Best constants in a class of Landau type inequalities

We first prove the best constant C in a Landau type inequality ‖ f ′ ‖ ∞ 5 ≤ C ‖ f ‖ ∞ 3 ‖ f ′ ′ ‖ ∞ ‖ f ′ ′ ′ ‖ ∞ , and with similar approach, we prove the best constants in a more general form ‖ f ′ ‖ ∞ 2 + η ≤ C η ‖ f ‖ ∞ 1 + η ‖ f ′ ′ ‖ ∞ 1 − η ‖ f ′ ′ ′ ‖ ∞ η for all 0 ≤ η ≤ 1 . Here we have the direct expression for each of the best constants C η = θ η η ( 1 − θ η ) 2 + η 2 ( 1 / 4 − θ η 2 / 3 ) 1 + η , where θ η = 3 + 3 η − − 3 η 2 − 6 η + 9 8 + 4 η . A Landau type inequality and another special case of Landau–Kolmogorov inequality (Shilov’s result) can also be deduced from this general form, with the best constant.