A humble (kenjōgo) verb meaning 'to say' — a polite, self-effacing equivalent of 言う used for greetings, apologies, statements, etc.
拙い作品ですが、ご覧いただければ幸いと申し上げます。
This is a humble work, but I would be grateful if you would take a look (I say).
心よりお礼を申し上げます。
I would like to extend my heartfelt thanks.
まずは簡単に自己紹介を申し上げます。
First, allow me to briefly introduce myself.
Used as an auxiliary after words beginning with お-/ご- (お／ご + noun + 申し上げる) to form a humble expression meaning 'to do' (e.g., ご説明申し上げる). Common in formal writing and speech.
ご報告申し上げます。先日の会議で決定したことは次の通りです。
I will humbly report: the decisions made at the previous meeting are as follows.
改めてご挨拶申し上げます。今後ともよろしくお願いいたします。
I would like to humbly greet you once again. I look forward to your continued support.