Let $X$ be a random variable following a $\mathrm{Binomial}(n,p)$ distribution, and let $$Y=\min\{X,n-X\}.$$ Ispired by the problem posed by C. Clement on https://math.stackexchange.com/questions/1696256/expectation-and-concentration-for-minx-n-x-when-x-is-a-binomial, I want to ask whether there exists some constant $c>0$ such that $\mathbb{E}(Y)\geq c\cdot \mathbb{E}(X)=c\cdot\min\{p,1-p\}\cdot n$ for all $0<p<1$. If this is not true, can we find some $p_0$ with $\frac{1+\sqrt{5}}{4}\leq p_0<1$ such that if $X\sim \mathrm{Binomial}(n,p_0)$ then $\mathbb{E}(Y)\geq (1-p)[(1+4p)n-8(1+p)]$?

Let $X$ be a random variable following a $\mathrm{Binomial}(n,p)$ distribution, and let $$Y=\min\{X,n-X\}.$$ Ispired by the problem posed by C. Clement on https://math.stackexchange.com/questions/1696256/expectation-and-concentration-for-minx-n-x-when-x-is-a-binomial, I want to ask whether there exists some constant $c>0$ such that $\mathbb{E}(Y)\geq c\cdot\min\{p,1-p\}\cdot n$ for all $0<p<1$. If this is not true, can we find some $p_0$ with $\frac{1+\sqrt{5}}{4}\leq p_0<1$ such that if $X\sim \mathrm{Binomial}(n,p_0)$ then $\mathbb{E}(Y)\geq (1-p)[(1+4p)n-8(1+p)]$?

Let $X$ be a random variable following a $\mathrm{Binomial}(n,p)$ distribution, and let $$Y=\min\{X,n-X\}.$$ Ispired by the problem posed by C. Clement on https://math.stackexchange.com/questions/1696256/expectation-and-concentration-for-minx-n-x-when-x-is-a-binomial, I want to ask whether there exists some constant $c>0$ such that $\mathbb{E}(Y)\geq c\cdot \mathbb{E}(X)=c\cdot np$ for all $0<p<1$. If this is not true, can we find some $p_0$ with $\frac{1+\sqrt{5}}{4}\leq p_0<1$ such that if $X\sim \mathrm{Binomial}(n,p_0)$ then $\mathbb{E}(Y)\geq (1-p)[(1+4p)n-8(1+p)]$?

Let $X$ be a random variable following a $\mathrm{Binomial}(n,p)$ distribution, and let $$Y=\min\{X,n-X\}.$$ Ispired by the problem posed by C. Clement on https://math.stackexchange.com/questions/1696256/expectation-and-concentration-for-minx-n-x-when-x-is-a-binomial, I want to ask whether there exists some constant $c>0$ such that $\mathbb{E}(Y)\geq c\cdot \mathbb{E}(X)=c\cdot\min\{p,1-p\}\cdot n$ for all $0<p<1$. If this is not true, can we find some $p_0$ with $\frac{1+\sqrt{5}}{4}\leq p_0<1$ such that if $X\sim \mathrm{Binomial}(n,p_0)$ then $\mathbb{E}(Y)\geq (1-p)[(1+4p)n-8(1+p)]$?