Throughout the problem, we denote for every integer $n \geqslant 1$, $H _ { n } = \sum _ { k = 1 } ^ { n } \frac { 1 } { k }$. We define $\psi ( x ) = \frac { \Gamma ^ { \prime } ( x ) } { \Gamma ( x ) }$ on $\mathbb{R}^{+*}$, and $\psi$ is increasing on $\mathbb{R}^{+*}$. Let $n$ be an integer $\geqslant 2$ and $x$ a real $> - 1$. We set $p = E ( x ) + 1$, where $E ( x )$ denotes the integer part of $x$. Prove that $$0 \leqslant \psi ( n + x + 1 ) - \psi ( n ) \leqslant H _ { n + p } - H _ { n - 1 } \leqslant \frac { p + 1 } { n }$$
Throughout the problem, we denote for every integer $n \geqslant 1$, $H _ { n } = \sum _ { k = 1 } ^ { n } \frac { 1 } { k }$. We define $\psi ( x ) = \frac { \Gamma ^ { \prime } ( x ) } { \Gamma ( x ) }$ on $\mathbb{R}^{+*}$, and $\psi$ is increasing on $\mathbb{R}^{+*}$.
Let $n$ be an integer $\geqslant 2$ and $x$ a real $> - 1$. We set $p = E ( x ) + 1$, where $E ( x )$ denotes the integer part of $x$. Prove that
$$0 \leqslant \psi ( n + x + 1 ) - \psi ( n ) \leqslant H _ { n + p } - H _ { n - 1 } \leqslant \frac { p + 1 } { n }$$