Some Inequalities for Modified Bessel Functions
© A. Laforgia and P. Natalini. 2010
Received: 15 October 2009
Accepted: 28 December 2009
Published: 24 January 2010
We denote by and the Bessel functions of the first and third kinds, respectively. Motivated by the relevance of the function , , in many contexts of applied mathematics and, in particular, in some elasticity problems Simpson and Spector (1984), we establish new inequalities for . The results are based on the recurrence relations for and and the Turán-type inequalities for such functions. Similar investigations are developed to establish new inequalities for .
have been established; see also 
(see also ). This generalized function and the classical one, , are widely used in the electronic field, in particular in radar communications [7, 8] and in error performance analysis of multichannel dealing with partially coherent, differentially coherent, and noncoherent detections over fading channels [7, 9, 10].
The results obtained in this paper are proved as consequence of the recurrence relations [11, page 376; 9.6.26]
and the Turán-type inequalities
The results are given by the following theorems.
In particular, for , the inequality holds also true.
In particular, for , the inequality holds also true.
2. The Proofs
Proof of Theorem 1.1.
To prove the lower bound in (1.12), we substitute the function given by (1.6) in the Turán-type inequality (1.8). We get, for ,
which is the desired result.
For , Jones  proved stronger result than (2.1) that the function decreases with respect to , when .
Proof of Theorem 1.2.
we get , for .
We substitute the function given by (1.7) in (1.9). We get
which is the desired result (1.13).
By means the integral formula [11, page 181]
Since when , only in this case the above upper bound for improves the (1.13) one.
in particular, for , we also have .
3. Numerical Considerations
Baricz obtained, for each , the following similar lower bound for the ratio (see [5, formula (5)])
where is the unique simple positive root of the equation . Inequality (3.1) is reversed when . It is possible to prove that, for , our lower bound in (1.12) for the ratio provides an improvement of (3.1).
Let be . Putting and , one has , for all .
From the inequality we obtain, by simple calculations, the following one which is satisfied for all when .
We report here some numerical experiments, computed by using mathematica.
In the first case we assume . In Figure 1 we report the graphics of the functions (solid line) and the respective lower bounds (short dashed line) and (long dashed line) on the interval .
For a survey on inequalities of the type (3.2) and (3.3) see .
By the values reported on Table 5 it seems that is a lower bound much more stringent with respect to for every (moreover we recall that (3.2) holds true also for ), while by the values reported on Table 6 it seems that is a lower bound more stringent with respect to for (but we recall that (3.3) holds true also for and ).
This work was sponsored by Ministero dell'Universitá e della Ricerca Scientifica Grant no. 2006090295.
- Bordelon DJ: Solution to problem 72–15. SIAM Review 1973, 15: 666–668.MathSciNetGoogle Scholar
- Ross DK: Solution to problem 72–15. SIAM Review 1973, 15: 668–670.Google Scholar
- Laforgia A: Bounds for modified Bessel functions. Journal of Computational and Applied Mathematics 1991, 34(3):263–267. 10.1016/0377-0427(91)90087-ZMATHMathSciNetView ArticleGoogle Scholar
- Baricz Á: Bounds for modified Bessel functions of the first and second kind. Proceedings of the Edinburgh Mathematical Society. In press Proceedings of the Edinburgh Mathematical Society. In press
- Baricz Á: Tight bounds for the generalized Marcum -function. Journal of Mathematical Analysis and Applications 2009, 360(1):265–277. 10.1016/j.jmaa.2009.06.055MATHMathSciNetView ArticleGoogle Scholar
- Baricz Á, Sun Y: New bounds for the generalized marcum -function. IEEE Transactions on Information Theory 2009, 55(7):3091–3100.MathSciNetView ArticleGoogle Scholar
- Marcum JI: A statistical theory of target detection by pulsed radar. IRE Transactions on Information Theory 1960, 6: 59–267. 10.1109/TIT.1960.1057560MathSciNetView ArticleGoogle Scholar
- Marcum JI, Swerling P: Studies of target detection by pulsed radar. IEEE Transactions on Information Theory 1960, 6: 227–228.View ArticleGoogle Scholar
- Nuttall AH: Some integrals involving the function. IEEE Transactions on Information Theory 1975, 21: 95–96. 10.1109/TIT.1975.1055327MATHMathSciNetView ArticleGoogle Scholar
- Simon MK, Alouini MS: Digital Communication over Fadding Channels: A Unified Approach to Performance Analysis. John Wiley & Sons, New York, NY, USA; 2000.View ArticleGoogle Scholar
- Abramowitz M, Stegun IA: Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, National Bureau of Standards Applied Mathematics Series. Volume 55. U.S. Government Printing Office, Washington, DC, USA; 1964:xiv+1046.Google Scholar
- Lorch L: Monotonicity of the zeros of a cross product of Bessel functions. Methods and Applications of Analysis 1994, 1(1):75–80.MATHMathSciNetGoogle Scholar
- Ismail MEH, Muldoon ME: Monotonicity of the zeros of a cross-product of Bessel functions. SIAM Journal on Mathematical Analysis 1978, 9(4):759–767. 10.1137/0509055MATHMathSciNetView ArticleGoogle Scholar
- Laforgia A, Natalini P: On some Turán-type inequalities. Journal of Inequalities and Applications 2006, 2006:-6.Google Scholar
- Baricz Á: On a product of modified Bessel functions. Proceedings of the American Mathematical Society 2009, 137(1):189–193.MATHMathSciNetView ArticleGoogle Scholar
- Soni RP: On an inequality for modified Bessel functions. Journal of Mathematical Physics 1965, 44: 406–407.MATHMathSciNetView ArticleGoogle Scholar
- Näsell I: Inequalities for modified Bessel functions. Mathematics of Computation 1974, 28: 253–256.MathSciNetView ArticleGoogle Scholar
- Jones AL: An extension of an inequality involving modified Bessel functions. Journal of Mathematical Physics 1968, 47: 220–221.MATHView ArticleGoogle Scholar
- Simpson HC, Spector SJ: Some monotonicity results for ratios of modified Bessel functions. Quarterly of Applied Mathematics 1984, 42(1):95–98.MATHMathSciNetGoogle Scholar
- Simpson HC, Spector SJ: On barrelling for a special material in finite elasticity. Quarterly of Applied Mathematics 1984, 42(1):99–111.MATHMathSciNetGoogle Scholar
This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.