my iphn5s is stuck on the apple logo after installing ios9. please help

Connect to computer iTunes and restore from there.

Try these steps for Recovery Mode:

Turn off your device.

Connect the device charging cable to the computer only (not to the iPhone or iPad yet).

Open iTunes.

Press and hold the power button on your device and don’t let go.

Connect the cable to the device while keeping the power button pressed down.

Wait until the computer detects the device and wait for it to restore.

Restore as New first and then restore from backup(s) if desired.

https://support.apple.com/en-us/HT204686

Or alternatively check this article out:

https://support.apple.com/en-us/HT201412