$L_infty $ Markov and Bernstein Inequalities for Freud Weights

Let $W(x): = e^{ - Q(x)} $, $x \in \mathbb{R}$ where $Q(x)$ is even and continuous in $\mathbb{R}$, $Q(0) = 0$, and $Q''$ is continuous in $(0,\infty )$ with $Q'(x) > 0$ in $(0,\infty )$, and for some $C_1 $, $C_2 > 0$, \[ C_1 \leqq {{(xQ'(x))'} / {Q'(x) \leqq C_2 ,\quad x \in (0,\infty ).}}\] For example, $Q(x): = | x |^\alpha ,\alpha > 0$, satisfies these hypotheses. This paper proves the Markov-type inequality \[ \left\| {P'W} \right\|_{L_\infty (\mathbb{R})} \leqq \left\{ {\int_1^{C_3 n} {{{ds} / {Q^{[ - 1]} }}(s)} } \right\}\| {PW} \|_{L_\infty (\mathbb{R})} ,\] degree $(P) \leqq n$. Here $C_3 $ is some constant and $Q^{[ - 1]} $ is the inverse function of Q. Further, we prove the Bernstein-type inequality \[\left| {(PW)'(x)} \right| \leqq C_4 \| {PW} \|_{L_\infty (\mathbb{R})} \left\{ {\int_{Q(\max \{ {1,| x |}\})}^{C_3 n} {{{ds} / {Q^{[ - 1]} (s)}}} } \right\},\quad | x | \leqq \eta a_n ,\] and \[\left| {(PW)'(x)} \right| \leqq C_5 \| {PW} \|_{L_\infty (\mathbb{R})} \left( {{n / {a_n }}} \right)\max \left\{ {n^{{{ - 2} / 3}} ,1 - \frac{{| x |}}{{a_n }}} \right\}^{{1 / 2}} ,\quad | x | \geqq \eta a_n .\] Here degree $(P) \leqq n$, $\eta $ is any number in $(0,1)$, $a_n $ an is the Mhaskar–Rahmanov–Saff number for W, and if $Q'(0)$ does not exist, x must be excluded near zero.