Convexity and coupon

There are two statements about convexity and coupon. Can someone explain them?

  1. For a given yield and maturity, the lower the coupon, the greater the convexity of a bond.
  2. For a given yield and modified duration, the lower the coupon, the smaller the convexity.

What sort of explanation are you looking to get?

Why with same modified duration, bond with lower coupon will has smaller convexity?

Convexity increases with maturity, all else equal.

I’m not sure if I understand your question. Lower coupon and longer maturity bonds have higher modified durations, which make them more sensitive to interest rate movements (price risk)