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A convexity property of zeros of Bessel functions

  • Árpád Elbert
  • Andrea Laforgia
Brief Reports

Abstract

Fork=1, 2,... letj vk andc vk be thek-th positive zeros of the Bessel function

$$C_v \left( x \right) = C_v \left( {\alpha ;x} \right) = J_v \left( x \right)\cos \alpha - Y_v \left( x \right)\sin \alpha , 0 \leqslant \alpha< \pi$$

whereY v (X) is the Bessel function of the second kind. Using the notationj =C vk withκ=k−α/π introduced in [3] we show that the functionj +f(v) is convex with respect toυ≥0 forκ≥0.7070..., wheref(υ) is defined in the theorem of section 2. As an application we find the inequality 0 >j +j − 2κπ > log 8/9, where κ≥0.7070....

Keywords

Mathematical Method Bessel Function Convexity Property Positive Zero Teorema 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Sunto

Perk=1,2,..., sianoj vk ec vk rispettivamente ilk-esimo zero della funzione di BesselJ v (X) di prima specie e della funzione cilindrica

$$C_v \left( x \right) = C_v \left( {\alpha ;x} \right) = J_v \left( x \right)\cos \alpha - Y_v \left( x \right)\sin \alpha , 0 \leqslant \alpha< \pi$$

doveY v (X) rappresenta la funzione di Bessel di seconda specie. Usando la notazionejC con κ =k−α/π, introdotta in [3], si dimostra che la funzionej +f(v) é convessa rispetto aυ≥0 perκ≥0.7070..., dovef(υ) é definita nel teorema del par. 2. Come applicazione troviamo la doppia disuguaglianza 0 >j +j − 2κπ > log 8/9, dove κ≥0.7070....

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References

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Copyright information

© Birkhäuser Verlag 1990

Authors and Affiliations

  • Árpád Elbert
    • 1
  • Andrea Laforgia
    • 2
  1. 1.Mathematical Institute of the Hungarian Academy of SciencesBudapest
  2. 2.Facoltà d'IngegneriaMonteluco di RoioL'AquilaItalia

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