Will airport authority create problems?
Similar to this question, airport authority couldn't care less. Customs might care, probably Bangladesh customs since that's where you're importing something. You can check the Bangladeshi customs exemptions rules here, I assume you can read it (the English part only covers tourists, you've mentioned Bangladesh is home). Generally, if you're not importing commercial quantities, I'd expect broken phones to be within exemptions limits, but confirm that.
Another aspect is security - at the departure airport the security personnel may want to confirm these are in fact phones. If they don't turn on and the security personnel cannot verify that they are in fact phones, you might not be allowed to board with them. So if the screen is completely broken and shows nothing when you turn it on, that may be a problem from their perspective.