Lower Bound on $E[\frac{1}{X}]$ for positive symmetric distribution

Question

Let $X$ be positive random variable and its distribution is symmetric about its mean value $m$. Then $$ E\left[\frac{1}{X}\right] \geq \frac{1}{m} + \frac{\sigma^2}{m^3}, $$ where $\sigma^2$ is variance of $X$. I can just prove that $$ E\left[\frac{1}{X}\right] \geq \frac{1}{m}, $$ using Jensen, but somehow can't incorporate symmetry and get also the second term.