What if the convertible note has both a discount and a cap?

For notes that include both a discount and a cap, the notes will typically specify that the conversion price will be the lower of the price per share determined by the discount to the next qualified priced round price per share OR the price per share determined by dividing the cap by the next qualified priced round pre-money valuation. The lower of means that the investor receives the better of the two possibilities (lower conversion price per share means that the note converts into more shares in the Series A).