Estimating the derivative of the Legendre polynomial. (English. Russian original) Zbl 1257.33009
Vestn. St. Petersbg. Univ., Math. 43, No. 4, 191-197 (2010); translation from Vestn. St-Peterbg. Univ., Ser. I, Mat. Mekh. Astron. 2010, No. 4, 3-9 (2010).
Let \(P_{n}(x)\) be the classical Legendre polynomials, \(-1\leq x\leq 1\). The authors determine the best constant \(A\) in the inequality
\[
(1-x^2)^{3/4}\,\Big|\frac{d\,P_{n}(x)}{dx}\Big|<A\,\sqrt{n+\frac{2}{3}},\quad n\geq 2.
\]
This is \(A=\max_{0\leq t<\infty}\sqrt{t}\,J_{1}(t)=0.825031\dots\), where \(J_{1}(t)\) is the usual Bessel function of the first kind.
Reviewer: Stamatis Koumandos (Nicosia)
MSC:
| 33C45 | Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.) |
| 26D05 | Inequalities for trigonometric functions and polynomials |
| 26D07 | Inequalities involving other types of functions |
| 26D10 | Inequalities involving derivatives and differential and integral operators |
References:
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| [2] | H. Bateman and A. Erdelyi, Higher Transcendental Functions, vol. 1 (McGraw-Hill, New York, 1953; Nauka, Moscow, 1965). |
| [3] | V. A. Antonov, E. I. Timoshkova, and K. V. Kholshevnikov, Introduction to the Theory of Newton Potential (Nauka, Moscow, 1988) [in Russian]. · Zbl 0677.31001 |
| [4] | L. Ya. Adrianova, Introduction to the Theory of Linear Systems of Differential Equations (St. Peterb. Gos. Univ., St. Petersburg, 1992) [in Russian]. |
| [5] | F. Tricomi, Differential Equations (Blackie & Son, London, 1961; URSS, Moscow, 2003). |
| [6] | G. M. Fichtengolz, Course of Differential and Integral Calculus, vol. 2 (Lan’, St. Petersburg, 2009) [in Russian]. |
| [7] | Yu. N. Bibikov, A Course in Ordinary Differential Equations (Vysshaya Shkola, Moscow, 1991) [in Russian]. |
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