A convexity property of zeros of Bessel functions

  • Árpád Elbert
  • Andrea Laforgia
Brief Reports


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....


Mathematical Method Bessel Function Convexity Property Positive Zero Teorema 
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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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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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