  In this paper, we derive some new combinatorial inequalities by applying well
known real analytic results like H\"{o}lder's inequality, Young's inequality,
and Minkowiski's inequality to the recursively defined sequence $f_n$ of
functions
  \begin{align*}
  f_0(x) & = \chi_{(-1/2, 1/2)} (x), \nonumber
  f_{n+1}(x) & = f_n(x+1/2)+ f_n(x-1/2), n \in \mathbb{N}\,\cup \,\{0\}.
\end{align*}
  Towards this goal, we derive the closed form of the aforementioned sequence $
(f_n)_{n\in \mathbb{N}\,\cup \,\{0\}}$ of functions and show that it is a
sequence of simple functions that are linear combinations of characteristic
functions of some unit intervals $ I_{n,i},\, i=0,1, ..., n $, with values the
binomial coefficients $ \binom{n}{i}$ on each unit interval $I_{n,i}$. We show
that $ f_n \in L^p(\mathbb{R})),\, 1\leq p \leq \infty $. Besides applying real
analytic methods to formulate some combinatorial inequalities, we also
illustrate the application of some combinatorial identities. For example, we
use the Vandermonde convolution (or Vandermonde identity), in the study of some
properties of the sequence of functions $ (f_n)_{n\in\mathbb{ N}\cup \{0\}}$.
We show how the $L^2$ norm of $f_n$ is related to the Catalan numbers.
