Kalpana Kalpana (Editor)

Turán's inequalities

Updated on
Edit
Like
Comment
Share on FacebookTweet on TwitterShare on LinkedInShare on Reddit

In mathematics, Turán's inequalities are some inequalities for Legendre polynomials found by Paul Turán (1950) (and first published by Szegö (1948)). There are many generalizations to other polynomials, often called Turán's inequalities, given by (E. F. Beckenbach, W. Seidel & Otto Szász 1951) and other authors.

If Pn is the nth Legendre polynomial, Turán's inequalities state that

P n ( x ) 2 > P n 1 ( x ) P n + 1 ( x )  for  1 < x < 1.


For Hn, the nth Hermite polynomial, Turán's inequalities are

H n ( x ) 2 H n 1 ( x ) H n + 1 ( x ) = ( n 1 ) ! i = 0 n 1 2 n i i ! H i ( x ) 2 > 0   ,

whilst for Chebyshev polynomials they are

T n ( x ) 2 T n 1 ( x ) T n + 1 ( x ) = 1 x 2 > 0  for  1 < x < 1   .

References

Turán's inequalities Wikipedia


Similar Topics