This excerpt is taken from Sheikh Ghulam Mustafa Zaheer Amanpuri's book Mas'alah Raf' al-Yadain.
① Sayyiduna Abdullah bin Umar (may Allah be pleased with them) narrates:
[كان رسول الله صلى الله عليه وسلم إذا قام للصلاة رفع يديه حتىٰ تكونا حذو منكبيه، ثم كبر]
When the Messenger of Allah ﷺ stood for prayer, he would raise both his hands up to the level of his shoulders, then say Allahu Akbar.
Reference: Sahih al-Bukhari: 736, Sahih Muslim: 390, and the wording is his
It is established that raising the hands up to the shoulders and then saying the takbir is permissible.
② Abu Qalabah, a Tabi'i, may Allah have mercy on him, narrates:
[إنه رأىٰ مالك بن الحويرث إذا صلىٰ كبر ورفع يديه، وإذا أراد أن يركع رفع يديه، وإذا رفع رأسه من الركوع رفع يديه، وحدث أن رسول الله صلى الله عليه وسلم صنع هٰكذا]
He saw Sayyiduna Malik bin Huwayrith (may Allah be pleased with him) praying; he would say Allahu Akbar and raise his hands, go into ruku', and when he raised his head from ruku', he would raise his hands again and stated that the Messenger of Allah ﷺ used to do the same.
Reference: Sahih al-Bukhari: 737, Sahih Muslim: 391
❀ In Sahih Muslim, the words are «كبر ثم رفع يديه» "He said Allahu Akbar and then raised his hands."
It is established that raising the hands is permissible even after Takbir al-Tahreemah. Also, it is established that the initial raising of the hands is part of the prayer.
③ Sayyiduna Malik bin Huwayrith (may Allah be pleased with him) narrates:
[إن رسول الله صلى الله عليه وسلم، كان إذا كبر رفع يديه حتىٰ يحاذي بهما أذنيه]
When the Messenger of Allah ﷺ said "Allahu Akbar," he would raise his hands until they were level with his ears.
Reference: Sahih Muslim: 391
Summary:
Raising the hands before saying Takbir, raising the hands after saying Takbir, and saying Takbir and raising the hands together—all three methods are permissible.