[proofplan]
We apply the standard explicit one-dimensional B-process estimate, whose proof combines Poisson summation for a smoothed sharp interval, stationary phase at the unique critical point of $f(x)-mx$, and first-derivative [integration by parts](/theorems/210) away from critical points. The strict positivity of $f''$ makes $f'$ a bijection from $[a,b]$ onto $[\alpha,\beta]$, so precisely the integers $m$ with $\alpha<m<\beta$ contribute stationary-phase main terms. The non-stationary Poisson frequencies and the removal of the endpoint smoothing give the logarithmic loss and the two endpoint losses involving $\|\alpha\|_{\mathbb R/\mathbb Z}$ and $\|\beta\|_{\mathbb R/\mathbb Z}$.
[/proofplan]
[step:Cite the external explicit B-process estimate]
We use the explicit one-dimensional van der Corput B-process theorem in the following form: see Graham and Kolesnik, Van der Corput's Method of Exponential Sums, Lemma 3.6, equivalently Iwaniec and Kowalski, Analytic Number Theory, Proposition 8.2 with the endpoint form recorded after the proof. In the notation of that result, if $g\in C^3([u,v];\mathbb R)$, $u<v$, $g''(x)>0$ on $[u,v]$, $\gamma:=g'(u)$, $\delta:=g'(v)$, and if there are constants $Z>0$ and $A_0\ge 1$ such that, for all $x\in[u,v]$,
\begin{align*}
A_0^{-1}Z^{-1}\le g''(x)\le A_0Z^{-1}
\end{align*}
and
\begin{align*}
|g'''(x)|\le A_0((v-u)Z)^{-1},
\end{align*}
then, for every integer $r$ with $\gamma<r<\delta$, the unique point $y_r\in(u,v)$ satisfying $g'(y_r)=r$ contributes the stationary-phase term
\begin{align*}
\frac{e(g(y_r)-ry_r+1/8)}{\sqrt{g''(y_r)}}.
\end{align*}
Moreover there are a complex number $R_g\in\mathbb C$ and a constant $C_{A_0}>0$, depending only on $A_0$, such that
\begin{align*}
\sum_{u<n\le v} e(g(n))=\sum_{\substack{r\in\mathbb Z,\gamma<r<\delta}}\frac{e(g(y_r)-ry_r+1/8)}{\sqrt{g''(y_r)}}+R_g
\end{align*}
and
\begin{align*}
|R_g|\le C_{A_0}\left(\log(\delta-\gamma+2)+\min\left\{\sqrt Z,\frac{1}{\|\gamma\|_{\mathbb R/\mathbb Z}}\right\}+\min\left\{\sqrt Z,\frac{1}{\|\delta\|_{\mathbb R/\mathbb Z}}\right\}\right).
\end{align*}
The cited theorem is an external input, not a lemma proved here; it is the precise result obtained there from Poisson summation, endpoint smoothing, [one-dimensional stationary phase](/theorems/6985), and first-derivative estimates for the non-stationary frequencies.
[guided]
The central analytic input is not being reproved in this theorem. We cite the explicit one-dimensional van der Corput B-process theorem in the form of Graham and Kolesnik, Van der Corput's Method of Exponential Sums, Lemma 3.6, equivalently Iwaniec and Kowalski, Analytic Number Theory, Proposition 8.2 with its endpoint error term. The cited result applies to a phase $g:[u,v]\to\mathbb R$ of class $C^3$ on a compact interval with $u<v$, positive [second derivative](/page/Second%20Derivative), and uniform scale bounds
\begin{align*}
A_0^{-1}Z^{-1}\le g''(x)\le A_0Z^{-1}
\end{align*}
and
\begin{align*}
|g'''(x)|\le A_0((v-u)Z)^{-1}
\end{align*}
for every $x\in[u,v]$.
Under these hypotheses, $g'$ is strictly increasing on $[u,v]$. Therefore, for each integer $r$ satisfying $\gamma<r<\delta$, where $\gamma:=g'(u)$ and $\delta:=g'(v)$, there is exactly one point $y_r\in(u,v)$ with
\begin{align*}
g'(y_r)=r.
\end{align*}
This is exactly the stationary point of the phase $x\mapsto g(x)-rx$. Because the cited theorem is normalized with $e(t)=\exp(2\pi i t)$ and positive second derivative, its stationary-phase main term is
\begin{align*}
\frac{e(g(y_r)-ry_r+1/8)}{\sqrt{g''(y_r)}}.
\end{align*}
The same cited theorem also gives the sharp half-open summation convention and the endpoint error estimate: there are $R_g\in\mathbb C$ and $C_{A_0}>0$, with $C_{A_0}$ depending only on $A_0$, such that
\begin{align*}
\sum_{u<n\le v} e(g(n))=\sum_{\substack{r\in\mathbb Z,\gamma<r<\delta}}\frac{e(g(y_r)-ry_r+1/8)}{\sqrt{g''(y_r)}}+R_g
\end{align*}
and
\begin{align*}
|R_g|\le C_{A_0}\left(\log(\delta-\gamma+2)+\min\left\{\sqrt Z,\frac{1}{\|\gamma\|_{\mathbb R/\mathbb Z}}\right\}+\min\left\{\sqrt Z,\frac{1}{\|\delta\|_{\mathbb R/\mathbb Z}}\right\}\right).
\end{align*}
Thus the only remaining work in this proof is to verify that the present phase $f$ satisfies these exact hypotheses with $u=a$, $v=b$, $Z=Y$, and $A_0=A$.
[/guided]
[/step]
[step:Verify that the hypotheses of the B-process estimate apply to $f$]
We apply the estimate from the previous step with
\begin{align*}
g:=f, \qquad u:=a, \qquad v:=b, \qquad Z:=Y, \qquad A_0:=A.
\end{align*}
The function $f:[a,b]\to\mathbb R$ belongs to $C^3([a,b];\mathbb R)$ by hypothesis. The inequality $f''(x)>0$ for every $x\in[a,b]$ implies that $f':[a,b]\to\mathbb R$ is strictly increasing. Its endpoint values are exactly
\begin{align*}
\alpha=f'(a), \qquad \beta=f'(b).
\end{align*}
The required second- and third-derivative bounds in the B-process estimate are precisely the assumed bounds
\begin{align*}
A^{-1}Y^{-1}\le f''(x)\le AY^{-1}
\end{align*}
and
\begin{align*}
|f'''(x)|\le A(NY)^{-1}
\end{align*}
because $N=b-a$.
[/step]
[step:Identify the stationary points and substitute the B-process main terms]
Since $f':[a,b]\to[\alpha,\beta]$ is continuous and strictly increasing, for each integer $m$ with $\alpha<m<\beta$ there is a unique point $x_m\in(a,b)$ satisfying
\begin{align*}
f'(x_m)=m.
\end{align*}
These are exactly the stationary points of the phases
\begin{align*}
\phi_m:[a,b]\to\mathbb R, \qquad x\mapsto f(x)-mx,
\end{align*}
because
\begin{align*}
\phi_m'(x)=f'(x)-m.
\end{align*}
Applying the analytic B-process estimate gives
\begin{align*}
\sum_{a<n\le b} e(f(n))=\sum_{\substack{m\in\mathbb Z, \alpha<m<\beta}}\frac{e(f(x_m)-mx_m+1/8)}{\sqrt{f''(x_m)}}+E
\end{align*}
for an error term $E\in\mathbb C$ satisfying
\begin{align*}
|E|\le C_A\left(\log(\beta-\alpha+2)+\min\left\{\sqrt Y,\frac{1}{\|\alpha\|_{\mathbb R/\mathbb Z}}\right\}+\min\left\{\sqrt Y,\frac{1}{\|\beta\|_{\mathbb R/\mathbb Z}}\right\}\right).
\end{align*}
Here the constant $C_A>0$ is the constant from the B-process estimate with $A_0=A$, and hence depends only on $A$.
[/step]
[step:Check the endpoint convention and finish the proof]
The sharp summation convention in the B-process estimate is the same as in the theorem:
\begin{align*}
\sum_{a<n\le b} e(f(n)).
\end{align*}
The distance notation is also the same, namely
\begin{align*}
\|t\|_{\mathbb R/\mathbb Z}=\inf_{k\in\mathbb Z}|t-k|.
\end{align*}
If $\alpha$ or $\beta$ is an integer, then by convention the reciprocal distance is infinite, and the corresponding minimum is interpreted as $\sqrt Y$. Therefore the preceding estimate is valid in all endpoint cases. This is exactly the asserted formula and error bound, so the proof is complete.
[/step]
