Askey–Gasper inequality
In mathematics, the Askey–Gasper inequality is an inequality for Jacobi polynomials proved by Richard Askey and George Gasper (1976 ) and used in the proof of the Bieberbach conjecture .
Statement
It states that if
β β -->
≥ ≥ -->
0
{\displaystyle \beta \geq 0}
α α -->
+
β β -->
≥ ≥ -->
− − -->
2
{\displaystyle \alpha +\beta \geq -2}
− − -->
1
≤ ≤ -->
x
≤ ≤ -->
1
{\displaystyle -1\leq x\leq 1}
∑ ∑ -->
k
=
0
n
P
k
(
α α -->
,
β β -->
)
(
x
)
P
k
(
β β -->
,
α α -->
)
(
1
)
≥ ≥ -->
0
{\displaystyle \sum _{k=0}^{n}{\frac {P_{k}^{(\alpha ,\beta )}(x)}{P_{k}^{(\beta ,\alpha )}(1)}}\geq 0}
where
P
k
(
α α -->
,
β β -->
)
(
x
)
{\displaystyle P_{k}^{(\alpha ,\beta )}(x)}
is a Jacobi polynomial.
The case when
β β -->
=
0
{\displaystyle \beta =0}
3
F
2
(
− − -->
n
,
n
+
α α -->
+
2
,
1
2
(
α α -->
+
1
)
;
1
2
(
α α -->
+
3
)
,
α α -->
+
1
;
t
)
>
0
,
0
≤ ≤ -->
t
<
1
,
α α -->
>
− − -->
1.
{\displaystyle {}_{3}F_{2}\left(-n,n+\alpha +2,{\tfrac {1}{2}}(\alpha +1);{\tfrac {1}{2}}(\alpha +3),\alpha +1;t\right)>0,\qquad 0\leq t<1,\quad \alpha >-1.}
In this form, with α a non-negative integer, the inequality was used by Louis de Branges in his proof of the Bieberbach conjecture .
Proof
Ekhad (1993 ) gave a short proof of this inequality, by combining the identity
(
α α -->
+
2
)
n
n
!
× × -->
3
F
2
(
− − -->
n
,
n
+
α α -->
+
2
,
1
2
(
α α -->
+
1
)
;
1
2
(
α α -->
+
3
)
,
α α -->
+
1
;
t
)
=
=
(
1
2
)
j
(
α α -->
2
+
1
)
n
− − -->
j
(
α α -->
2
+
3
2
)
n
− − -->
2
j
(
α α -->
+
1
)
n
− − -->
2
j
j
!
(
α α -->
2
+
3
2
)
n
− − -->
j
(
α α -->
2
+
1
2
)
n
− − -->
2
j
(
n
− − -->
2
j
)
!
× × -->
3
F
2
(
− − -->
n
+
2
j
,
n
− − -->
2
j
+
α α -->
+
1
,
1
2
(
α α -->
+
1
)
;
1
2
(
α α -->
+
2
)
,
α α -->
+
1
;
t
)
{\displaystyle {\begin{aligned}{\frac {(\alpha +2)_{n}}{n!}}&\times {}_{3}F_{2}\left(-n,n+\alpha +2,{\tfrac {1}{2}}(\alpha +1);{\tfrac {1}{2}}(\alpha +3),\alpha +1;t\right)=\\&={\frac {\left({\tfrac {1}{2}}\right)_{j}\left({\tfrac {\alpha }{2}}+1\right)_{n-j}\left({\tfrac {\alpha }{2}}+{\tfrac {3}{2}}\right)_{n-2j}(\alpha +1)_{n-2j}}{j!\left({\tfrac {\alpha }{2}}+{\tfrac {3}{2}}\right)_{n-j}\left({\tfrac {\alpha }{2}}+{\tfrac {1}{2}}\right)_{n-2j}(n-2j)!}}\times {}_{3}F_{2}\left(-n+2j,n-2j+\alpha +1,{\tfrac {1}{2}}(\alpha +1);{\tfrac {1}{2}}(\alpha +2),\alpha +1;t\right)\end{aligned}}}
with the Clausen inequality .
Generalizations
Gasper & Rahman (2004 , 8.9) give some generalizations of the Askey–Gasper inequality to basic hypergeometric series .
See also
References
Askey, Richard ; Gasper, George (1976), "Positive Jacobi polynomial sums. II", American Journal of Mathematics 98 (3): 709–737, doi :10.2307/2373813 , ISSN 0002-9327 , JSTOR 2373813 , MR 0430358 Askey, Richard; Gasper, George (1986), "Inequalities for polynomials" , in Baernstein, Albert; Drasin, David; Duren, Peter; Marden, Albert (eds.), The Bieberbach conjecture (West Lafayette, Ind., 1985) , Math. Surveys Monogr., vol. 21, Providence, R.I.: American Mathematical Society , pp. 7–32, ISBN 978-0-8218-1521-2 MR 0875228 Ekhad, Shalosh B. (1993), Delest, M.; Jacob, G.; Leroux, P. (eds.), "A short, elementary, and easy, WZ proof of the Askey-Gasper inequality that was used by de Branges in his proof of the Bieberbach conjecture", Theoretical Computer Science 117 (1): 199–202, doi :10.1016/0304-3975(93)90313-I ISSN 0304-3975 , MR 1235178 Gasper, George; Rahman, Mizan (2004), Basic hypergeometric series , Encyclopedia of Mathematics and its Applications, vol. 96 (2nd ed.), Cambridge University Press , ISBN 978-0-521-83357-8 MR 2128719
