Assume that there are two zero coupon bond with maturities $N_1$ and $N_2$ with prices $P_1 = \frac{CF_1}{(1+y)^N_1}$ and $P_2 = \frac{CF_2}{(1+y)^N_2}$ respectively. If we construct a bond portfolio by purchaing one each of the two ZCB, the price of the portfolio is $P=P_1+P_2$. Now, the convexity of the portfolio is

$\begin{align} {Convexity}_p &= -\frac{d^2P}{dy^2}\cdot\frac{1}{P} \\ &=\frac{1}{(1+y)^2}\left[\frac{CF_1}{(1+y)^{N_1}}N_1(N_1+1)+\frac{CF_2}{(1+y)^{N_2}}N_2(N_2+1)\right]\cdot\frac{1}{P} \\ &=\frac{1}{(1+y)^2}\left[\frac{P_1}{P}N_1(N_1+1)+\frac{P_2}{P}N_2(N_2+1)\right] \end{align}$.

Notice that $\frac{P_1}{P}+\frac{P_2}{P}=1$

Now, plot convexity against maturity ($N$),

We may see that the convexity of the barbell portfolio (in blue line) is above the convexity of the bullet portfolio (in black line).

Assume that there are two zero coupon bond with maturities $N_1$ and $N_2$ with prices $P_1 = \frac{CF_1}{(1+y)^{N_1}}$ and $P_2 = \frac{CF_2}{(1+y)^{N_2}}$ respectively. If we construct a bond portfolio by purchaing one each of the two ZCB, the price of the portfolio is $P=P_1+P_2$. Now, the convexity of the portfolio is

$\begin{align} {Convexity}_p &= -\frac{d^2P}{dy^2}\cdot\frac{1}{P} \\ &=\frac{1}{(1+y)^2}\left[\frac{CF_1}{(1+y)^{N_1}}N_1(N_1+1)+\frac{CF_2}{(1+y)^{N_2}}N_2(N_2+1)\right]\cdot\frac{1}{P} \\ &=\frac{1}{(1+y)^2}\left[\frac{P_1}{P}N_1(N_1+1)+\frac{P_2}{P}N_2(N_2+1)\right] \end{align}$.

Notice that $\frac{P_1}{P}+\frac{P_2}{P}=1$

Now, plot convexity against maturity ($N$),

We may see that the convexity of the barbell portfolio (in blue line) is above the convexity of the bullet portfolio (in black line).