, Bellingham , Oxford : Oxford University Press .
  • S. Bock and K. Gürlebeck, On an orthonormal basis of solid spherical monogenics recursively generated by anti-holomorphic Z -powers, AIP Conf. Proc. 1168 (2008), pp. 765–768.
  • Gürlebeck , K and Morais , J . 2009 . Bohr type theorems for monogenic power series . Comput. Methods Func. Theor. , 9 ( 2 ) : 633 – 651 .
  • Ingham , A . 1932 . The Distribution of Prime Numbers , London : Cambridge University Press .
  • Maharana , J . 1978 . On the zeros of absorptive diffraction cross sections . Commun. Math. Phys. , 58 : 195 – 203 .
  • Carleson , L and Gamelin , TW . 1993 . Complex Dynamics , New York : Springer-Verlag .
  • Rajagopal , C . 1941 . Carathéodory's inequality and allied results . Math. Stud. , 9 : 73 – 77 .
  • Rajagopal , C . 1947 . Carathéodory's inequality and allied results (II) . Math. Stud. , 47 : 5 – 7 .
  • Kresin , G and Maz'ya , V . 2002 . Sharp parametric inequalities for analytic and harmonic functions related to Hadamard–Borel–Carathéodory inequalities . Func. Differ. Equ. , 9 ( 1–2 ) : 135 – 163 .
  • Burckel , RB . 1979 . An Introduction to Classical Complex Analysis , Vol. 1 , New York, San Francisco : Academic Press .
  • Aizenberg , L , Aytuna , A and Djakov , P . 2000 . An abstract approach to Bohr's phenomenon . Proc. Amer. Math. Soc. , 128 ( 9 ) : 2611 – 2619 .
  • Chen , Y . 2004 . Borel–Carathéodory inequality and Schwarz lemma for analytic multifunctions . Complex Var. Theor. Appl. , 49 ( 10 ) : 747 – 757 .

Reprints and Corporate Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

To request a reprint or corporate permissions for this article, please click on the relevant link below:

Academic Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

Obtain permissions instantly via Rightslink by clicking on the button below:

If you are unable to obtain permissions via Rightslink, please complete and submit this Permissions form. For more information, please visit our Permissions help page.