# 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....

## References

1. [1]
Á. Elbert, L. Gatteschi and A. Laforgia,On the concavity of zeros of Bessel functions. Applicable Anal.16, 261–278 (1983).Google Scholar
2. [2]
Á. Elbert and A. Laforgia,Further results on the zeros of Bessel functions. Analysis5, 71–86 (1985).Google Scholar
3. [3]
Á. Elbert and A. Laforgia,On the square of the zeros of Bessel functions. SIAM J. Math Anal.15, 206–212 (1984).Google Scholar
4. [4]
Á. Elbert and A. Laforgia,Some consequences of a lower bound for the second derivative of the zeros of Bessel functions. J. Math. Anal. Appl.125, 1–5 (1987).Google Scholar
5. [5]
C. Giordano and A. Laforgia,Further properties of the zeros of Bessel functions. “Le Matematiche”XLII, 1–10 (1987).Google Scholar
6. [6]
S. J. Putterman, M. Kac and G. E. Uhlenbeck,Possible origin of the quantized vortices in He, II, Phys. Rev. Lett.29, 546–549 (1972).Google Scholar
7. [7]
G. N. Watson,A Treatise on the Theory of Bessel Functions, 2nd ed., Cambridge Univ. Press, Cambridge 1944.Google Scholar