Necessary and sufficient conditions for Landau–Kolmogorov type inequality via weighted majorization

The weighted majorization inequality is extended to ε ϵ -approximately convex functions. Using this weighted majorization tool, we prove that the Landau–Kolmogorov type inequality aligned ∏ ᵢ₌₀ᵐ f⁽ⁱ⁾ ∞^qᵢ≤ C(i;p,q)∏ ᵢ₌₀ᵐ f⁽ⁱ⁾ ∞^pᵢ aligned ∏ i = 0 m ‖ f ( i ) ‖ ∞ q i ≤ C ( i ; p , q ) ∏ i = 0 m ‖ f ( i ) ‖ ∞ p i holds for all f∈ L∞ ,∞ᵐ f ∈ L ∞ , ∞ m , if and only if aligned iq ₚi aligned i q ≺ p i for non-negative p , q . As the constant estimate C(i;p,q)<(π /2)^∑ ᵢ₌₀ᵐqᵢ C ( i ; p , q ) < ( π 2 ) ∑ i = 0 m q i </