I think I got it. Is it Chronicle of The Unhewn Throne series by Brian Staveley?
- Three Siblings; Kaden, Valyn and their sister Adare.
- Their father, the Emperor has been murdered.
- Kaden lives in a monastery with monks of the Blank God.
- Valyn trains with an Elite unit of troops (The Kettral) which uses black hawks as their mounts. Their HQ is located in Qirin Islands.
- Adare lives in the capital.
- The Imperial House claims descent from the goddess Intarra and cites this heritage as the reason for their blazing eyes. While All Emperors have such eyes, not all of the princelings have them. Kaden is heir to the throne because of possessing such eyes, despite being younger than Valyn.
- Four books in the series; The Emperor's Blades (2014), The Providence of Fire (2015), The Last Mortal Bond (2016). Fourth one, Skullsworn was published 2017 and is available on Amazon (Shout out to Eshier for pointing it out).
- There is an order of Courtesans called leina.
This is what Goodreads.com says about The Emperor's Blades (Book 1 of the series):
The Emperor has been murdered, leaving the Annurian Empire in turmoil. Now his progeny must bury their grief and prepare to unmask a
His son Valyn, training for the empire’s deadliest fighting force, hears the news an ocean away. He expected a challenge, but after
several ‘accidents’ and a dying soldier’s warning, he realizes his
life is also in danger. Yet before Valyn can take action, he must
survive the mercenaries’ brutal final initiation.
Meanwhile, the Emperor’s daughter, Minister Adare, hunts her father’s murderer in the capital itself. Court politics can be
fatal, but she needs justice.
And Kaden, heir to an empire, studies in a remote monastery. Here,
the Blank God’s disciples teach their harsh ways – which Kaden must master to unlock their ancient powers. When an imperial
delegation arrives, he’s learnt enough to perceive evil intent. But
will this keep him alive, as long-hidden powers make their move?
As for the eyes:
"Valyn [is older], by one year. It's a really common mistake. Kaden is
heir to the throne for his eyes, not his age."
Also, the Publisher's page confirms existence of birds used by the Elite troops:
In The Emperor's Blades by Brian Staveley, the emperor of Annur is
dead, slain by enemies unknown. His daughter and two sons,
scattered across the world, do what they must to stay alive and unmask
the assassins. But each of them also has a life-path on which their
father set them, destinies entangled with both ancient enemies and
Kaden, the heir to the Unhewn Throne, has spent eight years sequestered in a remote mountain monastery, learning the enigmatic
discipline of monks devoted to the Blank God. Their rituals hold
the key to an ancient power he must master before it's too late.
An ocean away, Valyn endures the brutal training of the Kettral,
elite soldiers who fly into battle on gigantic black hawks. But
before he can set out to save Kaden, Valyn must survive one horrific
At the heart of the empire, Minister Adare, elevated to her station
by one of the emperor's final acts, is determined to prove herself to
her people. But Adare also believes she knows who murdered her
father, and she will stop at nothing—and risk everything—to see that
justice is meted out.