§18.14 Inequalities

Equations (18.14.3) and (18.14.3_5) are Bernstein-type inequalities. For further inequalities of this type see Koornwinder et al. (2018, §1) and references given there.

For further inequalities see Abramowitz and Stegun (1964, §22.14).

Let $R_n(x)=P_n^{(\alpha ,\beta )}\left(x\right)/P_n^{(\alpha ,\beta )}\left(1\right)$. Then

See Nikolov and Pillwein (2015) for a variant of (18.14.11) when $\alpha =\beta \in (-1,0]$.

Let the maxima $x_{n,m}$, $m=0,1,\mathrm{\dots },n$, of $|P_n^{(\alpha ,\beta )}\left(x\right)|$ in $[-1,1]$ be arranged so that

When $(\alpha +\frac{1}{2})(\beta +\frac{1}{2})>0$ choose $m$ so that

Then

Also,

except that when $\alpha =\beta =-\frac{1}{2}$ (Chebyshev case) $|P_n^{(\alpha ,\beta )}\left(x_{n,m}\right)|$ is constant.

For extensions of (18.14.20) see Askey (1990) and Wong and Zhang (1994a, b).

Let the maxima $x_{n,m}$, $m=0,1,\mathrm{\dots },n-1$, of $|L_n^{(\alpha )}\left(x\right)|$ in $[0,\mathrm{\infty })$ be arranged so that

When $\alpha >-\frac{1}{2}$ choose $m$ so that

Then

Also, when $\alpha \le -\frac{1}{2}$

The successive maxima of $|H_n\left(x\right)|$ form a decreasing sequence for $x\le 0$, and an increasing sequence for $x\ge 0$.

for $\alpha +\beta \ge 0$, $\beta \ge -\frac{1}{2}$ or $\alpha +\beta \ge -2$, $\beta \ge 0$. The case $\beta =0$ of (18.14.26) is the Askey–Gasper inequality (18.38.3).