Real Analytic Methods in the Formulations of some Combinatorial Inequalities

Abstract

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.