For fixed $m = 0, 1, 2, ...$
$$f_m(k) = \prod_{j=1}^{m}(k+j).$$
Some examples of $f_m(k)$ are as following:
$$f_0(k) = 1, \quad f_1(k) = (k+1), \quad f_2(k) = (k+1)(k+2).$$

The $s_m(n)$ is defined as following:
$$s_m(n) = \sin\left(\frac{t}{2}\right)\sum_{k=0}^nf_m(k)\sin(k+0.5)t,\qquad t\in[0,\pi].$$

The $s_m(n)$  can also be defined as following:
$$s_m(n) = \sum_{j=0}^n\frac{(-4)^j}{(2j+1)!}\left(\sum_{k=j}^n\frac{f_m(k)(2k+1)(k+j)!}{(k-j)!}\right)x^{2j+2},\qquad x\in[0,1].$$

I want to prove
$$|s_m(n)| \le f_m(n), \forall x ~or ~t$$

I am sure the inequality holds but I am unable to prove it. I used MATLAB and verified the inequality for some values of $m$ and $n$ as presented below:

\begin{array}{ccccccccc}
  n & \max(s_0(n)) & f_0(n) & \max(s_1(n)) & f_1(n) & \max(s_2(n))& f_2(n) & \max(s_3(n)) & f_3(n)\\
  0 & 1.00         & 1      & 1.00         & 1      & 2.00        &  2     & 6.00         & 6     \\
  1 & 1.00         & 1      & 1.53         & 2      & 4.17        &  6     & 18.00        & 24    \\
  2 & 1.00         & 1      & 2.07         & 3      & 8.00        &  12    & 42.00        & 60    \\
  3 & 1.00         & 1      & 2.60         & 4      & 12.46       &  20    & 78.30        & 120   \\
  4 & 1.00         & 1      & 3.13         & 5      & 18.03       &  30    & 132.00       & 210 
\end{array}

Any help will be greatly appreciated.

**PS**: Please refer to [this question][1]. I asked for inductive proof so that I could use induction steps in the above inequality. But I did not get one.


  [1]: https://mathoverflow.net/questions/306118/inductive-proof-of-sn%E2%89%A4n1